Using Operations Research Methodologies to …Using Operations Research Methodologies to Improve...

Using Operations Research Methodologies

to Improve

Operating Theatre Scheduling

Kari Stuart Bachelor Applied Science (Mathematics)

Bachelor Applied Science Honours (Mathematics) Queensland University of Technology

~ CONFIDENTIAL ~

A thesis submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy 2010

Principal Supervisor: Professor Erhan Kozan

Queensland University of Technology

Discipline of Mathematical Sciences

Brisbane, Queensland, 4001, AUSTRALIA

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© Copyright by Kari Stuart 2010 All Rights Reserved

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Abstract

A hospital consists of a number of wards, units and departments that provide a

variety of medical services and interact on a day-to-day basis. Nearly every

department within a hospital schedules patients for the operating theatre (OT) and

most wards receive patients from the OT following post-operative recovery.

Because of the interrelationships between units, disruptions and cancellations within

the OT can have a flow-on effect to the rest of the hospital. This often results in

dissatisfied patients, nurses and doctors, escalating waiting lists, inefficient resource

usage and undesirable waiting times.

The objective of this study is to use Operational Research methodologies to enhance

the performance of the operating theatre by improving elective patient planning using

robust scheduling and improving the overall responsiveness to emergency patients by

solving the disruption management and rescheduling problem. OT scheduling

considers two types of patients: elective and emergency. Elective patients are

selected from a waiting list and scheduled in advance based on resource availability

and a set of objectives. This type of scheduling is referred to as ‘offline scheduling’.

Disruptions to this schedule can occur for various reasons including variations in

length of treatment, equipment restrictions or breakdown, unforeseen delays and the

arrival of emergency patients, which may compete for resources. Emergency

patients consist of acute patients requiring surgical intervention or in-patients whose

conditions have deteriorated. These may or may not be urgent and are triaged

accordingly. Most hospitals reserve theatres for emergency cases, but when these or

other resources are unavailable, disruptions to the elective schedule result, such as

delays in surgery start time, elective surgery cancellations or transfers to another

institution. Scheduling of emergency patients and the handling of schedule

disruptions is an ‘online’ process typically handled by OT staff. This means that

decisions are made ‘on the spot’ in a ‘real-time’ environment.

There are three key stages to this study: (1) Analyse the performance of the operating

theatre department using simulation. Simulation is used as a decision support tool

and involves changing system parameters and elective scheduling policies and

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observing the effect on the system’s performance measures; (2) Improve viability of

elective schedules making offline schedules more robust to differences between

expected treatment times and actual treatment times, using robust scheduling

techniques. This will improve the access to care and the responsiveness to

emergency patients; (3) Address the disruption management and rescheduling

problem (which incorporates emergency arrivals) using innovative robust reactive

scheduling techniques. The robust schedule will form the baseline schedule for the

online robust reactive scheduling model.

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Contents

ABSTRACT............................................................................................. 3

LIST OF FIGURES ................................................................................ 8

LIST OF TABLES ................................................................................10

PUBLICATIONS ARISING FROM THIS RESEARCH............ .....11

LIST OF ABBREVIATIONS ..............................................................13

STATEMENT OF ORIGINAL AUTHORSHIP ................... ............14

ACKNOWLEDGEMENTS..................................................................15

CHAPTER 1. INTRODUCTION .....................................................16

1.1 HEALTHCARE AND SURGERY IN AUSTRALIA................................... 16

1.2 SOME RESEARCH ISSUES ........................................................................ 19

1.3 A REAL LIFE EXAMPLE ............................................................................ 21

1.3.1 THE PRINCESS ALEXANDRA HOSPITAL .............................................. 21

1.3.1.1 ADMISSION PROCESSES .......................................................................... 21

1.3.1.2 OT LAYOUT................................................................................................. 22

1.3.1.3 OT SCHEDULING........................................................................................ 23

1.4 MOTIVATION, AIMS AND METHODOLOGY OF THE RESEARCH.... 26

1.4.1 GAPS IN THE RESEARCH.......................................................................... 26

1.4.2 AIMS OF THE RESEARCH......................................................................... 29

1.4.3 APPROACH .................................................................................................. 30

1.5 CONTRIBUTIONS OF THE THESIS .......................................................... 36

CHAPTER 2. LITERATURE REVIEW .........................................39

2.1 GENERAL OPERATING THEATRE STUDIES – ADMINISTRATIVE

ISSUES .......................................................................................................... 39

2.2 MULTI-OBJECTIVE PROGRAMMING..................................................... 42

2.3 SIMULATION............................................................................................... 43

2.4 SCHEDULING AND SEQUENCING.......................................................... 45

2.5 ALTERNATIVE APPROACHES................................................................. 50

2.6 ROBUST SCHEDULING ............................................................................. 50

2.7 REACTIVE SCHEDULING ......................................................................... 52

2.8 SINGLE MACHINE SCHEDULING ........................................................... 55

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CHAPTER 3. SIMULATION MODEL...........................................58

3.1 THE MODEL.................................................................................................58

3.1.1 DEVELOPING THE MODEL.......................................................................58

3.1.2 MODEL ASSUMPTIONS.............................................................................63

3.1.3 CHANGING PATIENT ARRIVALS............................................................63

3.1.4 SCHEDULING DISCIPLINES .....................................................................63

3.1.5 DATA INPUT................................................................................................64

3.1.6 SIMULATION MODEL VALIDATION PROCESS....................................66

3.2 SIMULATION RESULTS AND SENSITIVITY ANALYSIS.....................68

3.3 CONCLUSIONS............................................................................................81

CHAPTER 4. ROBUST SURGERY ASSIGNMENT MODELS..83

4.1 DATA ANALYSIS FOR ROBUST AND REACTIVE MODELS...............85

4.2 GENERATION OF RANDOM CASES FOR TESTING THE ROBUST

AND REACTIVE MODELS.........................................................................88

4.3 THE SUM OF INDEPENDENT BUT NOT NECESSARILY IDENTICAL

LOGNORMAL RANDOM VARIABLES ....................................................89

4.4 MODEL ASSUMPTIONS.............................................................................93

4.5 THE ROBUST SURGERY ASSIGNMENT MODELS ...............................94

4.5.1 THE ROBUST SURGERY ASSIGNMENT MODELS WITH

LOGNORMALLY DISTRIBUTED SURGICAL DURATIONS.................94

4.5.2 FLEXIBLE ROBUST SURGERY ASSIGNMENT MODEL WITH

LOGNORMALLY DISTRIBUTED SURGICAL DURATIONS.................98

4.6 PROBLEM COMPLEXITY ..........................................................................98

4.7 SOLUTION METHODS..............................................................................100

4.7.1 SOLUTION METHOD FOR THE ROBUST SURGERY ASSIGNMENT

MODEL WITH LOGNORMALLY DISTRIBUTED SURGICAL

DURATIONS...............................................................................................100

4.7.2 SOLUTION METHOD FOR THE FLEXIBLE ROBUST SURGERY

ASSIGNMENT MODEL WITH LOGNORMALLY DISTRIBUTED

SURGICAL DURATIONS..........................................................................104

4.8 ROBUST SCHEDULE RESULTS..............................................................104

4.8.1 RESULTS FOR THE ROBUST SURGERY ASSIGNMENT MODEL WITH

LOGNORMALLY DISTRIBUTED SURGICAL DURATIONS...............105

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4.8.2 RESULTS FOR THE FLEXIBLE ROBUST SURGERY ASSIGNMENT

MODEL WITH LOGNORMALLY DISTRIBUTED SURGICAL

DURATIONS............................................................................................... 112

4.9 CONCLUSIONS.......................................................................................... 115

CHAPTER 5. REACTIVE SCHEDULING..................................118

5.1 THE ROBUST REACTIVE ASSIGNMENT MODEL .............................. 119

5.1.1 SOLUTION APPROACH AND RESULTS ............................................... 123

5.2 THE BI-CRITERIA REACTIVE SCHEDULING MODELS .................... 129

5.2.1 THE MODELS ............................................................................................ 130

5.2.1.1 FIXED TREATMENT TIME MODEL....................................................... 133

5.2.1.2 ROBUST MODEL....................................................................................... 136

5.2.2 SOLUTION APPROACH ........................................................................... 140

5.2.3 SCHEDULE RESULTS AND CONCLUSIONS........................................ 152

5.3 CONCLUSIONS.......................................................................................... 159

CHAPTER 6. CONCLUSIONS AND FUTURE WORK ............161

6.1 CONCLUSIONS.......................................................................................... 161

6.2 FUTURE WORK......................................................................................... 167

REFERENCES....................................................................................169

APPENDIX A ......................................................................................178

APPENDIX B ......................................................................................179

APPENDIX C ......................................................................................182

APPENDIX D ......................................................................................188

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List of figures

Figure 1. Patient flow diagram................................................................................24

Figure 2. Stages of research.....................................................................................30

Figure 3. Elective patient flow.................................................................................59

Figure 4. Emergency patient flow...........................................................................60

Figure 5. Program block..........................................................................................61

Figure 6. Elective patient arrival ............................................................................61

Figure 7. Re-direction of elective consultant to emergency patient.....................62

Figure 8. Elective patient cancellation decision.....................................................62

Figure 9. Distribution of simulation emergency patient waiting time.................69

Figure 10. Distribution of emergency patient wait taken from actual data........70

Figure 11. Utilisation of theatres A1, B3 and C1...................................................71

Figure 12. Emergency patient wait –10%.............................................................73

Figure 13. Emergency patient wait +10%.............................................................73



Figure 16. Elective patient wait –10%...................................................................75

Figure 17. Elective patient wait +10%..................................................................75



Figure 20. Effect of admissions on waiting time....................................................76

Figure 21. Effect of admissions on cancellations ...................................................76

Figure 22. Effect of admissions on utilisation rate................................................77

Figure 23. Effect of scheduling disciplines on cancellations.................................80

Figure 24. Effect of scheduling disciplines on waiting time..................................80

Figure 25. Fitted distribution versus actual data for specialty 1 .........................87

Figure 26. Gantt Chart illustrating results for one schedule .............................128

Figure 27. Process chart for reactive scheduling model ....................................132

Figure 28. BE&T LOS distribution fitting results ... ...........................................179

Figure 29. CARD LOS distribution fitting results ..............................................179

Figure 30. COLO LOS distribution fitting results..............................................179

Figure 31. FMAX LOS distribution fitting results... ...........................................179

Figure 32. ENT LOS distribution fitting results .................................................179

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Figure 33. HPB LOS distribution fitting results ................................................. 179

Figure 34. NSUR LOS distribution fitting results .............................................. 180

Figure 35. PLAS LOS distribution fitting results ............................................... 180

Figure 36. OPHT LOS distribution fitting results .............................................. 180

Figure 37. RTPT LOS distribution fitting results............................................... 180

Figure 38. ORTH LOS distribution fitting results.............................................. 180

Figure 39. UGI LOS distribution fitting results.................................................. 180

Figure 40. UROL LOS distribution......................................................................181

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List of Tables

Table 1. List of surgical categories .........................................................................65

Table 2. Surgical duration distribution fitting results ..........................................65

Table 3. Simulation results ......................................................................................68

Table 4. Changing patient arrivals .........................................................................72

Table 5. Alternative scheduling disciplines............................................................78

Table 6. Data for Specialties....................................................................................88

Table 7. Schedule results comparing LDSD and NDSD models ........................105

Table 8. Simulation results comparing robust assignment models ..................108

Table 9. Example comparison matrix for alternative objectives. .....................109

Table 10. AHP Objective weights for alternative priorities ...............................109

Table 11. AHP Results for alternative priorities.................................................111

Table 12. Schedule details for top 5 AHP selected schedules .............................111

Table 13. Comparison of fixed and flexible assignment methods.....................113

Table 14. Comparison of constructive heuristics ...............................................114

Table 15. LVF results when SPT logic is used to assign additional patients ...115

Table 16. Results of reactive schedules................................................................126

Table 17. Comparison of initial patient sequences for fixed processing time and

robust models..................................................................................................153

Table 18. Comparison of final patient sequences for fixed processing time and

robust models..................................................................................................154

Table 19. Comparison of performance measures for fixed and robust schedules

against robust assignment model ..................................................................157

Table 20. Percentage of patients requiring more time than assigned ...............158

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Publications arising from this research

Refereed Journal Articles

1. Stuart, K., Kozan, E., Sinnott, M. and Collier, J (2010). An Innovative Robust

Reactive Surgery Assignment Model, ASOR Bulletin, 29.3, 48-58.

ERA Rank: C

Refereed Conference Papers

1. Stuart, K. and Kozan, E. (2007) Modelling of Operating Theatres using

Simulation, The 8th Asia-Pacific Industrial Engineering and Management

Systems Conference, Kaohsiung, Taiwan.

ERA Rank: C

2. Stuart, K. and Kozan, E. (2009) Online Scheduling in the Operating Theatre, The

10th Asia-Pacific Industrial Engineering and Management Systems Conference,

Kitakyushu, Japan.

ERA Rank: C

Non-Refereed Conference Papers

1. Stuart, K. and Kozan, E. (2007) Optimal Offline Robust Surgical Schedules, The

Australian Society for Operations Research Annual Conference 2007,

Melbourne, Australia.

Papers Submitted to Journals

1. Stuart, K. and Kozan, E. Comparing robust elective surgery assignment models.

Submitted to European Journal of Operations Research, April, 2010.

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2. Stuart, K. and Kozan, E. A new methodology and constructive heuristics for

elective surgery scheduling. Submitted to Computers and Mathematical

Modelling, 25th January, 2010.

3. Stuart, K. and Kozan, E. Bi-criteria reactive scheduling model for the operating

theatre. Submitting to Flexible Services and Manufacturing Journal, September,

2010.

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List of abbreviations

TERM ABBREVIATION

Breast, Endocrine & Thoracic BE&T

Classification and regression tree CART

Clinical priority assessment criteria CART

Elective booking officer EBO

Emergency department ED

First come first served FCFS

Highest variance first HVF

Intensive care unit ICU

Integer program IP

Least flexible job LFJ

Length of surgery LOS

Longest processing time LPT

Longest time first LTF

Lowest variance first LVF

Non-deterministic polynomial time NP

Operations research OR

Operating room management information system ORMIS

Operating theatre OT

Polynomial P

Post-anaesthesia care unit PACU

Princess Alexandra hospital PAH

Room for improvement RFI

Surgical care unit SCU

Shortest processing time SPT

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Statement of original authorship

I acknowledge that the work contained in this thesis has not been previously

submitted to meet requirements for an award at this or any other higher education

institution. To the best of my knowledge and belief, no material previously

published or written by another person except where due reference is made is

contained in this thesis.

Signed,

Kari Stuart

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Acknowledgements

Firstly, I am grateful for the support and guidance of my principal supervisor

Professor Erhan Kozan! Thank you for everything!

To the staff at the Princess Alexandra Hospital, I wish to acknowledge and thank you

for your cooperation and contribution to this study. In particular, for their insight

and knowledge on the running of the emergency department and their assistance in

providing collaborations with operating theatre staff, thank you to Dr James Collier

and Dr Michael Sinnott of the emergency department. Thank you also to Ms

Margaret Walker for spending invaluable time providing information on the running

of the operating theatre and explaining the processes occurring within the

department.

Thank you to the financial supporters of this project, the Australian Research

Council, the Princess Alexandra Hospital and also to the Mathematical Sciences

Discipline, Queensland University of Technology, for support of this project.

To my associate supervisor, Professor Vo Anh and the other members of this

research project, Mel Diefenbach and Andy Wong, thank you for your

collaborations, knowledge and most importantly, support. I would also like to

express my gratitude to the panel members and examiners of this thesis. Your input

is immeasurable.

A very big thank you to Ms Mary-Anne Ramis of the Queensland University of

Technology for liaising with PAH staff members, organising meetings with staff,

collecting data and generating databases from patient records and for your assistance

with editing this thesis.

Finally and possibly most importantly, thank you to my family - Damian, Jarrod and

Nathan. I could not have completed this research without your support, love and

patience!! And to the One whose love and faithfulness endures forever, thank you

for being my rock.

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Chapter 1. Introduction

1.1 Healthcare and surgery in Australia

“Healthcare is the preservation of mental and physical health by preventing or

treating illness through services offered by the health profession” (Miller, Fellbaum,

Tengi & Wakefield, 2006).

The healthcare system in Australia consists of both private and public hospitals,

medical practitioners, and funding by insurance schemes. Federal and state

governments cover approximately 70% of healthcare costs, with the remainder

covered by private sources. In 2004-05, approximately $87.3 billion was spent on

healthcare in Australia, a quarter of which was spent on public hospitals. Funding

sources were federal ($9.8 billion), state, territory and local governments ($10.6

billion) and private sources ($1.7 billion). These costs comprise staff salaries and

superannuation, fees, equipment, medical supplies and support services. The costs of

treating conditions depend on individual requirements including anaesthetics,

medications and facilities (Australian Government Department of Health and

Ageing, 2008).

Medicare was introduced to Australia’s healthcare system in 1984. This

comprehensive system facilitates access to free or low cost medical, optometric and

public hospital care. Funding for this system comes from individuals, based on their

income through a taxation levy. Patients with private health insurance have access

to both public and private hospitals. These patients may choose the treating doctor

but are charged for hospital accommodation and other expenditures including theatre

fees. Some or all of these outstanding fees are covered by private health insurance

(Australian Government: Department of Foreign Affairs and Trade, 2008).

There are three ways in which a patient may enter a hospital; as an outpatient,

emergency patient or inpatient. Ranges of specialist medical and surgical services

are provided in outpatient clinics. Most specialist outpatient clinics are open

Monday to Friday 8:30am to 4:30pm and are closed on public holidays. Obtaining

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an appointment for an outpatient clinic requires a referral from a general practitioner

or other medical practitioner. Free emergency care is provided in a public hospital’s

emergency department (ED) for treatments varying from life-threatening illnesses to

less severe symptoms or injuries that are potentially serious. Patients are categorised

(triaged) according to the standards defined by the Australasian College of

Emergency Medicine. Each triage category has a maximum time by which patients

should be seen for treatment by a doctor or nurse with the more urgent triage

categories treated first. Inpatients may be further categorised as scheduled or non-

scheduled. Non-scheduled inpatients are immediately admitted by a specialist in

either the outpatient clinic or ED. Scheduled inpatients are elective patients taken

from a waiting list. The wait for admission depends on many factors including

priority for treatment and the length of the waiting list. There are various reasons for

public hospital admissions including surgery, diagnostic tests, or treatment for severe

medical conditions. In 2005-06, there were almost 4.5 million admissions to public

hospitals (61% of all patient admissions); 65% of these were for acute medical care

and 18% for surgery (Australian Government Department of Health and Ageing,

2008).

According to the State of our Public Hospitals June report, approximately 43%

(908,000) of all admissions for surgery during 2006-07 were in public hospitals

(Australian Government Department of Health and Ageing, 2009). The proportion

fell by 2% in 2007-08 but the actual number of surgical procedures rose to 920,000

(Australian Government Department of Health and Ageing, 2009).

In 2007-08, the number of emergency cases increased to 223,000 and elective

surgeries were up by 9500 cases. Of the emergency surgery cases, 87% were carried

out in public hospitals (Australian Government Department of Health and Ageing,

2009). In 2006-07, approximately 23.7% were emergency cases (or 215,000

procedures) and more than 556,000 were admitted from elective surgery waiting lists

(Australian Government Department of Health and Ageing, 2008). The rest of the

procedures requiring surgery are those that are not normally assigned an urgency

category, such as emergency caesarean surgeries.

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Elective surgery waiting lists are managed with the aim of ensuring patients are seen

within a recommended time appropriate to the urgency of their condition (Australian

Government Department of Health and Ageing, 2008). Public hospitals follow the

Policy Framework for Elective Surgery Services (Queensland Health, 2004) when

prioritising elective surgery. This policy provides a structured approach to

scheduling elective surgeries. Clinical urgency category is the fundamental decision

for scheduling elective patients (Category 1 before Category 2, Category 2 before

Category 3). The most recent definitions for the clinical urgency categories

according to the June 2009 Report for the state of our public hospitals (Australian

Government Department of Health and Ageing, 2009) are as follows:

Category 1 Admission within 30 days desirable for a condition that has the

potential to deteriorate quickly to the point that it may become an

emergency.

Category 2 Admission within 90 days desirable for a condition causing some pain,

dysfunction, or disability but which is not likely to deteriorate quickly or

become an emergency.

Category 3 Admission within 365 days for a condition causing minimal or no pain,

dysfunction or disability, which is unlikely to deteriorate quickly and

which does not have the potential to become an emergency.

Within each clinical urgency category, other factors such as waiting time, previous

postponement, operating theatre (OT) management and effective bed management

must be considered when selecting patients from the elective surgery waiting list.

In 2007-08, 31% of public patient admissions for elective surgery were assigned to

Category 1, 39% to Category 2 and 30% to Category 3. Variations in these

distributions occur at state levels. Timeframes for the provision of elective surgery

are key performance measures for public hospitals. For 2008-09, 89% of Category 1

patients, 74% of Category 2 patients and 92% of Category 3 patients were seen

within the recommended timeframes. (Australian Government Department of Health

and Ageing, 2009).

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1.2 Some research issues

The efficient running of a hospital is a concern for hospital administrators,

politicians, healthcare providers and patients alike. Although each of these has a

vested interest in a hospital’s success, they may have different and/or conflicting

measures of efficiency. For example, hospital administrators and politicians measure

efficiency by cost and achieving high utilisation of resources. On the other hand,

healthcare providers measure by appropriateness of care while patients desire prompt

satisfactory care. High utilisation of resources such as beds, has been linked to

increased patient waiting times (Sier, 2004) and for those waiting for treatment (and

possibly in pain), the costs imposed are the least of their concerns.

Operating theatre scheduling

A hospital consists of a number of wards, units and departments that provide a

variety of medical services, which interact on a day-to-day basis. Nearly every

department within a hospital schedules patients for the operating theatre (OT) and

most wards receive patients from the OT following immediate post-operative

recovery. This link between the OT and other facilities contributes not only to its

importance but also its costs (Cardoen, Demeulemeester & Belien, 2009a). Due to

the interrelationships between hospital units, disruptions in the OT department are

not limited to the OT. Elective surgery cancellations have a cascading effect,

resulting in dissatisfied patients and escalating waiting lists (Cheng & Newman,

2005). Compounding this, is that hospital wards act autonomously and scheduling of

each ward is usually done without considering the actions and needs of other wards,

each of which acts on behalf of its patients (Vermeulen, Bohte, Somefun & La

Poutre, 2006). This may lead to inefficient usage of resources and undesirable and

unnecessary waiting times. Improving the interaction between hospital units

however, is a difficult task. Hughes, Carson, Morgan and Silvester (2005) showed

that as the degree of inter-dependency between hospital units increases, the process

of admitting patients becomes more difficult.

OT scheduling involves assigning surgical patients and scarce resources to the

theatres in a manner that maximises resource utilisation and minimises theatre idle

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time. Two types of patients are considered: elective and emergency. Elective

patients are selected from a waiting list and scheduled in advance based on resource

availability and a set of objectives. This type of scheduling is referred to as ‘offline

scheduling’. Disruptions to this schedule occur for various reasons, including

variations in length of treatment, equipment restrictions or breakdown, unforeseen

delays and the arrival of emergency patients, which may compete for resources.

Emergency patients consist of acute patients requiring surgical intervention or in-

patients whose conditions have deteriorated. These may or may not be urgent and

are triaged accordingly. Most hospitals reserve theatres for emergency cases, but

when these or other resources are unavailable, elective theatres may be used. This

can lead to disruptions to the elective schedule such as delays in surgery start time,

elective surgery cancellations or transfers to another institution. Scheduling of

emergency patients and the handling of schedule disruptions is an ‘online’ process

typically handled by OT staff.

Scheduling operations for the OT is a difficult task. Complicating the problem is the

natural variability of patients, including the type and severity of the presenting

illness, which leads to variability in duration of surgery and resource requirements.

If demand for resources is higher than the available capacity, bottlenecks in the

system result, which generally have a cascading effect, affecting the efficiency of the

entire hospital. In the same way, bottlenecks in other departments can affect the OT.

For example, some surgeries require a post-operative intensive care unit (ICU) bed.

This needs to be considered when planning surgeries, as a bed needs to be booked in

advance in the ICU. If the required bed is unavailable prior to surgery, this may

necessitate the cancellation of the surgery. When problems like this occur, decisions

have to be made to determine which patients will be operated on and which will not.

Improving the system

There are generally two methods for improving a system; investment of funds to

provide more resources or optimisation of the already available resources.

Operations research (OR) techniques will be applied to enhance the performance of

the system by optimising the use of operating theatres’ capacity, via improving

elective patient planning using robust scheduling techniques, and increasing the

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overall responsiveness to emergency patients by solving the disruption management

and re-scheduling problem.

1.3 A real life example

1.3.1 The Princess Alexandra Hospital

The Princess Alexandra Hospital (PAH), located in Woolloongabba, Brisbane, is one

of three tertiary teaching Hospitals in Queensland. The hospital currently has 712

available beds and has an extended capacity for 825 beds. With the exception of

obstetrics, care is provided for all major adult specialties with particular expertise in

spinal injuries and solid organ transplantation. Other services offered by the hospital

include acute care, medical, rehabilitation, surgical, mental health and allied health.

The surgical services provided by the hospital include cardiac surgery, spinal injury,

liver and renal transplantation and varying elective procedures relating to

hepatobiliary, ophthalmology, urology, gynaecology, breast, endocrine, thoracic, ear,

nose and throat (ENT), plastics, dermatology, orthopaedics, oncology and vascular

care.

1.3.1.1 Admission processes

The majority of patients at the PAH are booked admissions that are either ‘day

patients’ or ‘in patients’. An ‘in patient’ is one that stays overnight or for a few days

at the hospital. Day patients receive tests or treatments on the day of their admission

and do not stay overnight.

Not all patients arriving at hospital need to be admitted. The PAH is just one of

Australia’s public hospitals that provide a range of specialist medical and surgical

services in outpatient clinics. The PAH also provides a free emergency care service

open 24 hours a day for seriously ill cases and those requiring immediate treatment.

Treatment is issued on a priority basis, using a FCFS discipline within priority

categories. Emergency patients may be admitted to hospital through the ED if

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necessary. The wait for admission depends on many factors including priority for

treatment and the length of the waiting list.

Patients waiting for elective surgery are placed on the elective surgery waiting list.

The PAH follows the guidelines set out in the Policy Framework for Elective

Surgery Services (2005) when prioritising elective surgery. For details on this policy

see Section 1.1 on healthcare and surgery in Australia. Within each clinical urgency

category, other factors such as waiting time, previous postponement, OT

management and effective bed management must be considered when selecting

patients from the elective surgery waiting list.

1.3.1.2 OT layout

There are two entry points to the OT department depending on the patient’s source.

The main entry point is for both emergency and elective patients. Associated with

this entry point is a holding bay where beds may be wheeled prior to surgery if the

scheduled theatre is unavailable on arrival. The second entry point is for the surgical

care unit (SCU). Patients undergoing day procedures who are generally discharged

the same day and those that are admitted on the day of their surgery, but may require

admittance to a ward following surgery, originate from the SCU. The SCU has its

own holding bay that functions in the same way as the main entrance holding bay.

The PAH contains five OT pods (A, B, C, D & E) with each containing 2-6 theatres

depending on the theatre size. Currently, 18 functioning elective theatres and two

emergency theatres are in use. Each OT contains a standard set of equipment

necessary for routine surgical procedures and any specialized equipment according to

the types of surgeries performed there. Depending on size, equipment may or may

not be movable to different theatres. For example, due to the largeness of the

specialized equipment, cardiac procedures are limited to the larger cardiac theatres.

The OT layout is given in Appendix A. Prior to surgery, elective patients wait in

wards and other departments until they are ‘called for’ to prevent bottlenecks.

Bottlenecks occur when patients are brought up from the ED before a theatre is ready

and there is no room for the patients in the holding bays. Post-operative bottlenecks

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occur particularly when a patient is waiting to go back to a ward and there is no bed

available.

Attached to each Theatre is an anaesthesia workroom where any necessary

anaesthetics are administered before entering the actual theatre room. This allows

for ease of patient flow and reduces the possibility of bottlenecks to the system.

Following surgery, patients are sent to one of three destinations depending on their

originating source and health status; the ICU, the Post Anaesthesia Care Unit

(PACU) or the SCU. A patient flow diagram (Figure 1) was generated with the

assistance of hospital OT and ICU medical staff.

1.3.1.3 OT scheduling

Arriving patients are classified as elective, day surgery and emergency. Elective

surgeries are only performed during business hours, Monday to Friday, and are

selected in advance. Emergency patients may be broadly defined as those requiring

surgery in the immediate future and are not on the scheduled elective surgery list.

They arrive at random from other wards and departments within the hospital,

complicating the scheduling process as they compete for the scarce resources.

Generally, emergency patients are treated in the emergency theatres; however, when

necessary they may be treated in an elective theatre. In the case of cardiac

emergencies, for example, these patients would require the use of cardiac theatres for

equipment reasons. Allocation of emergency patients to theatres is the responsibility

of the OT staff. They are added to the schedule at appropriate times according to the

triage category assigned by the requesting surgeon and resource availability.

Emergency cases are listed on an emergency white board in the nurses’ station. The

emergency white board lists the patient’s name, the required procedure, triage

category and previous cancellations. Emergency cases that require day surgery are

sent to the SCU. Many day surgery emergency cases are cancelled due to

insufficient theatre capacity. Minimising disruptions to the original schedule such as

delays in start times and elective cancellations are important considerations when

making any changes.

Figure 1. Patient flow diagram

Currently a block scheduling system is implemented for elective theatre scheduling

in which a theatre block is assigned to a given surgeon or specialty. OT blocks for

elective procedures are generally AM (0800 hours to 1200 hours), PM (1300 hours to

1700 hours) or all day (0800 hours to 1500 hours or 0830 hours to 1530 hours).

Depending on the duration of the surgical case, however, some procedures will start

early (0730 hours). A four-week cyclical surgical roster is used for the elective

theatres, with the exception of the eye theatres, which work on a 12-week cyclic

roster.

Medical officers are responsible for selecting their own patients and sequencing

them. Generally, a longest processing time (LPT) scheduling discipline is used. The

longer surgeries are typically more variable and prone to running overtime resulting

in cancellations of the remaining cases for that particular theatre. In this way,

scheduling the more variable surgeries first minimises the incidence of overtime.

Surgical duration is affected by the surgeon’s caseload and abilities. OT bookings

are entered on the Operating Room Management Information System (ORMIS). The

Elective Booking Officer (EBO) notifies all patients by phone (and booking letter if

time permits) of the booking date.

The goal of the elective schedule is to maximise patient throughput and theatre

utilisation. Operating room utilisation is the percentage of time that a service uses its

allocated operating room time (Abouleish, Hensley, Zornow & Prough, 2003).

Currently a meeting is held each Thursday to review the forthcoming week’s surgery

lists. At these meetings, staffing issues including absences (especially nursing),

caseload, equipment requirements and the types of surgeries to be performed, are

discussed. In the case of a surgeon on leave, the list may be given to another surgeon

to perform. In some cases however, the surgeon will cancel the list.

The following non-exhaustive list of events may result in surgical cancellations;

· Late arrival of surgeon. In the case of a surgeon being late, another surgeon

may not fill the time left unutilised;

· Unexpected complications during surgical procedure resulting in tardy

completion, which results in insufficient theatre capacity for the remaining

cases;

26

· The unavailability of a post operative bed, for example in the ICU, prior to a

surgery commencing;

· Arrival of a category 1 or 2 emergency case requiring an elective theatre;

· Insufficient resources or equipment failure; and

· Non-allowance of theatre swapping.

In the event of an elective cancellation that frees up theatre capacity, the available

capacity may be used by subsequent cases on the medical officer’s list or emergency

cases listed on the white board.

1.4 Motivation, aims and methodology of the research

1.4.1 Gaps in the research

Literature devoted to OT scheduling to date, has focused primarily on the

deterministic process of assigning patients to theatres and the patient sequencing

problem to maximise resource utilisation (Belien & Demeulemeester, 2007b, Guinet

& Chaabane, 2003, Hans, Wullink, Van Houdenhoven & Kazemier, 2008, Jebali,

Hadj-Alouane & Ladet, 2003). Advancements in technology, however, have

demonstrated the ineffectiveness of empirical, deterministic and stochastic

scheduling models during project execution as they fail to capture the dynamic and

variable nature of the OT environment prompting a move towards robust scheduling

techniques (Daniels & Kouvelis, 1995). Daniels and Kouvelis (1995) stated that ‘a

schedule which is optimal with respect to a deterministic or stochastic scheduling

model yields quite poor performance when evaluated relative to the actual processing

times’. The objective of robust scheduling is to perform better than stochastic

scheduling or using empirical distributions alone.

Robust scheduling considers the stochastic element of surgery durations when

developing a schedule by incorporating buffer (or slack) times to allow for the

variability in surgical durations. This buffer forms a ‘hedge’ against poor schedule

performance. The use of buffers absorbs some of the variation in surgical duration

making schedules more robust to variations in actual treatment time that occur during

27

project execution. Hans et al. (2008) applied robust scheduling techniques to the

patient assignment problem. In their paper, the sum of the surgical durations

assigned to a theatre is assumed normally distributed and the properties of their

summation are used to determine an amount of slack for each theatre. Their model

however, cannot be applied to the case where the random variables used to model

surgical durations, are not normally distributed. In particular, this thesis addresses

the problem of lognormally distributed surgical durations, which are shown to be a

better choice for modelling surgical durations than the normal distribution.

By using a statistical distribution that better ‘fits’ the data, (such as the lognormal

distribution rather than the normal distribution), these ‘buffers’ should be improved.

The reason for this is because the mode of a normal distribution would be

quantitatively greater than that of a lognormal distribution for the same set of data.

One could see this in the shapes of the distributions. This means that the amount of

time assigned for the patients using the normal distribution would be greater than that

for the lognormal distribution. If more time is assigned for a list of patients than is

necessary then fewer patients may actually be assigned than is efficient. Therefore,

by using a more appropriate statistical distribution, the efficiency of the robust

schedule can be improved.

In addition to addressing lognormally distributed surgical durations for robust

scheduling, in this thesis, a measure called ‘deviation’ is introduced that considers

both earliness and tardiness. Studies have shown that costs of running the operating

theatre occur not only from theatres running into overtime, but also due to theatres

completing surgery early and therefore leaving unused capacity (Dexter, Macario &

O'Neill, 2000, Dexter, Macario, Traub, Hopwood & Lubarsky, 1999). This is a

further extension over the work by Hans et al. (2008) who consider only the tardiness

of the schedule.

Emergency (or non-scheduled) operations are responsible for most of the uncertainty

in operating room scheduling and yet the problem has been largely ignored by the

literature (Cardoen, Demeulemeester & Belien, 2010). To the author’s knowledge,

there is no application of online reactive scheduling for the OT other than for disaster

events. Reactive schedules update the elective schedule at regular time intervals,

28

taking into account cancellations, emergency arrivals, expected delays and early

completion times. The objective of reactive scheduling is to minimise cancellations

and deviations from the initial schedule. Bard and Purnomo (2005) applied reactive

techniques to nurse scheduling. Before the beginning of each shift, the schedule was

re-solved to consider updated staffing requirements with the smallest deviation from

the initial schedule. A similar approach will be applied to the OT. The elective

theatre schedule will be regularly re-solved to incorporate disruptions to the original

schedule, such as arrival of emergency patients, equipment failure and post-operative

bed availability, generating an updated assignment of patients that is as close as

possible to the original schedule. Schedules generated with the robust scheduling

model will form the baseline schedule for the robust reactive scheduling model. By

using this as a baseline schedule, the need for re-scheduling due to variations in

surgical duration is reduced.

29

1.4.2 Aims of the research

The objective of this thesis is to use operations research methodologies to improve

operating theatre scheduling by analysing the system, determining offline elective

patient schedules and developing an online reactive scheduling model.

1. Analysis of an OT department system with the aid of simulation. A

simulation model will be used as a decision support tool for demonstrating the

effects of changes on the system and simulation will be used for comparing the

results of the analytical models. Simulation is beneficial because it allows the

analyst to realistically represent complex systems and investigate hypothetical

scenarios without continual interruptions to the real system (Kim, Horowitz,

Young & Buckley, 2000, Kozan & Gillingham, 1997, McHardy, Kozan &

Cook, 2004). This is important for healthcare systems, where patient’s lives

may be at stake and therefore testing new techniques without prior predictions

on performance outcomes could be devastating (Kim et al., 2000).

2. Determination of offline elective patient schedules using a robust scheduling

technique. Variability in scheduled arrivals, can reduce access to care and

impair the overall responsiveness to emergencies (McManus et al., 2003). On a

more general scale Sier (2004), found that to improve a unit’s performance the

variability in the system needs to be reduced. It is therefore important to

implement effective elective scheduling policies to reduce the effects of

variability in surgical durations. Advancements in technology have

demonstrated the ineffectiveness of deterministic and stochastic scheduling

models upon implementation as they fail to capture the dynamic and variable

nature of the OT environment, prompting a move towards robust scheduling

techniques.

3. Development of online reactive scheduling models. These models will handle

disruptions to the offline schedule such as delays in surgery start time,

emergency patient arrivals requiring elective theatres and variations in

treatment times. These models are completely innovative and have not (to the

author’s knowledge) been addressed in the literature before.

30

1.4.3 Approach

This PhD thesis will:

- analyse the performance of the OT using simulation;

- improve elective theatre scheduling using a robust scheduling technique; and

- address the disruption management problem using innovative robust reactive

scheduling techniques.

Figure 2 is an overview of the different stages of the research including the

interactions between the stages.

Figure 2. Stages of research

31

Simulation will be used in this thesis, as a decision support tool and to analyse the

results of the analytical scheduling and sequencing models. Firstly, a simulation

model will be used to compare the current sequencing method implemented at the

hospital with two alternatives and secondly to observe the effects of changing patient

arrivals on the system’s performance measures.

The first sequencing heuristic to be tested is one that has been previously examined

in the literature, SPT. The second method, least flexible job (LFJ), is normally used

in machine scheduling and has not yet been applied to the patient sequencing

problem. The second sensitivity analysis will provide evidence for hospital planners

of the effects of hypothetical ‘what if’ scenarios and introduces other possible

avenues of investigation such as the benefits of opening the elective theatres for

longer hours and/or on weekends compared with the extra staffing costs. The

simulation model, once developed, will be validated against historical data.

Methodologies from robust scheduling and approaches to handling surgical duration

variance are brought together to create schedules more robust to variations in patient

treatment times that exhibit during project execution. Robust (or preventive)

scheduling serves as the basis for planning resource commitments.

Robust scheduling addresses the feasibility of the offline schedule in the online

environment. This is achieved by assigning an amount of ‘slack’ or ‘buffer’ to the

schedule to absorb the variation in the treatment times of the patients. This slack is

based on the variance of the expected value of the surgical duration. In addition to

addressing quality robustness, the robust schedule will also address the flexibility in

patient assignment. The development of an optimal operating theatre schedule

requires the flexibility of surgeons (Calichman, 2005) and hence increasing the

flexibility of elective scheduling is investigated.

Robust schedules will be used as the basis for reactive scheduling techniques applied

to address the disruption management problem. The reactive scheduling models will

address solution robustness in the online environment and routinely repair the

baseline schedule following activity disruptions. Such disruptions include resource

or staffing unavailability, emergency patient arrivals, patient delays or cancellations,

32

and variability in treatment times. Efficient reactive schedules should incorporate

changes that occur during project execution with minimal changes to the original

schedule. Difficulties in reactive schedules arise due to the time dependent

component of the model. Automating the scheduling procedure negates the need to

be heavily reliant upon human experts to manually perform the rescheduling process

and has the potential for great cost saving benefits for the hospital.

The models developed in this dissertation will be adaptable to varying levels of

patient flexibility. Essentially, this means the models can be easily applied to other

hospitals that utilise similar scheduling systems. At the PAH for example, surgeons

are assigned specific operating rooms and times on a rostered basis. In addition to

this, patient swapping is not performed at the PAH. More general forms of

scheduling models for the OT assume that the operating rooms are assigned to

surgical specialties rather than specific surgeons and allow numerous surgeons to

treat a given specialty (and therefore any patient within the specialty assuming skill

levels are met). The resulting schedules may be compared and the differences

presented to hospital staff to illustrate what would happen if patient assignment were

more flexible, allowing patients to be treated by more than one consultant.

In order to determine the solution approach used for each of the mathematical

models, the complexity of the problem must be known. In computational theory, a

complexity class is a set of problems of related complexity. The complexity of a

problem is the degree of difficulty in providing efficient algorithms for specific

computational problems.

One of the most fundamental complexity classes is polynomial (P). P contains all

decision problems that can be solved in polynomial time on a deterministic Turing

machine. An algorithm with a time complexity function O(p(k)), where p is some

polynomial and k is the input length of an instance, is said to be a polynomial time

algorithm. This class is generally taken to contain computational problems that are

tractable.

A problem is assigned to the non-deterministic polynomial time (NP) class if its

solution may be verified in polynomial time on a non-deterministic Turing machine,

33

but the problems themselves have no known polynomial time algorithm. A P

problem is always also NP. A problem is NP-hard if an algorithm for solving it can

be translated into one for solving any other NP problem. A problem that is both NP

and NP-hard is called NP-complete. Some NP complete problems are termed NP

hard in the ordinary sense because they may be solved in pseudo-polynomial time.

These become more difficult to solve with the size of the input.

The relationship between the complexity classes P and NP is an unsolved question in

theoretical computer science. If P and NP are not equivalent, then the solution of

NP-problems requires (in the worst case) an exhaustive search, while if they are, then

asymptotically faster algorithms may exist.

If a problem that is NP-hard or NP-complete remains so when its numerical

parameters are bounded by a polynomial in the length of the input, then it is said to

be strongly NP-hard or strongly NP-complete. Numerical parameters are usually

given in binary form such that numerical values may be represented by two symbols

(0-1). Therefore the size of the parameters may be exponential in n if the problem

has input size n. Natural numbers may also be represented in unary, in which any

natural number N, is given by any arbitrary symbol repeated N times. If a problem

originally defined in binary is re-defined so that its parameters are given in unary

notation, then the parameters must be bounded by the input size. So strongly NP-

hard or strongly NP-complete problems are NP-hard or NP-complete problems re-

defined in unary form.

The robust assignment models are strongly NP hard (see Section 4.6 for the proof of

this). For optimisation problems of this complexity, polynomial time optimisation

algorithms for obtaining the optimal solution are not known. Heuristic algorithms

are used to find near optimal solutions in a reasonable amount of time. A heuristic is

a method based on intellectual “guesswork” rather than a pre-established formula for

solving a problem (Blazewicz, Ecker, Pesch, Schmidt & Weglarz, 1996).

Many general heuristics exist that can be applied to various problem types, where

some suit particular problem structures better than others. The alternative is to create

problem specific heuristics. The disadvantage is that developing problem specific

34

heuristics may be more time consuming, however because they exploit the nature of

the problem, they may result in better solutions.

A meta-heuristic is a heuristic method, which combines other heuristics in an

efficient way to solve computational problems. These are generally applied when

there is no satisfactory algorithm or heuristic for a specific problem (Van Hentenryck

& Michel, 2005). Local search is an example of a meta-heuristic. They move

through the solution space from solution to solution until either a solution deemed

optimal is found, or a time bound elapses. Choosing the next possible solution to

move to, is based solely on information about the solutions in the neighbourhood of

the current solution. In optimisation problems, the measure of the quality of a

solution is an expression of the objective of the problem that differentiates between

solutions with the same objective value (Van Hentenryck & Michel, 2005).

Neighbourhood search is a local search algorithm that starts with some initial

solution and moves from one neighbour solution to another while improving the

objective function. This technique underpins many local search algorithms

(Blazewicz et al., 1996). Methods of moving to new solutions depend on

information about the solutions in the neighbourhood of the current solution. Hill

Climbing, for example, moves to the first closest solution that improves on the

current one. Steepest ascent hill climbing, on the other hand, compares all possible

successors, choosing the closest one to the current solution. Other methods

randomly select a neighbour solution and decide whether to move to that neighbour

or choose another (Van Hentenryck & Michel, 2005).

For the robust assignment models, innovative constructive heuristics and a problem

specific neighbourhood search are proposed. These methods will exploit the nature

of the problem at hand and the performance of the proposed constructive heuristics

will be compared with traditional constructive heuristics.

There are two types of reactive models to solve, both of which are NP hard in the

ordinary sense. The first is an assignment model that will be solved with an

enumerative method. The second group of reactive scheduling models build on the

first by sequencing the patients. The scheduling models treat the problem as a single

35

machine scheduling problem, with a common due date and weighted jobs, and are

analogous to a binary knapsack problem.

The knapsack problem has been widely studied in the literature and an in-depth

discussion of the problem may be found in (Martello & Toth, 1990). Knapsack

problems are decision problems in which the number of items that can be fit into a

knapsack of given capacity must be decided, based on the weight consumed and the

benefit obtained by including the object. The most basic form of the problem is the

0-1 knapsack, in which an item is either packed or it is not. This problem may be

mathematically formulated as:

Maximise ∑=

n

iii xb

1

Subject to:

cxwn

iii ≤∑

=1

=otherwise 0

included is i item if 1ix

The binary variables ix determine whether or not an item is included in the knapsack

and each item has an associated benefit of being packed, given by ib , and weight that

it consumes, iw . The total capacity of the knapsack is given by c. Solution

approaches for such problems include dynamic programming, branch and bound

algorithms, enumerative algorithms and various approximation algorithms (Martello

& Toth, 1990). The reactive models will be solved using enumerative algorithms

and branch and bound algorithms.

Enumerative algorithms, as the name suggests, systematically enumerate all possible

candidate solutions to various optimisation problems. One such popular method is

the branch and bound algorithm. The branch and bound method uses two

fundamental procedures; branching and bounding. Branching is the process of

dividing a large problem into two or more sub-problems. These sub-problems in turn

36

continue to be partitioned into smaller problems. Large subsets of solutions may

then be discarded by the use of upper and lower solution bounds; a process referred

to as bounding (Blazewicz et al., 1996).

1.5 Contributions of the thesis

Simulation is used in this thesis to model the highly complex system that is the OT

department of the PAH, using the simulation software package EXTEND.

Simulation has been chosen because of its ability to immediately observe the effects

that changes to parameters and inputs have on a system and also its use as a visual

tool for explaining model outputs to decision makers. An important contribution of

the simulation model is its potential implementation as decision support for surgical

planners, by predicting the effects changes in the model’s parameters may have on a

system.

In particular, the simulation model contributes to the literature by exploring the

effects that increasing patient arrivals and alternative elective patient admission

disciplines will have on the performance measures. Specifically, waiting times of

patients, elective cancellations and theatre utilisation rates will be observed.

While simulation provides a way to examine scenarios without actually impacting

the real life system, it is also important to develop mathematical models, which

provide methods for optimising the system. For this reason, theoretical models based

on operation research techniques are used for elective scheduling and to develop a

new tool for the dynamic online scheduling problem that incorporates emergency

patients.

A robust scheduling approach will be used for elective surgery scheduling. Buffers

are used to absorb variations in treatment times, where the amount of slack planned

for each block, is based on the variations in expected surgical durations of the

patients scheduled in the block. In particular, the robust assignment model

developed in this thesis is based on the assumption that surgical durations are

modelled with the lognormal distribution (for justification of this choice, see Section

37

4.1), whose sum is not modelled by a normal distribution. This is a point of

innovation for robust patient assignment in the operating theatre.

Further innovation is achieved by addressing the issue of flexible patient assignment.

In the real life, surgical consultants are generally responsible for selecting and

sequencing their own patients. Introduced is the idea of adapting the robust patient

assignment model to allow patients to be selected directly from a waiting list to

optimise planning of the available capacity. When patient assignment is more

flexible, the benefits of such a schedule are expected to be greater and the results can

be compared with current practices and any differences presented to hospital staff.

An alternative performance measure to tardiness, namely deviation, is also

considered. The deviation of a surgery block is defined as the difference between

used and available surgical block capacity. Costs are incurred from both over and

under use of the available theatre capacity. For this reason, deviation is introduced

because it considers both earliness and tardiness.

Innovative constructive heuristics based on surgical duration variance and meta-

heuristics are proposed for solving the model and then compared with traditional

approaches. Performance measures for making comparisons between the heuristic

methods include deviation of the schedule, the average number of surgeries assigned

to a surgery time block, and the number of surgery blocks required to generate the

schedules.

An important aspect of the robust models is that they are applicable irrespective of

whether patient assignment is surgeon specific or not. This means that the method

can be applied when theatre capacity is assigned to specific surgical consultants or

surgical teams, or in an open scheduling system where capacity is not assigned in this

manner. As a result, a generalised model may be easily adapted to similar hospital

systems and can easily incorporate hospital specific requirements.

Reactive models are developed to deal with disruptions that occur in the online

environment. These models can be used for the real life daily disruption

management problem, and have not been addressed in the literature before. The

38

reactive scheduling models are capable of handling both theatre and patient

disruptions. Theatre disruptions occur when a theatre becomes unavailable for some

period of time. Patient disruptions on the other hand occur when treatment times are

less than, or greater than, the assigned surgery time.

Two types of models are developed, that use the results of the offline robust surgery

assignment models as baseline schedules. The first is a reactive assignment model

that determines the number of patients that can be assigned to the available theatre

capacity. This model is solved with an exact solution method and implemented

using Visual Basic.

The second reactive approach improves and expands on the first by addressing

patient sequencing. Two approaches to estimating surgical durations are used. The

first assumes surgical durations are based on the expected duration alone. The

second uses the robust theory approach and incorporates a portion of slack in the

surgical duration calculations. Both models are developed using single machine

scheduling techniques. Two criteria are proposed for the models; the first is to

minimise the weighted number of tardy jobs and the secondary constraint is the

minimisation of the tardiness. A branch and bound approach is proposed to solve

both models.

39

Chapter 2. Literature review

In order to understand any system under study, it is important to use a holistic

approach that encompasses as many aspects of the problem as possible. This means

that not only must theoretical approaches be investigated, but also administrative

issues should be considered. The following review covers general OT studies,

multiple objective programming, simulation applied to healthcare, scheduling

techniques, alternative scheduling techniques and robust, reactive and single machine

scheduling. Cardoen, Demeulemeester and Belien (2010) recently reviewed

operational research methodologies applied to operating room planning and

scheduling.

2.1 General operating theatre studies – Administrative issues

There is a plethora of literature discussing OT administrative issues including

elective surgery waiting lists, admission and discharge decisions and costs of the OT.

Hospital waiting lists provide a method for rationing scarce resources and are almost

exclusively used for non-emergency patients (Cromwell, 2004). Mullen (2003)

describes many benefits of waiting lists including allowance for recovery without

surgical intervention, balancing case loads, improving resource usage efficiency and

providing appropriate case mix for teaching purposes. Dew et al. (2005) evaluated

the use of clinical priority assessment criteria (CPAC) that was introduced to make

rationing of elective surgery more explicit. Explicit rationing systems aim to make

scheduling fairer but may be susceptible to political manipulation and lack of

flexibility. CPAC was found to receive limited acceptance by surgeons due to a

perceived lack of flexibility.

The existence or sizes of waiting lists, are largely considered a measurement of

resource inadequacy or resource mismanagement (Ghotb, 1993). Ghotb (1993)

argues that this assumption is not justified due to inaccuracies and the inexistence of

standards for waiting times. Inaccuracies are a result of many causes, including

patients remaining on waiting lists when they have already received treatment and

40

doctors using waiting lists to manage patients whose conditions will improve with

time.

Numerous studies investigate issues dealing with waiting lists. Mullen (2003)

examines the design of priority scoring systems for waiting list selection, concluding

that they are not purely technical and require clarity about political and clinical

objectives that are often conflicting and poorly articulated.

Long waiting times for hospital admission have generally become unacceptable to

patients and the need for finding ways of reducing waiting times has increased. As a

result, many studies aim at decreasing waiting times for patients (Cardoen et al.,

2010). Hilkhuysen, Oudhoff, Rietberg, Van der Wal and Timmermans (2005)

assessed the physical, psychological and social consequences of waiting for surgery.

Responses differ from patient to patient, depending on type of illness and personal

characteristics and these may be used as an aide for caregivers and waiting time

decisions.

The governments of some countries have online information services that allow

patients requesting elective surgery to compare waiting times for surgical units.

Cromwell (2004) investigated the accuracy of waiting time estimates for patients

about to join a waiting list compared with using clearance time functions.

Vissers, Adan and Dellaert (2007) examined extreme admission service concepts

compared with the current admission concept, based on maximising resource use,

with the aid of simulation to address long wait times for admission. Two alternative

concepts were considered based on the ‘zero wait time’ principle and the ‘booked

admissions’ principle. The study looked at the effects of admissions principles on

occupancy levels, waiting times and cancellations. Different admission policies can

be used to meet alternative performance measures. Kusters and Groot (1996)

developed a decision support tool for admission planners for predicting the effects of

decisions on resources. The models led to an improvement in operating room

utilisation without detrimentally impacting bed occupancy.

41

Adan and Vissers (2002) developed a linear integer programming (IP) model for

admission planning. Admission planning not only considers patient numbers but

also patient mix. Patient mix is important because different categories of patients

indicate different resource requirements, including beds, OT capacity, nursing

capacity and intensive care unit (ICU) beds. The model satisfied a given target

patient throughput and utilisation whilst observing resource constraints. Demeester,

Souffriau, De Causmaecker and Vanden Berghe (2010) developed an admission

scheduling algorithm that supports operation decisions within a hospital. Patient

assignment to the appropriate department is considered whilst also maintaining a

balance in the load across the departments. A computerised information system for

characterisation and analysis of hospital admission flow was developed by de Mello,

Almeida and Pereira (2001).

In addition to admission planning, the discharge process has also been studied.

Chang, Cheng and Su (2004) used case-based reasoning and analytical hierarchy

process to establish a continuing care information system for discharge planning.

Huber and McClelland (2003) investigated the importance of patient and family

participation for discharge planning. Inadequate discharge planning has been

associated with hospital readmission, lack of adherence to treatment and psychologic

stress.

Another important issue for hospital administrators is the cost of running a hospital.

The operating room is a major cost centre and revenue generator of a hospital

(Cardoen et al., 2010, Cardoen, Demeulemeester & Belien, 2009b, Denton, Viapiano

& Vogl, 2007, Pham & Klinkert, 2008) and the salaries of the operating room staff

account for most of these costs (Dexter et al., 1999). In order to minimise costs,

labour productivity must be maximised. This means using the fewest number of staff

possible to care for the patients. Since staffing numbers do not change on a day-to-

day basis with variations in patient numbers, the best way to maximise labour

productivity is to achieve maximum use of the available resources (Dexter et al.,

1999). Staffing costs were shown by Dexter, Macario and O’Neill (2000) to increase

if the operating room time goes unused or if overtime must be used. The optimal

allocation of theatre time therefore minimises unused time and overtime and hence

staffing costs.

42

Since staffing costs are affected by utilisation of theatre capacity, another avenue for

minimising costs is efficient scheduling methods (Breslawski & Hamilton, 1991).

Manual scheduling systems include first come first served (FCFS), block scheduling,

dynamic block scheduling, longest time first (LTF) and shortest time first (STF). For

a more detailed review of these systems see Section 2.4 on scheduling and

sequencing.

In order to achieve efficient use of theatre time, resource allocation must be co-

ordinated with patient scheduling. The inefficient allocation of resources may lead to

unused resource capacity for several reasons including unbalanced demand and

supply of resources, inefficient resource allocation timing leading to peaks and

troughs in workload and imbalances in the coordination of different resources that

are required simultaneously (Vissers, 1998); This issue was addressed by developing

an approach to resource allocation according to demand, whilst also balancing

resource utilisation.

In addition to addressing efficiency of resource planning and scheduling approaches,

there are many human factors affecting the efficiency of the operating theatre

department. Applegeet (1995) investigated several such key issues for OR

scheduling. Some specific problems identified included non-emergency surgical

procedures being scheduled as emergencies and issues relating to late surgeons.

2.2 Multi-objective programming

Improving the performance of any hospital department is complicated by its political

nature. Admission decisions are made by surgeons and administrators who often

have conflicting performance objectives (Gladish, Parra, Terol & Uria, 2005,

Hamilton & Breslawski, 1994, Kim et al., 2000, Sier, Tobin & McGurk, 1997). In

addition, patient satisfaction is an important consideration, which again may differ

from the perspective of the surgeon and administrator.

The goals of OT scheduling as identified by Breslawski and Hamilton (1991) are

- effective use of OT suite;

- satisfy surgeons;

43

- satisfy patients;

- satisfy OT staff;

- simplicity in generating schedule;

- effective use of post anaesthesia care unit (PACU); and

- low incidence of cancellations.

Positive relationships exist between some of these criteria, such as a low incidence of

cancellations and patient satisfaction. However, others may be conflicting and

counter-intuitive. For example, hospital administrators focus on achieving high

utilisation of resources such as bed utilisation, which has been shown to lead to

longer patient waiting times (Sier, 2004), while patients desire prompt and effective

service and to experience little waiting time. In support of this, Vissers et al. (2007)

demonstrated that high occupancy levels produce longer wait times, higher

cancellation percentages and frequent excess of target capacity levels. Balancing

conflicting criteria between decision makers was the focus of several studies

(Gladish et al., 2005, Kim et al., 2000, Sier et al., 1997). Sier et al. (1997) treated

surgical scheduling as a non-linear mixed-integer programming problem with

weighted constraints. Heuristic techniques were used to generate solutions that

balanced conflicting criteria; however a feasible solution could not always be

generated. Gladish et al. (2005) applied a possibilistic linear multi-objective

programming technique to annual surgical planning. The effect that different

performance measures would have on surgical waiting lists and the interactions of

conflicting criteria were demonstrated. Cardoen et al. (2009a) proposed a ‘room for

improvement’ (RFI) calculation for handling multiple criteria objective functions.

Finding a trade-off between objectives with different units, or even for objectives in

the same unit, is subjective and leaves the result open to argument. RFI is a unit-less

calculation based on the values of the best and worst schedules. Relevance weights

assigned by the planner may also be incorporated into this calculation.

2.3 Simulation

Simulation is defined as the act of imitation (Imagine That, 2002) and is a practical

tool used for finding practical solutions to complicated problems (Brailsford, 1995).

Mathematical models are built to act like a system of interest to mimic certain

44

important aspects (Dexter et al., 1999). By changing variables, the effects may be

observed and predictions can be made about the behaviour of the system without

having to implement such changes in the practical setting (Kim et al., 2000, Kozan &

Gillingham, 1997, McHardy et al., 2004). This is important for healthcare systems,

where patient’s lives may be at stake and therefore testing new techniques without

prior predictions on performance outcomes could be devastating (Kim et al., 2000).

Before modelling any system, it is fundamental that the modeller understands the

system including constraints, causes of variability, accuracy of data and the

objectives to be met (Brailsford, 1995). The other crucial aspect is the relationship

between the modeller and the decision makers. If this relationship is poor, then the

results of the simulation model, regardless of the modeller’s expertise is likely to be

unusable (Brailsford, 1995).

Simulation has been used for many applications in healthcare including the ICU, OT,

maternity suite, ED and admissions planning. For the ICU and OT of a major public

hospital, McHardy et al. (2004) used simulation to investigate the effect of changes

in parameters on efficiency measures. The model was used as a decision support

tool. Further simulation studies on the ICU include the evaluation of bed reservation

schemes and their associated effects of performance objectives (Kim et al., 2000) and

analysis of surgical ICU bed requirements (Troy & Rosenberg, 2009). Simulation

has been used to address many aspects of the OT including scheduling policies

(Dexter et al., 2000, Ferrin, Miller, Wininger & Neuendorf, 2004) operating room

allocation (Wullink et al., 2007), capacity of the surgical suite (Ballard & Kuhl,

2006) and resource availability (VanBerkel & Blake, 2007).

For the emergency room, Badri and Hollingsworth (1993) evaluated changes such as

scheduling practices, resource numbers and patient demand patterns. Kozan and

Gillingham (1997) tested the performance of a maternity system and found the

number of beds and patient arrival rates affect patient wait. Sier (2004)

demonstrated the use of simulation for calculating performance measures of a

hospital unit or department. Vissers et al. (2007) investigated alternative admission

policy choices for admission planners when choosing a policy to optimise specific

performance measures. Very high occupancy levels produced longer waiting times,

45

higher cancellation percentages and higher incidences of excess capacity and

reserving capacity for emergency cases reduced the incidence of cancellations.

In the past, simulation has faced much scrutiny in the academic field. One of the

issues behind this opposition is the concern that commercial simulation softwares

may not be flexible enough to handle non-standard situations when compared with

simulations written from scratch (Brailsford, 1995). Since the report by Brailsford

(1995) was written however, simulation software has advanced. For example, the

simulation software used for this thesis, EXTEND, addresses this issue by providing

the modeller the opportunity to use both standard simulation ‘blocks’ and ones that

can be built from scratch. In this way the model can be built to better fit the system

being studied. Recent approaches in simulation have also addressed the issue of

acceptance by the academic field, by moving toward simulation based optimisation

(Cardoen et al., 2010).

2.4 Scheduling and sequencing

Scheduling in healthcare is applied to many areas including resource planning (both

material and human) and also patient scheduling within hospital departments such as

the OT, ICU, ED and other wards. For example, Vermeulen et al. (2009) developed

a dynamic resource allocation model for computer tomography scans (CT-scans) at a

radiology department. Resources are typically assigned to several patient groups and

the allocation process must be flexible due to fluctuations in demand.

Chern, Chien and Chen (2008) propose a heuristic for solving the binary integer

programming model for health examination assignment. The model assigns health

procedures to doctors to minimise both waiting time of patients and doctors. In this

paper, both sequence of examination procedures and the availability of resources are

taken into consideration. Belien and Demeulemeester (2006) developed an exact

branch and price algorithm to solve the trainee scheduling problem. Carter and

Lapierre (2001) used integer programming to generate emergency room physician

schedules that incorporate scheduling rules and individual preferences.

46

Elective surgical planning has been described in the literature as a two (Guinet &

Chaabane, 2003, Jebali, Hadj-Alouane & Ladet, 2006, Jebali et al., 2003), three

(Belien & Demeulemeester, 2007b, Cardoen et al., 2009a, Santibanez, Begen &

Atkins, 2007), or more recently four-stage process (Cardoen et al., 2009b). The four-

stage process is defined as;

1. Strategic level case mix planning. The process of defining surgical blocks and

allocation of time to surgical specialties.

2. Development of a master surgery schedule. This involves assigning the surgical

teams and/or specialties to the surgical blocks defined in the first step.

3. Scheduling and sequencing of individual cases.

4. Monitoring of schedule in online environment.

Stage one is generally a management decision involving budget allocation. Dexter et

al. (1999) used simulation to investigate the allocation of surgical block time to

surgeons that maximised operating room use. They showed block time allocation

could be improved by moving control of the surgical date decision to the operating

room suite rather than leaving it to the surgeon and patient. Matching up operating

room caseloads with the days on which full-time operating room personnel are

scheduled to work and assigning caseloads based on operating room capacity were

the key elements for effective allocation of surgical block time.

Santibanez et al. (2007) addresses the second stage of surgical planning by proposing

a mixed IP model for assigning surgical specialties to operating room time whilst

considering operating room availability and post surgical resource constraints across

a network of hospitals. The 12 hospitals were considered as components of a single

system. Belien and Demeulemeester (2007b) propose and evaluate models for the

construction of the master surgery schedule with levelled resulting bed occupancy.

The problem is constrained by the number of blocks required by surgeons (or

surgical teams) and the total available blocks. Surgical block requirements depend

on the type of surgery, which in turn determines the length of stay of each operated

patient and the number of patients that may be treated per block. Mixed IP based

heuristics and a meta-heuristic were developed to minimise the expected number of

bed shortages.

47

Dexter and Traub (2000) investigated stage three of the surgical planning process,

which involves sequencing of surgical cases where limited equipment in different

operating rooms are required on the same day. Decision theory is applied to estimate

the probability that one case will have a longer duration than another when

sequencing two cases. The desired outcome was to decrease the impact of

equipment and personnel constraints on operating room scheduling. The derived

methods are useful for decision makers wishing to set a ‘threshold’ on the risk of an

overlap between surgical cases. Pham and Klinkert (2008) treat surgical case

scheduling as a generalised job shop scheduling problem. The multi-model blocking

job-shop is formulated as a mixed integer linear programming problem. They argue

a holistic approach should be applied to surgical case scheduling connecting surgical

stages and coordinating multiple resources during any surgical step. Galvin (1997)

treated the surgical scheduling problem as a cutting stock problem. Resource

utilisation is improved by reducing the expected slack time in theatres. Persson and

Persson (2005) use optimisation modelling to synchronise allocation of different

resources for operating room planning. Sier et al. (1997) developed a tool for

scheduling operations that considered bed availability, efficient theatre utilisation,

minimising schedule deviations and emergency arrivals. Cardoen et al. (2009b)

propose an exact branch and price approach for sequencing surgical cases in a day-

care environment. The peak use of recovery beds, the occurrence of recovery

overtime and the violation of various patient and surgeon preferences are minimized.

Guinet and Chaabane (2003) consider stages two and three by assigning patients to

theatres over a medium term and addressing the daily rescheduling problem to

coordinate human and material resources. This decision tool optimised OT overload

and patient waiting time. Jebali et al. (2006) describes an assignment model that

optimises room usage and minimises patient wait. Two strategies are used for

patient sequencing, one that re-considers patient assignment and one that does not.

Incorporating time spent in the recovery room is an innovative step of this paper that

adapts a two-stage hybrid flow shop to the problem where the first stage represents

the operating room and the second is the recovery room. Roland, Di Martinelly,

Riane & Pochet (2009) solve the surgical scheduling problem as a single model

rather than using a two-stage approach. Surgeries are assigned to both human and

material resources to minimise the number of operating rooms used and their

48

associated overtime. The two types of human constraints considered are renewable,

such as nurses and specific, i.e. surgeons. A genetic algorithm is used to solve the

computationally complex problem.

Constructive heuristics usually applied to machine scheduling are also commonly

applied to patient sequencing. Breslawski and Hamilton (1991) give a

comprehensive examination of such techniques in the OT setting such as longest

processing time first (LPT) and shortest processing time first (SPT). LPT is

commonly used to balance the load on the theatres, minimise overtime and ensure

the entire workload is completed (Breslawski & Hamilton, 1991). Harper (2002)

showed that LPT increased the throughput of patients without the need for extra

resources and reduced the chance of closing theatres early. LPT however results in

unreliable start time estimates for the later procedures and may result in surgeon

dissatisfaction as certain surgeries consistently receive the ‘prime’ early morning

spots (Breslawski & Hamilton, 1991). SPT is used in practice to maintain the load

on the PACU (Hamilton & Breslawski, 1994). This technique, however, often

results in an increased incidence of overtime as the longer and more variable

surgeries are performed last (Breslawski & Hamilton, 1991).

Other common scheduling systems for the OT include FCFS, block scheduling and

dynamic block scheduling. FCFS is the simplest form of sequencing, however it

commonly results in a high cancellation rate due to overbooking. Block scheduling

allocates surgeons to specific surgical blocks. Benefits of this system include

reduced surgeon competition, reduced administrative work and foreknowledge of

surgical start times. One problem that may occur is surgeons keeping their block

even when they have no cases to schedule. This problem is reduced by dynamic

block scheduling in which an individual surgeon’s use of his/her block is regularly

reviewed (Breslawski & Hamilton, 1991).

Variations of the traditional two or three stage OT planning also exist. Belien and

Demeulemeester (2007a) use a novel branch and price approach for integrating nurse

and surgery scheduling. Dexter et al. (2000) used simulation to explore ways of

assigning patients that cannot be completed during regular surgical block times to

‘overflow block time’ whilst balancing staffing costs and flexibility in choosing

49

surgical dates and times. They showed that staffing costs are minimised when

surgical date and time preferences are not considered however demonstrated a

strategy that allows for some flexibility whilst only slightly raising the costs above

the minimum possible.

Various studies in the literature consider resource availability in the scheduling

phase. Roland et al. (2009) schedule the operating theatre under human resource

constraints. The problem is formulated as a mixed integer program and solved with a

proposed genetic algorithm. Augusto, Xie and Perdomo (2010) investigate

scheduling of the operating room with patient recovery in both operating rooms and

recovery beds. The problem is modelled as a 4-stage hybrid flowshop that takes into

account the availability of operating rooms, porters and recovery beds. The problem

is investigated for an open scheduling policy rather than a block scheduling policy.

Fei, Meskens and Chu (2010) solve the operating theatre planning and scheduling

problem in two phases. The first stage solves the planning problem of assigning

surgeries to operating rooms using a set-partitioning integer-programming model.

The second stage determines the sequence of operations taking into account the

availability of recovery beds using a two-stage hybrid flow shop approach.

A recent study has recently investigated the problem of elective surgery scheduling

from a waiting list (Min & Yih, 2010). The work extends on that by Gerchak, Gupta

and Henig (1996) by including patient priority. Lamiri, Grimaud and Xie (2009)

present a stochastic integer programming model for surgery planning when OR

capacity is shared among elective and emergency surgery.

One of the benefits of any systematic approach to surgery scheduling is a potential

increase in transparency and fairness in allocating time to surgeons (Santibanez et al.,

2007). Studies have also shown that poor scheduling of elective patients reduces

access to care and impairs the overall responsiveness to emergency patients

(McManus et al., 2003). This type of variability linked with operation scheduling

due to poor scheduling policies is referred to in the literature as ‘artificial variability’

(Belien & Demeulemeester, 2007a, McManus et al., 2003).

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2.5 Alternative approaches

Distributed multi-agent systems provide an alternative technique to OR, for

scheduling processes in the ED, or other multiple treatment pathways. Contrary to

OR techniques, multi-agent systems view a hospital as a decentralised system, where

each department essentially has authority over its own schedule, as is the case in a

hospital setting. Agents act on behalf of resources or patients to derive effective

schedules by exchanging schedule positions based on interaction rules (Paulussen,

Jennings, Decker & Heinzl, 2003, Paulussen et al., 2004, Vermeulen et al., 2006).

Paulussen et al. (2003) and Paulussen et al. (2004) used a market mechanism to

allocate patients to the resources. A market mechanism aims to efficiently distribute

the scarce resources, based on what the patient agents are willing to ‘pay’ for the

resources. The decision for distributing the scarce resources was driven by cost

functions based on health state. Vermeulen et al. (2006) argued against the use of

market mechanisms for agent negotiations stating the difficulty in generating a

general solution, questionability of the scalability of the system and controversy

between doctors and patients when quantifying the relative patient well-being. A

Pareto appointment exchanging technique was used in which exchanges between

agents only occurs if no patient is worse off than before. The approach by

Vermeulen et al. (2006) performed ‘nearly as well’ when tested against centralised

heuristics (an OR technique).

2.6 Robust scheduling

Surgical durations are variable and depend on the type and severity of a patient’s

illness and the performing surgeon (Guinet & Chaabane, 2003). Oversimplification

of service distribution estimates, such as average occupancy, do not capture the

variability in patient arrivals, which may lead to an underestimation of resource

requirements and overestimation of resource availability (McManus, Long, Cooper

& Litvak, 2004, Sier, 2004).

Various approaches to estimating service distributions have been used. Guinet and

Chaabane (2003) considered surgical duration estimates in one-hour blocks. Jebali et

51

al. (2006) used a log-normal distribution to generate random surgical durations. Sier

(2004) used stochastic modelling to capture the variability of patient arrivals and

length of stays. Harper (2002) discusses the problem of segregating patients into

statistically and clinically meaningful groups using classification and regression tree

(CART) analysis. Although many different techniques have been applied to service

distribution estimates, Kim et al. (2000) argued that even if arrival distribution

assumptions are very different, the same fundamental conclusions and policies are

unchanged despite these differences.

Denton et al. (2007) developed a stochastic optimisation model for generating

operating room schedules that hedge against surgical duration uncertainty. It was

suggested that scheduling longer and more complex cases before shorter surgeries

may have a significant negative impact on operating room performance measures.

Daniels and Kouvelis (1995) explains some of the reasons why experienced

schedulers are moving away from deterministic and stochastic optimisation

scheduling models towards robust scheduling models. Deterministic and stochastic

optimisation scheduling models can easily produce poor actual performance results

due to the inability to predict processing times for jobs, which are highly uncertain.

It is impossible to accurately predict these times even with the use of stochastic

modelling. Robust scheduling addresses this issue by the inclusion of ‘buffers’ that

absorb the variations in treatment times that occur during project execution.

Schedule robustness is defined by Daniels and Kouvelis (1995) as the determination

of a schedule whose performance is relatively insensitive to the potential realisations

of the task parameters. Essentially, robust schedules are makespan (due date)

protective or quality robust (Van de Vonder, Demeulemeester, Herroelen & Leus,

2005).

Kuroda, Shin and Zinnohara (2002) applied robust scheduling techniques to an

advanced planning and scheduling environment. To minimise the need to fix

ongoing schedules, due date buffers were applied to the customer order problem.

The size of the due date buffers was determined by a monotone decreasing time

function. A simulation based scheduling algorithm was developed and examined.

52

The model assumes that orders are flexible and become fixed as processing

progresses.

Hans, Wullink, Van Houdenhoven and Kazemier (2008) introduce assigning

surgeries and planned slack time to the operating room days to prevent overtime.

Surgical duration is based on historical data from an academic hospital in the

Netherlands. The planned slack time on each operating day is based on the expected

variance of the surgical durations planned for that day. The problem is formulated to

minimise expected overtime and planned slack time, thereby theoretically freeing up

operating room capacity. Planned slack time is minimised by exploiting the portfolio

effect of portfolio theory, which clusters surgeries with similar variability on the

same operating room day. This typically resulted in patients receiving the same type

of treatment being scheduled on the same day, which may have an additional benefit

of reducing surgeon operating time due to the repetitive nature of the operations.

Robust scheduling is an example of a preventive scheduling approach that serves as a

baseline schedule for online production scheduling. Effective preventive schedules

are important since they form the basis for resource commitment decisions. When

used in conjunction with reactive scheduling models, they improve the performance

of online scheduling. Other preventive scheduling approaches include stochastic

based approaches, fuzzy programming, sensitivity analysis and parametric

programming (Li & Ierapetritou, 2008).

2.7 Reactive scheduling

Proactive schedules are constructed through anticipating disruptions that occur in the

online scheduling environment. However, the actual duration of an activity cannot

be known for certain until its completion. As new information becomes available,

disruptions may cause deviations from the predictive schedule and even make it

infeasible. Methods that address solution robustness are therefore required. Solution

robustness is addressed by reactive scheduling or disruption management, which is

used to repair the baseline schedule following activity disruptions, by including

changes whilst minimising disruptions from the original schedule (Van de Vonder et

al., 2005, Yu & Qi, 2004).

53

Various disruption management policies exist including post disruption management

and predictive disruption management. Post disruption management deals with the

unpredictable disruptions and therefore the schedule may only be updated after the

event. Only the remaining patients need be considered in this case. Predictive

management is used for disruptions that may be anticipated such as an emergency

arrival that requires treatment at some time in the future. At the time when the new

schedule starts, for either type of disruption management policy, time is re-set to zero

and the expected completion and start times of the remaining patients are updated.

The remaining n patients must also be re-indexed from 1 to n.

In addition to choosing a disruption management policy, the solution approach must

also be decided upon. The first approach minimises the deviation from the original

schedule. This technique finds a new feasible solution that is as close as possible to

the original schedule. The second approach does a full reschedule of the tasks,

subject to a new objective function that captures the disruption. This approach does

not consider the deviation from the old schedule (Li & Ierapetritou, 2008,

Sabuncuoglu & Bayiz, 2000). .

These two goals may be combined to find solutions that address both objectives.

One technique used for solving multi-objective reactive models is goal programming.

Goal programming handles multiple criteria decision problems by determining an

expected value for each criterion and minimising the gap between each criterion and

its expected value (Yu & Qi, 2004).

Li and Ierapetritou (2008) give a review of the main methodologies for handling

uncertainty in production scheduling and Sabuncuoglu and Bayiz (2000) studied

reactive scheduling problems in a stochastic manufacturing environment. Reactive

scheduling is described as a process to modify the created schedule during the

manufacturing process to adapt to changes in the production environment (Li &

Ierapetritou, 2008). Event based scheduling is described as rescheduling triggered by

an unexpected event (Sabuncuoglu & Bayiz, 2000). Disruptive events include rush

order arrivals (emergency patients), order cancellations (surgery cancellations) or

machine breakdowns (equipment failure or resource unavailability) (Li &

Ierapetritou, 2008).

54

A number of recent studies address online scheduling for the machine scheduling

environment. Tan and Yu (2008) and Hurink and Paulus (2008) studied online

scheduling problems for the two parallel machine environment. Competitive ratio

was used to measure the performance of the online algorithms that were used to

minimise the maximum machine makespan. Qi, Bard and Yu (2006) addressed

reactive scheduling in the machine environment following random or anticipated

disruptions. Both the original objective function and a measure of deviation from the

original schedule were considered.

Predictive-reactive scheduling is considered by Yang and Geunes (2008) for the firm

that must compete with other firms for future jobs. In this scenario the scheduler

must plan idle time in the predictive schedule for uncertain jobs and their position in

the schedule. When the planned idle time does not match up with the actual

requirements, disruptions occur. The costs of expected tardiness, schedule

disruptions and wasted idle time were minimised. Four approaches to handling

disruptions were identified; completely reactive approaches, predictive-reactive

scheduling, robust scheduling approaches and knowledge based scheduling.

There is very little literature for the reactive scheduling problem for the operating

theatre. Recently, Nouaouri, Nicolas & Jolly (2010) developed an approach for

inserting unexpected patients into the operating theatre schedule in the case of

disasters such as terrorist attacks. The problem was modelled as a three-stage integer

linear programming model. Bard and Purnomo (2005) present a new methodology

for reactively scheduling nurses considering shift-by-shift imbalances in their supply

and demand. The problem was formulated as an integer program (IP) and solved

within a rolling horizon. When commercial techniques such as CPLEX could not

solve the model, a branch and price algorithm was developed. Two heuristics were

developed, namely Tabu search and a set covering IP heuristic, which was the more

effective of the two. The problem analysed begins with a midterm schedule and

addresses the issue of daily adjustments.

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2.8 Single machine scheduling

Single machine scheduling is the problem of assigning and sequencing a group of

jobs to a single machine so that one or multiple objectives may be optimised.

Various objectives may be used from due date related objectives to flowtime. The

problem may be constrained in many ways including (but not limited to) precedence

constraints imposed upon the job orders, ready times or release times of the jobs and

job due dates. Processing times of the jobs may also vary from unit processing times

to stochastic.

Vast literature exists for the study and classification of scheduling problems

(Blazewicz et al., 1996, Brucker, 2004, Pinedo, 1995). Three fields are used for the

notation; α | β | γ. The first field describes the machine environment. For the single

machine environment a “1” is used. The second field describes task and resource

characteristics. For example, pre-emption is indicated by pmtn but no pre-emption

may be indicated by either nopmtn or leaving the field empty. In this dissertation,

the latter is used. Other parameters for this field include resources, res, precedence

constraints, prec, and ready times r j. The third field indicates the objective function

(Blazewicz et al., 1996).

A multitude of literature exists on single and multi machine scheduling. Stankovic

Spuri, Di Natale & Buttazzo (1995) give a summary of classical scheduling results

for single and multiple machine scheduling with their applications and algorithms for

the ‘real world’. Static scheduling is described as the scheduling problem for which

complete knowledge on the task set and constraints is known. On the other hand, a

dynamic schedule changes over time and new arrivals may arrive in the future that

are not known at the time of scheduling.

Considerable literature exists on single machine scheduling with due dates and their

complexity. The problem for minimising the number of unweighted tardy jobs, is

solved in polynomial time using Hodgson’s algorithm (Blazewicz et al., 1996).

Extending this problem by weighting the jobs, is NP hard even with equal due dates

(Blazewicz et al., 1996, M'Hallah & Bulfin, 2007). Several authors demonstrated

that the problem with equal due dates is equivalent to a knapsack problem and may

56

be solved with the weighted shortest processing time (WSPT) rule (Blazewicz et al.,

1996, Pinedo, 1995). This rule sequences jobs in increasing order of ratio of

processing time to weight, however, in arbitrary cases it can produce poor results

(Pinedo, 1995).

Jang and Klein (2002) investigate single machine scheduling for minimising tardy

jobs and evaluate the effect of variance of processing times when they are drawn

from a normal distribution. A dynamic scheduling policy is developed for jobs with

a common due date. The heuristic is based on a myopically optimal solution (i.e. a

greedy algorithm). Seo, Klein and Jang (2005) consider the same problem but

transform the original stochastic problem into a deterministic non-linear integer

programming model with relaxation. Soroush (2007) studied the problem that

minimises the sum of weighted early and tardy jobs where jobs have random

processing times and due dates are known but not necessarily equal. The general

problem is NP hard and it is nearly impossible to generate an optimal sequence in

polynomial time (Seo et al., 2005). Under certain conditions however, as developed

in the paper by Soroush (2007) an exact solution is possible. In fact, Pinedo (1995)

showed the problem with random variables is tractable only with exponentially

distributed processing times.

The problem of position dependent processing times is also discussed in the

literature. Bachman and Janiak (2004) investigated the problems when job

processing time is characterised by an increasing, or decreasing, function dependent

on the position of the job in the sequence. The makespan minimisation problem was

shown to be strongly NP hard, assuming jobs are ready at their processing times.

Gordon and Strusevich (2009) also examined positional dependent processing times

for single machine scheduling with due dates. They developed polynomial time

dynamic programming algorithms for the problems.

There is also considerable literature for bi-criterion single machine scheduling and

there are generally three approaches to such problems (Huo, Leung & Zhao, 2007).

The first technique minimises a secondary criterion, subject to the constraint that the

primary criterion is minimised (Huo et al., 2007). Examples of this approach are

given in (Huo et al., 2007, Jolai et al., 2007, Wan & Yen, 2009) and will also be

57

adopted in this dissertation. Jolai et al. (2007) develop a genetic algorithm for the

single machine scheduling problem that minimises the maximum earliness and

number of tardy jobs. Wan and Yen (2009) minimise total weighted earliness

subject to the minimal number of tardy jobs. Solution approaches include a heuristic

and a branch and bound algorithm. Huo et al. (2007) minimise the number of tardy

jobs and maximum weighted tardiness. Chen and Sheen (2007) use the second

approach for bi-criterion models proposing a pareto-optimal solution algorithm with

the objective of minimising the sum of weighted earliness and tardiness and the

number of tardy jobs. The third approach is to use a linear combination of both

criteria in the cost function (Huo et al., 2007).

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Chapter 3. Simulation model

An advanced model for the OT department of the PAH was developed using Extend

(Imagine That, 2002) simulation software. The model can be used as a decision

support tool for hospital planners. The development of the model is presented in

Section 3.1. This includes the model assumptions, a brief discussion of the

sensitivity analysis that will be performed on the system, the inputs for the

simulation model and the validation process. Section 3.2 details the results of the

simulation model including the effects of changing patient arrival rates and elective

scheduling disciplines on the system’s performance measures. The findings of the

simulation model and identified areas of future work are summarised in the

concluding section of this chapter.

3.1 The model

3.1.1 Developing the model

The simulation software called Extend was chosen for this project due to its technical

ability to represent highly complex systems and analyse model outputs and also as a

visual tool for explaining model outputs to decision makers. It has the capacity to

immediately observe the effects that changes to parameters and inputs have on the

system by the use of plotters and outputs. Model observations form decision support

by predicting the effects changes in the model’s parameters may have on a system.

A model in Extend is built using an array of blocks, some of which generate items

and others, which process these items. Text files may be used for importing and

exporting large amounts of data into and out of a model. Information regarding

treatment items is collected, analysed and reported at the end of each simulation run

(Imagine That, 2002).

Modelling of the OT department was achieved through the use of blocks. Each

theatre is represented as an activity block (server) in parallel capable of serving one

patient at a time. Each theatre has an individual list of patients, determined by the

59

type of surgery to be performed. For example, the cardiac theatres only receive

cardiac patients. Figure 3 is a screen print of the simulation model of a single

operating theatre but does not demonstrate the hidden detail within the simulation

model. The animated blocks are called ‘hierarchical’ blocks, which are used as

‘containers’ for holding other blocks. These are useful for complex models with

thousands of blocks for hiding the detail by grouping areas of the model, making the

model aesthetically easier to view.

Figure 3. Elective patient flow

Figure 3 shows a generalised overview of the logic of elective patient flow through

the model; the patient arrives and is joined by a surgical team, a decision is made as

to whether there is enough time to operate on the patient, if not the patient exits, if

there is, then the patient proceeds to theatre. Figure 4, similarly shows the general

pathway of emergency patients through the model. There are two emergency

theatres for general emergency patients and patients go to the first available. Cardiac

and ophthalmology patients however are re-routed to the appropriate theatres.

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Figure 4. Emergency patient flow

Within the hierarchical block (the block containing other blocks) that shows patients

arriving, patients (items) are generated using a program block that specifies the time

of arrival and type of surgery to be performed by use of an identification number and

‘end of day’ time as illustrated in Figure 5. ‘End of day’ time was used for

cancellation decisions to prevent an elective patient from waiting in the system

longer than one day.

Upon arrival, patients are segregated by identification number and allocated a

random surgical duration. Following this, the patient must wait for a surgical team

consisting of 2 nurses and an anaesthetist. Assignment of the resources is on a FCFS

basis with priority, where emergency patients have priority over elective patients.

Figure 6 illustrates the arrival of an elective patient and its path as it queues for a

surgical team.

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Figure 5. Program block

Figure 6. Elective patient arrival

Following selection of a surgical team, the patient waits in queue for a consultant

(surgeon). Only one consultant is available in an elective theatre at any one time.

Once the consultant finishes treating a patient, either they continue their shift and

begins treating the next patient, or they complete their shift and the next consultant

takes over. In some cases, when there are no available emergency consultants,

elective consultants are required to treat emergency patients. In the model, it is

assumed that an emergency patient will have precedence over an elective case and

that emergency consultants are not able to treat elective patients. Figure 7 illustrates

the re-direction of an elective consultant to the emergency theatres.

62

Figure 7. Re-direction of elective consultant to emergency patient

If the queue for a consultant results in the patient waiting in the operating theatre

suite after the operational hours, then the patient is cancelled, which is illustrated in

Figure 8. This was especially important for sensitivity analysis, for demonstrating

whether alternate sequencing disciplines or to what degree of increasing arrivals

would result in increased cancellations. Once a surgery has begun, however, it could

not be pre-empted by the arrival of a more urgent patient or cancelled due to the

theatre running into overtime. Emergency cases are not cancelled under any

circumstance and theoretically will wait indefinitely for treatment.

Figure 8. Elective patient cancellation decision

Following completion of surgery, the surgical team is released. The patient then

exits the system and a decision is made as to whether the surgeon has completed their

shift. If not, they are sent back to their rostered theatre. If the shift has finished, the

surgeon exits the system. Nurses and anaesthetists are treated as recyclable

resources.

63

3.1.2 Model assumptions

A number of assumptions were made for the model.

1. Historical data collected and analysed was representative of a typical period.

2. The number of staff comprising a surgical team remained unchanged throughout

the simulation.

3. Theatre operating hours were fixed at 0800 hours to 1200 hours and 1300 hours

to 1700 hours for all elective theatres with the exception of ophthalmology

theatres A2 and cardiac theatre E2. Theatres A2 and E2 remained open

throughout the simulation as these were required by emergency patients after

hours.

4. Length of simulation was sufficient to dilute the effects of the warm up period.

5. Emergency operations are restricted to the emergency theatres with the exception

of ophthalmology and cardiac patients

6. Elective patient waiting time is limited by an upper bound.

7. Emergency patient wait time is unlimited.

3.1.3 Changing patient arrivals

Performance comparisons were made when the number of patients arriving to the

system was (i) decreased 10%; (ii) increased 10%; (iii) increased 20%; and (iv)

increased 30%.

3.1.4 Scheduling disciplines

Theoretically, different scheduling disciplines may be employed to optimise OT

objectives. Three alternative scheduling disciplines are implemented and compared,

namely longest processing time (LPT) first, shortest processing time (SPT) first and

least flexible job (LFJ) first. LPT and SPT are common scheduling disciplines used

in the OT environment. For a discussion on these, and how they have been

previously tested in the literature, refer to Section 2.4 on scheduling and sequencing.

The LFJ rule is optimal for minimising the maximum completion time in the parallel

machine environment without pre-emption (Pinedo, 1995) and has not yet, to the

64

author’s knowledge, been applied to the OT environment. The patient with the

smallest set of possible theatres to choose from is selected when a machine becomes

available, i.e. the least flexible job. The concept behind this technique is scheduling

the more flexible patients last to provide greater flexibility toward the end of the

schedule for evening out the load over the theatres. For the application of this

heuristic, theatre blocks are assumed to be assigned to the surgical specialties, not

specific consultants. The list of specialties that may be performed in each theatre is

identified and used for the schedule derivation.

3.1.5 Data input

Based on historical data taken at the PAH during 2005, an emergency patient arrival

distribution and lengths of surgery distributions for all surgical categories were

determined with the statistical distribution fitting software Stat::Fit. Table 1 is a list

of surgical categories used to calculate surgical duration distributions. The results of

the distribution fitting for surgical duration are presented in Table 2. Appendix B

shows graphs of the selected distributions against the actual data.

Actual elective schedules used during a two month period during 2005 were used as

a basis input for comparisons with the sensitivity tests. This same list of patients was

also used to generate schedules based on the LPT, SPT and LFJ scheduling

disciplines. For the LPT and SPT tests, the list of patients treated in each of the

theatres were ordered according to longest and shortest processing time respectively.

Determination of LFJ schedules was more involved as this allowed patients to be re-

assigned to another theatre. Possible theatre sets were identified for surgical

specialties and those with the smallest set were assigned first. Assignment aimed to

even out the load over the elective theatres. For the change in patient arrival

sensitivity tests, the number of elective arrivals was altered according to the test

parameter based on the original schedule. The proportion of patients from each

specialty was maintained regardless of the number of arrivals. Creating the elective

schedules was a particularly time consuming stage, as these patient lists were

manually generated and then input from excel spreadsheets.

65

Table 1. List of surgical categories

Surgical Specialty Abbreviation

Breast, endocrine & thoracic

Cardiac Surgical Unit

Colorectal

Ear, nose & throat

Facial/Maxillary

Hepatobiliary

Neurosurgery

Ophthalmology

Orthopaedics

Plastics

Renal Transplant

Upper gastro-intestinal

Urology

Vascular

BE&T

CARD

COLO

ENT

FMAX

HPB

NEURO

OPHT

ORTHO

PLAS

RTPT

UGI

UROL

VASC

Table 2. Surgical duration distribution fitting results

SPECIALTY DISTRIBUTIONBE&T Beta (35., 1320, 2.86, 43.8)CARD Weibull (34., 2.69, 240)COLO Pearson 6 (23., 2180, 1.53, 24.4)ENT Pearson 6 (26., 56.3, 1.97, 1.71)FMAX Loglogistic (19., 1.99, 34)HPB Lognormal (56., 4.53, 0.987)NSUR Pearson 6 (40., 614., 2.76, 12.5)OPHT Loglogistic (12., 2.69, 46.1)ORTH Lognormal (12, 4.42, 0.791)PLAS Loglogistic (9., 1.8, 49.6)RTPT Lognormal (18., 4.09, 0.888)UGI Pearson 6 (27., 1.68, 68.2)UROL Loglogistic (7., 2.02, 56.5)

Analysis of historical data showed emergency patients arrived according to an

exponential distribution with an average inter-arrival time of 113 minutes. This was

used as a standard measure for generating emergency patients for the basis model

and each of the scheduling discipline models. For the change in patient arrival

sensitivity tests, emergency patient inter-arrival times were altered accordingly.

66

Upon generation, both elective and emergency patients were allocated a random

length of surgery (LOS) based on their surgical category (refer Table 2). These were

selected from a distribution according to the data analysis performed with Statfit (see

Table 2).

Extend records an extensive set of statistics obtained from each simulation run. This

data can be saved in a text file or exported to Microsoft Excel spreadsheets for

manipulation and analysis. The key performance measures and outputs of the model

were patient waiting times, number of elective surgery cancellations and theatre

utilisation rates. An important outcome of the model was to demonstrate how

improving one of those performance measures might have a detrimental effect on

others.

3.1.6 Simulation model validation process

To assist with model validation, patient flow was repeatedly analysed and compared

with the actual flow of patients depicted in the patient flow diagram in Figure 2 (see

page 45). The actual data collected in 2005 was initially entered into the simulation

model. Taking out data variability enabled the simulation process to be compared

with the observed period. This meant the actual arrival time and length of surgery

for each patient (both electives and emergencies) was explicitly entered. Patient

priority and the possibility for surgeon re-assignment remained variable, as this data

was not available for validation. The model was run and compared with the actual

results and the following observations were made:

1. Validation was based on elective patient wait time within the surgical suite for

the following reasons;

- Information regarding the number of patients cancelled, and the original schedule

compared with what actually resulted was not available.

- The average waiting time for emergency patients was recorded as less than 1

minute in the supplied data. Any waiting times experienced in the ED or other

wards prior to surgery were not recorded in the data.

67

2. All emergency patients were assumed to have priority over electives.

Realistically, some ‘emergent’ but not necessarily urgent patients have lower or

the same priority as electives and are triaged accordingly. As a result, in the

simulation these patients may receive treatment before an elective patient,

thereby reducing the waiting times of emergency patients and increasing that of

electives.

3. The observed order in which patients arrived to theatre for the real life data was

not necessarily the order in which they were served. The ability to change order

was omitted from the model because the data supplied did not contain any

indications of` patient re-scheduling or the initial schedule.

4. Assigning emergency patients to elective theatres was not initially considered in

the simulation model to simplify the patient modelling process. During the

validation stage however, it became apparent that the simulation was producing

excessive emergency waiting times. As a result emergency cardiac patients and

ophthalmology patients were serviced in the elective cardiac and ophthalmology

theatres respectively, as is often the case in the real life for resource purposes.

All other emergency patients were serviced in the emergency theatres. This had

the intended effect of reducing the number of elective surgeons being re-assigned

to emergency patients, had little effect on average elective patient wait time and

reduced the average waiting time of emergency patients. It did, however,

produce a slight increase in the incidence of elective cancellations, as would be

reasonably expected.

5. For simplification, the two elective theatres used for emergency cases remained

open throughout the simulation without any scheduled downtimes. This resulted

in a lowered utilisation rate for these theatres.

6. The data showed that most theatres were in fact opened earlier than the scheduled

8:30am starting time. This was incorporated into the model resulting in fewer

cancellations and a better representation of the real system.

68

Initially, some patients experienced excessive waiting times in the suites (remaining

in the suites overnight), when in reality they would have been re-scheduled or

cancelled. To overcome this, elective patients would renege if their wait time

reached 495 minutes ensuring no patient remained in a suite ‘overnight’. Emergency

patients had no limit placed on their waiting time as it was assumed that they could

not be cancelled.

3.2 Simulation results and sensitivity analysis

By application of the central limit theorem, confidence intervals for the performance

measures were calculated. The number of simulation runs required for a tolerance

level (at the 95% confidence level) was determined. The population standard

deviation was approximated by the sample standard deviation.

The simulation run length was set to two months (87840 minutes) with the

assumption that the length of the simulation run was sufficient to ‘dilute’ the effects

of the warm up period. Performance measures for patient waiting times, number of

cancellations and theatre utilisation rates were gathered after each run. The

simulation was initially run using the actual data as a base scenario and the results

are given in Table 3.

Table 3. Simulation results

Basis Results Sample mean Standard deviation 95% Confidence Interval Best Case Worst Case

Emergency Patient Wait Time (minutes) 50.24 10.90 (28.88, 71.61) 0.00 532.91Average Elective Patient Wait Time (minutes) 126.40 3.15 (120.24, 132.57) 0.00 544.19Number of Cancellations 215.15 23.63 (168.83, 261.47) 161 267Emergency Theatre Utilisation rate 49.77% 2.27% (45.32%, 54.23%) 55.26% 43.42%Average Elective Theatre Utilisation rate 88.62% 0.14% (88.34%, 88.89%) 88.83% 88.31%

Emergency patient wait

The waiting time before entering the OT department is not provided in the historical

data. The time of entering the OT suite however is given and so the data provided

includes the anaesthesia preparation time.

69

The simulation model indicates that based on historical data (basis model)

emergency patients that use the dedicated emergency theatres currently wait an

average of 50.24 minutes for surgery, with the best case having no wait and the worst

case being 532.91 minutes (8.88 hours). The actual historical data indicates that, for

the same set of patients, the average wait from when an emergency patient arrives to

the theatre suite to entering the theatre is 22.91 minutes (standard deviation 31.11

minutes) with best case 0 minutes and worst case 415 minutes. For both the

simulation and the actual data there is a high level of skewness evident from the

standard deviation and the extreme values.

Figure 9 illustrates the distribution of emergency patient wait for each of the

emergency theatres, B3 (blue) and C1 (red), which shows the majority of these

patients do not wait for surgery. The distribution of emergency patient wait for all

emergency patients taken from the actual data is presented in Figure 10. Comparison

of these two figures indicates more patients in the simulation have zero waiting time

and there is also an increase in the number of patients with higher waiting times

leading to a larger average wait. The increase in zero waiting times may be due to

the model assumptions, which ignore priority levels for emergency patients and

assume all emergencies have precedence over electives. The source of the high

waiting times appears to occur when patients wait for a surgical team. This is due to

the fact that elective and emergency patients compete for the nurses and

anaesthetists.

0 129.2703 258.5406 387.811 517.08130

82.75

165.5

248.25

331

Min Bin Range

EntriesHistogram

Result Result DataData

Figure 9. Distribution of simulation emergency patient waiting time

70

Histogram

0

100

200

300

400

059

.29

118.

57

177.

86

237.

14

296.

43

355.

71M

ore

Fre

qu

ency

Figure 10. Distribution of emergency patient wait taken from actual data

Elective patient wait

The average elective patient wait time (across all elective theatres) was 126.40

minutes with standard deviation 3.15 minutes compared to the actual data giving

135.82 minutes with standard deviation 38.05 minutes. Although the average wait is

very similar there is a noticeable difference in volatility between simulation and

reality. This indicates that the simulation is producing less volatility in the average

wait than the real life data.

Cancellations

On average, 3.53 patients per day were cancelled in the simulation model. There was

no information in the original data to verify whether this result was accurate or

otherwise.

Theatre utilisation

The average elective theatre and emergency theatre utilisation for the base scenario

was 88.62% and 49.77% respectively. Figure 11 illustrates the utilisation in one of

the elective theatres (A1) and the two emergency theatres B3 and C1. The utilisation

can be seen to stabilise around 10980 minutes, showing theatre C1 around the 40%

level and B3 around 60% resulting in the combined average given in Table 3.

71

0 21960 43920 65880 878400

0.25

0.5

0.75

1

Time

ValuePlotter, Discrete Event

A1 B3 C1 Black

Figure 11. Utilisation of theatres A1, B3 and C1.

The model was developed such that arriving patients could go to either theatre,

however, if both theatres are available, the first theatre, B3, is selected, therefore

producing the discrepancy in utilisation between the two theatres.

The following sensitivity analysis was performed on the simulation:

• Change the number of patient arrivals;

1. decreased 10%;

2. increased 10%;

3. increased 20% and

4. increased 30%.

• Implementation of alternative admission disciplines;

1. SPT;

2. LPT; and

3. LFJ.

Changing patient arrivals

The results of changing patient arrivals are presented in Table 4. Some of the

underlying model assumptions - restricting emergency operations to the emergency

theatres, with the exception of ophthalmology and cardiac patients; using an upper

bound for elective patient waiting time and; unlimited emergency patient wait time -

have revealed interesting results for the system, which are detailed in this section.

All tests for significance were performed at the 95% confidence level.

72

Table 4. Changing patient arrivals

Emergency Elective Emergency ElectiveSample mean 50.24 126.40 215.15 49.77% 88.62%

Standard deviation 10.90 3.15 23.63 2.27% 0.14%95% CI (28.88, 71.61) (120.24, 132.57) (168.83, 261.47) (45.32%, 54.23%) (88.34%, 88.89%)

Worst case 532.91 544.19 267 43.42% 88.31%Best Case 0.00 0.00 161 55.26% 88.83%

Sample mean 42.32 112.96 194.38 46.15% 87.96%Standard deviation 7.67 3.09 19.03 1.87% 0.14%

95% CI (27.3, 57.34) (106.9, 119.02) (157.08, 231.67) (42.48%, 49.81%) (87.68%, 88.23%)Worst case 435.06 559.69 256 41.52% 87.57%Best Case 0.00 0.00 157 49.69% 88.24%


95% CI (38.5, 93.34) (124.76, 138.38) (273.04, 359.11)(50.95%, 59.99%) (88.83%, 89.37%)Worst case 672.46 638.06 362 50.84% 88.80%Best Case 0.00 0.00 272 60.68% 89.39%


95% CI (57.76, 118.16) (146.36, 158.24) (421, 535.11) (57.77%, 66.52%) (88.96%, 89.51%)Worst case 908.68 567.93 564 57.48% 88.85%Best Case 0.00 0.00 425 67.84% 89.50%



+ 20%

+ 30%

Waiting TimeRESULTS Cancellations

Utilisation rate

Base case

+ 10%

- 10%


Decreasing patient arrivals 10% decreased the average emergency patient waiting

times approximately 15% from 50.24 to 42.32 minutes. Average emergency patient

waiting times increased 15.68, 37.72 and 96.2 minutes respectively when arrivals

were increased 10%, 20% and 30%. Figures 12 – 15 illustrate these changes in

emergency patients wait. These illustrate the increase in the proportion of patients

that wait longer as the number of patients arriving increases.

73

Figure 12. Emergency patient wait –10%

Figure 13. Emergency patient wait +10%



0 150.1206 300.2412 450.3619 600.48250

86.5

173

259.5

346

Min Bin Range

EntriesHistogram


0 147.4922 294.9844 442.4766 589.96870

83

166

249

332

Min Bin Range

EntriesHistogram


0 117.5182 235.0364 352.5546 470.07280

80.25

160.5

240.75

321

Min Bin Range

EntriesHistogram


0 224.4195 448.839 673.2585 897.6780

87.5

175

262.5

350

Min Bin Range

EntriesHistogram


74


There was a steadier impact on elective patient waiting time across all changes in

patient arrivals compared with emergency patient wait. Waiting time reduced from

126.40 to 112.96 minutes for a 10% decrease in arrivals, and increased 5.17, 25.9

and 41.74 minutes for the three incremental increase in arrivals respectively. Due to

the underlying model assumptions, elective patient wait for individual patients was

limited and therefore there is not a sharp increase in the worst case results. Figures

16-19 illustrate the changes in elective patient wait time as patient arrivals are varied.

The shift to the right indicates the proportion of patients waiting longer increases as

the number of patients arriving increases. However, this increase is limited as

resources are already stretched and there is an upper limit on waiting time.

Cancellations

The number of cancellations per day was reduced from 3.53 to 3.19 when arrivals

were decreased 10%. Increasing arrivals by 10%, 20% and 30% increased the

average number of cancellations by 1.65, 4.31 and 9.19 patients per day respectively.

Changing the percentage of arrivals appears to have a linear effect on elective patient

time whilst the impact on the number of cancellations and the emergency patient wait

time however, appears to be exponential. This can be explained by the underlying

model assumptions. Emergency patients are assumed to wait indefinitely for an

available theatre, thereby exponentially increasing the experienced waiting time. In

contrast, by limiting the waiting time of elective patients, this reduces bottlenecks in

the theatre suites, which would occur due to the build of up patients waiting for

resources to become available. The increase in the experienced waiting time

therefore is limited; however the number of cancellations increases exponentially as

arrivals increase, much like the wait for emergency patients. Figure 20 and Figure

21 present an illustration of these relationships.

75

Figure 16. Elective patient wait –10%

Figure 17. Elective patient wait +10%



0 162.6821 325.3643 488.0464 650.72860

10

20

30

40

Min Bin Range

EntriesHistogram

Result Data DataData

0 120.25 240.5 360.75 4810

7.25

14.5

21.75

29

Min Bin Range

EntriesHistogram


0 130.4812 260.9624 391.4436 521.92470

7.5

15

22.5

30

Min Bin Range

EntriesHistogram


0 191.5 383 574.5 7660

12.75

25.5

38.25

51

Min Bin Range

EntriesHistogram


76

0

50

100

150

200

250

-10% Benchmark + 10% + 20% + 30%

Wai

tin

g T

ime

(min

s)

Emergency wait lower EmergencyEmergency wait upper Elective wait lowerElective Elective wait upper

Figure 20. Effect of admissions on waiting time

0

200

400

600

800

1000

-10% Benchmark + 10% + 20% + 30%Nu

mb

er o

f C

ance

llati

on

s

95% CI lower bound Average cancellations95% CI upper bound Best caseWorst case

Figure 21. Effect of admissions on cancellations

77

Theatre utilisation

The elective theatre utilisation rate was unchanged for all increased patient arrival

tests and decreased from 88.62% to 87.96% for the 10% decrease in arrivals. This

result suggests that regardless of the degree of the increase in elective patient arrivals

there will be no significant increase in elective theatre utilisation rate. It follows then

that the elective OTs are currently running at maximum utilisation and cannot

tolerate an increase in elective arrivals. The emergency theatre utilisation rate

decreased 3.62% for the 10% decrease in arrivals and increased 5.70%, 12.38%

21.60% respectively when arrivals increased 10, 20 and 30%. Unlike the elective

theatres, the emergency theatres could handle an increase in emergency arrivals of at

least 30%. The relationship between changing the percentage of arrivals and theatre

utilisation rates is presented in Figure 22.

One point to note is that an increase in emergency arrivals only was not considered.

All increases in arrivals were across all patient types. An interesting avenue for

further exploration could be the extent to which the emergency theatres could handle

increased emergency arrivals and whether this impacts elective cancellation as they

compete for resources.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

-10% Benchmark + 10% + 20% + 30%

% Change in Arrivals

Utilis

atio

n R

ate

Emergency lower Emergency Emergency upper

Elective lower Elective Elective upper

Figure 22. Effect of admissions on utilisation rate

78

Table 5. Alternative scheduling disciplines

Emergency Elective Emergency ElectiveSample mean 50.24 126.40 215.15 49.77% 88.62%

Standard deviation 10.90 3.15 23.63 2.27% 0.14%95% CI (28.88, 71.61) (120.24, 132.57) (168.83, 261.47) (45.32%, 54.23%) (88.34%, 88.89%)

Worst case 532.91 544.19 267 43.42% 88.31%Best Case 0.00 0.00 161 55.26% 88.83%




95% CI (29, 77.29) (114.25, 124.85) (160.72, 210.22) (46.45%, 54.17%) (88.36%, 88.86%)Worst case 511.93 563.12 207 46.40% 88.36%Best Case 0.00 0.00 156 55.37% 88.88%



Waiting Time

LFJ

Base case

LPT

SPT

Cancellations Utilisation Rate

RESULTS

Alternate scheduling disciplines

Implementation of the three scheduling disciplines (LPT, SPT and LFJ) was

compared. The results of implementing each scheduling discipline are presented in

Table 5. All of the findings are presented at a 95% confidence level.


The scheduling disciplines were found to have no significant impact on emergency

patient wait. The average emergency patient wait times were 52.0, 53.15 and 52.27

minutes respectively for LPT, SPT and LFJ. This was expected, as the sequencing

discipline of emergency patients was unchanged. Any slight variations in value are

therefore due to the variable nature of simulation models.


Scheduling disciplines LFJ and SPT both improved elective patient waiting time with

LFJ performing better than SPT. Conversely, LPT was found to increase elective

patient waiting time. The average elective patient wait times were 130.16, 119.55

79

and 107.31 minutes respectively for LPT, SPT and LFJ. The effects of the LPT and

SPT disciplines on elective patient waiting time were somewhat intuitive.

Performing the longer, more variable surgeries first (LPT) results in a longer waiting

time for the shorter surgeries while scheduling the shorter surgeries first, will

decrease the wait for the remaining activities. More noteworthy however, is that LFJ

outperformed SPT. By scheduling the least flexible patients first, the remaining

patients can be juggled around to optimise the resource usage towards the latter end

of the schedule, thereby improving the average elective waiting time.

Cancellations

The average number of elective cancellations was 3.66, 3.04 and 3.40 patients per

day for LPT, SPT and LFJ respectively. At the 95% confidence level, the LFJ and

SPT disciplines reduced the number of patient cancellations, with SPT performing

better than LFJ. LPT significantly increased the number of cancellations. While it

would be expected (or at least hoped) that LFJ would reduce cancellations due to

scheduling the more flexible surgeries last, the results for SPT and LPT conflict with

literature (see literature review). One would expect that LPT reduce cancellations by

scheduling the longer more variable surgeries first. However, perhaps the long

surgeries are taking much more time than the expected duration, resulting in fewer

surgeries being completed in time. Correspondingly, more short surgeries can be

performed in the same space of time compared with long surgeries. So subsequently,

with LPT, more surgeries are cancelled at the end. Figures 23 and 24 demonstrate

the effect on the number of cancellations and patient waiting time for the alternative

scheduling disciplines respectively.

Theatre utilisation

At the 95% confidence level there was no evidence to suggest an improvement or

otherwise in elective or emergency theatre utilisation rates for SPT and LPT. This

means that despite LPT increasing the number of cancellations, there was no

corresponding change in utilisation. This again contradicts literature that found LPT

increased throughput and utilisation (Harper, 2002). For LFJ, elective theatre

utilisation increased 0.47% and emergency theatre utilisation was unchanged.

80

0

50

100

150

200

250

300

Base case LPT SPT LFJ

Alternative Scheduling Disciplines

Nu

mb

er o

f Can

cella

tion

s

95% CI lower bound Best case Average95% CI upper bound Worst case

Figure 23. Effect of scheduling disciplines on cancellations

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

Base case LPT SPT LFJ

Alternative Scheduling Disciplines

Wai

tin

g T

ime

(min

s)

Emergency lower Emergency wait timeEmergency upper Elective lowerElective wait time Elective upper

Figure 24. Effect of scheduling disciplines on waiting time

81

3.3 Conclusions

Simulation was used to model the OT department of the Princess Alexandra

Hospital. Service distribution and emergency patient inter-arrivals were generated

based on historical patient data. Efficiency measures calculated for the real system

were compared with simulation outputs to validate the model. The model was then

used to explore the effects of changing patient arrivals and alternative elective patient

admission disciplines would have on the performance measures.

Emergency patient waiting times and the number of elective cancellations appeared

to increase exponentially as the number of arrivals was increased. The relationship

between elective patient waiting time, elective theatre utilisation rate and emergency

theatre utilisation rate with changing arrivals appeared linear and less pronounced.

There was no significant impact on elective theatre utilisation rate for any change in

arrival, which suggested that the elective theatres were running at near maximal

activity. This claim may be supported by the exponential effect seen on the number

of cancellations when arrivals increased with no accompanied effect on elective

theatre utilisation rate.

LFJ improved patient waiting times and reduced the number of cancellations. SPT

improved patient waiting times, had a lesser impact on elective patient waiting times,

but was found to increase both elective patient waiting times and the number of

cancellations. The improvement in emergency patient waiting times for all

disciplines suggested this was a result of reducing elective scheduling variability.

The scheduling disciplines did not have a significant effect on theatre utilisation

rates.

An important result of this study is its potential implementation as decision support

for surgical planners wishing to adopt an alternative scheduling discipline or for

predicting the effects than an increase in patients will have on performance criteria.

Traditionally, theatre utilisation rate has been considered the only important

performance measure for the OT. However, when the theatres are already running at

maximum capacity, improvements in OT performance are not seen in theatre

utilisation rates. In fact there was little change in elective theatre utilisation rates at

82

all. This study therefore demonstrates the need to consider the effects on patient

waiting times and elective cancellations as an adjunct measure of performance.

Although this study indicates the likely effects changes may have on the OT system’s

performance measures, it does not produce any quantifiable measures of resources

for optimising some particular objective. This is a possible avenue for future

research. In addition, it would be interesting to examine an increase in emergency

arrivals without changing electives and to what extent the system could handle such

an increase. Thirdly, this study analyses the system from a macro perspective, taking

average performance measures across all electives. Further study (although much

more time consuming) could include an examination of the elective theatres on an

individual basis.

83

Chapter 4. Robust surgery assignment models

Simulation was used in the previous chapter to analyse the real life problem and

investigate various ‘what if’ scenarios and their implications on the performance of

the OT department. While simulation provides a way to examine scenarios without

actually impacting the real life system, it does not optimise a system. Optimisation

of a system is usually achieved with mathematical models. These approaches are not

designed to conflict with one another, but rather are complementary. Simulation will

further be used in this chapter to generate patient treatment times from statistical

distributions to assist with model validation and testing.

Theoretical models based on operation research techniques are used in many

applications to determine ‘optimal’ (or sufficiently good, in complex cases) solutions

to problems. The literature review provided in Chapter 2 has already shown that

operations research models have been applied to many areas in healthcare including

the OT. In this dissertation, simulation is not only used to examine the system, but

theoretical models are also used to improve elective scheduling procedures and also

develop a new tool for the dynamic online scheduling problem incorporating

emergency patients. This chapter begins with a look at elective scheduling methods.

Surgical durations are variable and depend on many factors including the type and

severity of illness and the experience of the surgeon performing the operation

(Guinet & Chaabane, 2003). Even with detailed statistical analysis it is very difficult

to precisely predict how long a procedure will take. When the assigned time for a

surgery is less than the actual time, this can lead to either delayed starts for

remaining procedures, cancellations or overrun theatres, which is an added cost to

the running of the department. Robust patient assignment is a technique used to

address this issue faced by schedulers.

A statistically calculated amount of ‘extra’ time is assigned to each operation based

on the proportion of surgeries expected to be completed on time, which is determined

by the scheduler or decision maker. So for the schedule that wants to minimise

84

overrun surgeries, the buffer would be larger than that for a schedule with a higher

tolerance of overruns.

As seen in the literature review, robust patient assignment is first introduced for the

operating theatre by Hans et al. (2008). In their paper, historical data is analysed for

surgical duration mean and variance estimates. The sum of the surgical durations

assigned to a theatre is assumed normally distributed and the properties of their

summation are used to determine an amount of slack for each theatre. Possible

future work highlighted by Hans et al. (2008), included addressing the inapplicability

to the sum of random variables not normally distributed.

The robust patient assignment models developed are offline non-linear integer-

programming models, where the amount of slack planned for each block is based on

the variations in surgical durations of the patients scheduled in the block. In each of

the models, surgical durations are modelled with the lognormal distribution, which is

a common choice for surgical durations. The summation of lognormal variables

however, is not modelled by a normal distribution, as with the summation of some

other types of random variables. This is a point of innovation for robust patient

assignment in the operating theatre. The results of the model are compared with the

case of normally distributed surgical durations.

Further innovation is achieved by addressing the issue of flexible patient assignment.

In the real life setting, surgical consultants are generally responsible for selecting and

sequencing their own patients. Introduced is the idea of adapting the robust patient

assignment model to allow patients to be selected directly from a waiting list to

optimise planning of the available capacity. When patient assignment is more

flexible, the benefits of such a schedule are expected to be greater and the results can

be compared with current practices and any differences presented to hospital staff.

An alternative performance measure to tardiness, namely deviation, is considered.

The deviation of a surgery block is defined as the difference between used and

available surgical surgery block capacity. Studies have shown that the costs of

running the operating theatre occur not only from theatres running into overtime, but

also due to theatres completing surgery early and therefore leaving unused capacity

85

(Dexter et al., 2000, Dexter et al., 1999). Deviation considers both earliness and

tardiness.

The developed models in this dissertation are applicable irrespective of whether

patient assignment is surgeon specific or not. The main benefit of this is that a

generalised model may be easily adapted to similar hospital systems and incorporate

hospital specific requirements.

This chapter begins with a section on data analysis for the robust and reactive models

that illustrates why the lognormal distribution is used for modelling surgical

durations. This is immediately followed with the process used for generating random

cases for testing the robust and reactive models. The theory of the sum of

independent but not necessarily identical lognormal random variables, which

underpins the developed models, is also provided. Before the robust assignment

models are presented in the Section 4.5, the assumptions of the models are outlined.

The remaining sections include the computational complexity of the models, the

solution methods, results and conclusions respectively.

4.1 Data analysis for robust and reactive models

The simulation model of Chapter 3 was based on the entire Operating Theatre

Department of the Princess Alexandra Hospital. In the following two chapters, the

robust and reactive models will be based on the smaller Surgical Care Unit. Using

the SCU is a satisfactory representation of the whole department because both

elective and emergency patients are treated there. The benefit to using the SCU is

the reduction in problem size and hence calculations. The results obtained for the

SCU can be extended and applied to the entire OT department.

The study of the real life problem in Section 1.3 highlighted the point that elective

surgery patients may be either day surgery or non-day surgery patients. Day surgery

patients are treated in the four assigned day surgery theatres in the Surgical Care Unit

(SCU), namely theatres A1 – A4. These patients generally arrive and/or are released

on the day of their surgery.

86

Since the SCU was selected as a sample for the whole problem, specialties that are

only treated in the SCU were sought after. The reason for this was to get a closer

representation of the real life problem. If a theatre only treats a particular specialty

then the total capacity is dedicated to those patients. In other words, there is no need

for adjusting total capacity available to those patients. If a number of different

specialties are treated in a theatre, then either all the specialties need to be

considered, or the capacity must be adjusted for the selected specialties. Of the

specialties treated in the SCU, the Ophthalmology patients are almost exclusively

treated in theatres A2 and A3. Of the specialties treated in these two theatres, the

majority are also Ophthalmology. Most other specialties (treated in the SCU) are

also spread across the non-day surgery theatres. For this reason, the developed

robust and reactive models are applied to the Ophthalmology patients in theatres A2

and A3 and it is assumed that all capacity available is dedicated to those patients.

This assumption may be changed according to a scheduler’s requirements. It is

important to note that the developed models may also be adjusted to incorporate any

patient specialty and any theatre.

Historical data for the Ophthalmology patients was analysed for surgical duration

estimates. The times available in the data provided were ‘time in suite’, ‘in

anaesthesia’, ‘in OR’ and ‘Out OR’. These times indicate the time the patient enters

the operating theatre suite (which is composed of the operating rooms and their

anaesthesia workrooms, day surgery unit, ICU, PACU etc), the time the patients

enters anaesthesia and the times of entering and exiting the operating room

respectively. The time of anaesthesia may be calculated as the difference between

‘in OR’ and ‘in anaesthesia’. Likewise, the time in the OR is the difference between

‘Out OR’ and ‘in OR’. These varied according to the type of surgery being

performed. The total treatment time of a patient was assumed to include anaesthesia

preparation time, however this assumption can easily be changed if desired.

The Ophthalmology specialty was broken down into sub-specialties for the

calculation of surgical duration estimates. Six surgical sub-specialties were

determined based on the type of surgery performed. A histogram of actual data

suggested a possible lognormal distribution for each of the surgical sub-specialties.

Goodness of fit tests supported the hypothesis that surgical durations may be

87

described with the lognormal distribution but rejected the hypothesis of a normal

distribution. An example plot of the fitted distribution against actual data for the first

specialty is provided in Figure 25.

Figure 25. Fitted distribution versus actual data for specialty 1

The lognormal distribution is a continuous distribution, based on the normal

distribution. It is used to describe many applications including physicians’

consultation time, lifetime distributions, the long-term return rate on a stock

investment and weight and blood pressure of humans. It has also been used in the

literature for describing surgical durations (Jebali et al., 2006, Strum, May & Vargas,

2000).

The distribution is described with 3 parameters; the mean of the included normal µ,

the standard deviation of the included normal σ and the minimum value or location

parameter γ. The three parameter probability density function is given by

( )2

2

ln( )

2

2

1( )

( ) 2

x γ µ

σf x ex γ πσ

− −−

=−

Parameters for the lognormal and normal approximations were determined for each

of the surgical specialties based on the data analysis and are presented in Table 6.

88

The Lognormal distribution is described with 3 parameters, i.e. the mean of the

included normal µ, the standard deviation of the included normal σ and the minimum

value or location parameter γ. For data analysis, surgical durations were given in

minutes.

Table 6. Data for Specialties

4.2 Generation of random cases for testing the robust and reactive models

In order to test the robust and reactive models, patient lists needed to be generated

that could be used as inputs for the models. When generating these patient lists, it

was necessary to consider the future implication of these models in the real life.

Thus, it was important to use an approach that could easily be translated in the

practical setting. More precisely, a patient list was considered to be the list of

possible patients for a particular surgical consultant or surgical team for the near

future, (this means that each surgeon or surgical group may have their own

individual patient list. This takes into account different qualifications, etc). The

length of the future period is not fixed and depends on the particular consultant. For

example, some surgeons may be assigned two surgical blocks in any given week, on

a weekly basis whereas another may only be assigned one surgical block every

second week. This kind of detail is not necessary at this planning stage and can be

omitted.

For the purpose of testing the models, there are six surgical specialties that the

patients may belong to. At this stage, restrictions on which specialties a consultant

can treat, are not examined (because it is an easy extension to omit particular

specialties from a patient list) and it is assumed any surgical category may be treated.

Lognormal Normal Specialty γ µ σ2 µ σ2

1 18 2.78 0.674 3.773663 1.757204 2 20 3.38 0.561 3.508642 1.427401 3 16 3.42 0.779 3.64058 2.890148 4 12 3.89 0.777 4.91161 6.694422 5 55 4.0 0.79 8.258025 10.23665 6 29 3.82 0.452 4.981818 3.398303

89

For a patient list, the random number generator function in excel, was used to

generate the number of patients of a particular specialty that must be assigned to a

surgical block. It was decided that there is no need to use an empirical approach

from historical data for generating these numbers. This is because despite the

controllable nature of schedule generation, the surgeons at the PAH believe that there

is no real pattern to the mix of elective patients that present to the operating theatre

over time and therefore patient mix is generally considered to be highly

unpredictable. For this reason, using random numbers to assign patients to

categories is justified.

The generated patient lists will be used as inputs for the robust assignment models.

These will determine the number of patients that must be assigned to the surgical

blocks. The patient list will also specify the maximum number of patients from a

particular specialty that can be assigned. This is used for the flexible assignment

models seen in chapter 4.5.2. The results of the flexible assignment models are used

as inputs for the reactive scheduling models developed in Chapter 5.

4.3 The sum of independent but not necessarily identical lognormal random

variables

The following theory on the sum of independent but not necessarily identical

lognormal random variables comes from (Nie & Chen, 2007, Weisstein, 1999-2010).

Let NP,...,P1 be N independent but not necessarily identical lognormal random

variables with parameters iµ and iσ . Let ∑==

N

iiN pP

1 and the mean of NP is NM

with

90

[ ]

2

2

2

1

2

1

1

2

1

ieie

ieie

ii

e

PEM

N

i

N

i

N

i

N

iiN

σ

=

µ

σ

=

µ

=

σ+µ

=

∑=

∑=

∑=

∑=

and NV is the variance of NP

[ ]( )[ ]

∑

−=

∑ −=

=

σ+µσ

=

N

i

N

iiiN

iieie

PEPEV

1

2

1

2

21

2

The third and fourth order central moments, NS and NT are respectively given by

[ ]( )[ ]2

22

12

2

33

1

1

3

iieieie

PEPES

N

i

N

iiiN

σ+µσ

=

σ

=

+∑

−=

∑ −=

[ ]( )[ ]

∑

−++

−=

∑ −=

=

σ+µσσσσ

=

N

i

N

iiiN

iieieieieie

PEPET

1

24234

1

4

23

23

22

21

2

The skewness 1γ

and kurtosis 2γ

are respectively

231 /N

N

V

S=γ

22N

N

V

T=γ

91

The sum of independent but not necessarily identical lognormal random variables is

assumed to be modelled with a lognormal random variable and used to approximate

the sum of lognormally distributed surgical durations. NM and NV are used to

calculate the mean, Nµ , and variance, Nσ , of this lognormal distribution. Nie and

Chen (2007) discuss the limitations of using the lognormal distribution for modelling

the sum of lognormal random variables, with respect to the skewness and kurtosis

parameters. However, in this thesis, for the purpose of simplification, this

assumption will be held. Based on this assumption, the amount of time, k, that needs

to be assigned to the surgeries can be calculated.

For a lognormally distributed variable, D, the probability that D exceeds some

threshold k is given by ( )

σµ−Φ−=≥ )kln(

kDP 1 where Φ is the cumulative

distribution function of the standard normal distribution. If

−σ

µ)ln(k= x, then

( ) [ ]xkDP Φ−=≥ 1 . Since xk =

−σ

µ)ln(, then µσ += xek .

From the properties of the lognormal distribution, NM can be rearranged to obtain

and expression for Nµ in terms of NM and Nσ .

2

2N

NN eM

σµ +

=

( )2

ln2N

NNMσµ +=

( )2

ln2N

NN Mσµ −=

Now substitute this into NV and rearrange to obtain an expression for2Nσ in terms of

NV and NM

222

1NNN

N eeVσµσ +

−=

92

( ) 22

2

ln22

1

N

NNM

NN eeV

σσ

σ+

−

−=

( )NMNN eeV ln2

2

1

−=σ

( )

2

ln21

N

NMN e

e

V σ=+

( )2

ln21ln NNM

N

e

V σ=

+

( )( )

2ln2

ln2ln NNM

NMN

e

eV σ=

+

22ln

2ln

ln NNM

NM

N

e

eV σ=

+

22

2ln N

N

NN

M

MV σ=

+

Substituting 2Nσ into Nµ gives

( )

+−=

2

2ln

2

1ln

N

NNNN

M

MVMµ

( )

+−=

2

2lnln

N

NNNN

M

MVMµ

+=

2

2ln

N

NN

NN

M

MV

Mµ

+=

2

2ln

NN

NN

MV

Mµ

93

Substituting 2Nσ and Nµ into k gives

++

+

=2

2ln

2

2ln

NMNV

NM

NM

NMNVx

ek

+

+

=2

2ln

2

2

ln

NMNV

NM

NM

NMNVx

eek

+

+=

2

2

ln

2

2NM

NMNVx

NN

N eMV

Mk

Variations in the choice of x affect the value of k. In a practical sense, since k

represents the amount of time assigned to a surgery block, the larger the x value, the

lower the overtime expectation. For ease of calculations however, choose x = 1 then

( ) [ ]11 Φ−=≥ kDP = 1 – 0.8413 = 0.1587. This means that the probability that the

surgical duration d exceeds some time k is approximately 15.87%. Then

+

+=

2

2ln

2

2NM

NMNV

NN

N eMV

Mk

4.4 Model assumptions

The following assumptions are held for the model:

1. In the interest of analytical tractability, surgical durations are assumed to be

mutually independent. This implies that treatment times are not reduced by

repetition of surgical operations of the same type despite this being the case in

practice.

2. Staff and equipment are assumed readily available and do not constrain the

model. Additional resource constraints can be easily added to the model by

considering their availability at each patient’s assignment, however are not

considered in this dissertation.

3. Data collected was representative of a typical period.

94

4. Overtime capacity is equal for all surgery blocks.

5. The distribution of surgical duration estimates for specialty type i is the same

regardless of the treating surgeons or surgical team assigned to a surgery block.

4.5 The robust surgery assignment models

The robust surgery assignment models covered are:

i. The robust surgery assignment model with lognormally distributed surgical

durations

ii. Flexible robust surgery assignment model with lognormally distributed surgical

durations

4.5.1 The robust surgery assignment models with lognormally distributed

surgical durations

Indices

i: Specialties considered within a surgical category, i = 1, …, I

j: Available surgery blocks during the scheduling period, j = 1, …, J

Parameters

Cj: Capacity of surgery blocks j

Ri: The required number of patients of specialty i

µi, σi : Lognormal distribution parameters for surgical specialty i

di: Expected surgical duration of specialty i

si2: Expected surgical duration variance of specialty i

95

Decision Variables

Xij: The number of patients of specialty i to assign to surgery block j

Uj: The used time of surgery block j

Tj: The tardiness of surgery block j

Ej: The earliness of surgery block j

Dj: The deviation of surgery block j

=otherwise 0

used block if 1 jYj

Objective function

Minimise the maximum surgical surgery block deviation

jJj

DZ minimax∈

= (4-1)

Constraints and equations

All patients must be scheduled exactly once.

∑ ∀=j

iij iRX , (4-2)

The expected surgical duration, id and the variance, 2is are respectively given by

2

2i

ii ed

σµ +

= (4-3)

2222 1 iiii ees

σµσ +

−= (4-4)

96

where iµ and iσ are lognormal random variable parameters determined by analysis

of historical data. The sum of the expected durations and the variance of the patients

assigned to a surgery block are given respectively by

2

1

in

i

iijj eeXM

σµ∑==

(4-5)

∑

−=

=

+n

i

iiiijj eeXV

1

2221

σµσ (4-6)

The amount of time that is planned for each schedule to ensure the probability that

surgeries run overtime is less than 15.87% is given by

+

+=

2

2

ln

2

2jM

jMjV

jj

jj e

MV

MU (4-7)

The tardiness of surgery block j is greater than or equal to the sum of the planned

time of the assigned surgeries, less the capacity of the surgery block. In this model,

there are a number of available surgery blocks, but not all of them are necessarily

required. For this reason variables Yj are required, indicating whether or not a

surgery block is used. The reason is because due to the solution methods, discussed

in Section 4.7, the number used varies. To ensure a schedule can be produced there

must be enough blocks available at the beginning.

( ) jjjj YCUT −≥ , j∀ (4-8)

In scheduling theory, tardiness must be positive therefore

,0 jT j ∀≥ (4-9)

The earliness of surgery block j is greater than or equal to the capacity of the surgery

block less the planned time of the assigned surgeries.

97

( ) jjjj YUCE −≥ , j∀ (4-10)

Earliness must also be positive

,0 jE j ∀≥ (4-11)

The deviation of a surgery block is the difference between the total allocated time of

the surgeries assigned to a surgery block and the available capacity of the surgery

block. This is equivalent to the amount of capacity unused (earliness) or overused

(tardiness). The deviation of surgery block j is the sum of the tardiness and earliness

of the block.

, j j jD T E j= + ∀ (4-12)

To ensure that the Yj variables are not all zero and consequently all deviation

variables equal to zero, the following constraint must be used

, j jU MY j≤ ∀ (4-13)

This equation ensures that when Yj = 0, then 0jU ≤ and 0jU > only if Yj = 1.

The final constraints on the problem are that the number of patients assigned to each

theatre is a positive integer and variables jY are binary integers

ijX Z+∈ (4-14)

0 or 1jY = (4-15)

98

4.5.2 Flexible robust surgery assignment model with lognormally distributed

surgical durations

The flexible model is essentially the same as the previous model with the following

adaptations.

Additional Parameters

iAi specialty of patients available ofnumber The:

Additional Constraints

Equation 4-2 is replaced with the following two constraints.

∑ ∀≥j

iij iRX , (4-16)

∑ ∀≤j

iij iAX , (4-17)

4.6 Problem complexity

The offline robust surgical assignment models are analogous to a parallel machine

environment with extensible machine capacity. For the non-preemptive parallel

machine scheduling problem that minimises the maximum lateness, all problems that

are NP hard under the minimised makespan (Cmax) criterion remain NP hard under

the Lmax criterion. The problem P2||Cmax has been shown to be NP-hard and

therefore so is P2||Lmax. Since Pm||Cmax is known to be NP-hard for the m = 2

machine case, the problem is not any easier to solve when there are more than 2

machines. Therefore, this is also true for the Lmax problem (Blazewicz et al., 1996).

The following theorems can be used then to show that when P||Lmax and P||Emax are

NP-hard then P||Dmax is also NP-hard.

99

Theorem I

A schedule that is optimal with respect to Lmax is also optimal with respect to Tmax.

Proof

max 1 2

1 2

max

max(max( ,0),max( ,0),...,max( ,0))

max( , ,... ,0)

max( ,0)

n

n

T L L L

L L L

L

===

Therefore minimise Lmax will also minimise Tmax.

The minimise P||Emax problem may also be shown to be NP-hard when P||Lmax is NP-

hard.

Theorem II

A minimise P||Emax problem is NP-hard when P||Lmax is NP-hard.

Proof

)0,max( iii CDL −=

where iD is the due date and iC is the completion time of job i.

)0,max( iii DCE −=

)0,min( ii LE −=

Therefore the minimise P||Emax problem is NP-hard when P||Lmax is NP-hard.

Since Dmax = Emax + Lmax, then P||Dmax is also NP-hard when P||Lmax is NP-hard.

From this relationship between lateness and earliness for the parallel machine

scheduling environment, the offline robust surgical assignment models with

extensible machine capacity are strongly NP-hard. Approximation algorithms

therefore should be applied to find near-optimal solutions to such problems in

reasonable time.

100

4.7 Solution methods

As shown in Section 4.6 the robust assignment models are computationally difficult

to solve and therefore approximation methods are required to obtain solutions. A

number of constructive and local search heuristics are developed for the models and

will be discussed.

Traditional constructive heuristics for scheduling in the operating theatre include

SPT and LPT (see Section 2.4). Innovative constructive heuristics combined with

assignment and local search heuristics are proposed to obtain solutions for the

schedules. These are HVF (highest variance first) and LVF (lowest variance first).

Rather than sequencing patients by surgical duration length, patients are ordered by

non-increasing surgical duration variance and non-decreasing surgical duration

variance respectively. In practice, longer surgeries are typically considered more

variable than surgeries of shorter duration. Analysis of historical data however

demonstrated that a LPT (SPT) sequence is not strictly the same as a HVF (LVF)

sequence. Whether sequencing based on LVF and HVF have improvements over

current methods, SPT and LPT, is investigated.

The logic of the constructive heuristics is used throughout the assignment and local

search algorithms to exploit the idea of clustering surgeries with the same surgical

duration variance together to minimise the amount of slack required for a schedule.

In this section the assignment and local search heuristics for each of the robust

assignment models are described. Each of the initial assignment and local search

heuristics builds on the previous heuristics, adapting to the changes between the

models. The algorithms used to solve the models are given in Appendix C.

4.7.1 Solution method for the robust surgery assignment model with

lognormally distributed surgical durations

The robust schedules are solved with an initial assignment heuristic and

neighbourhood search algorithm. The logic behind these algorithms is applied to

both models regardless of whether surgical durations are modelled with lognormal or

101

normal distributions. It is noted however, that due to the differences in these

distributions, the calculations of treatment times are obviously different.

The initial assignment algorithm aims to assign the surgeries to the surgery blocks

such that the maximum deviation of the surgery blocks is minimised. In addition, it

also tries to minimise the number of surgery blocks actually required to satisfy

patient demand. The time phasing of surgery blocks is not considered in this model.

It is assumed that the number of surgery blocks used for a schedule correspond to the

next consecutive set of surgery blocks assigned to a surgeon or surgical group. For

example, if a surgeon is assigned 4 surgery blocks over a month, then the first four

surgery blocks (or any set of four of the required surgery blocks) correspond to one

month’s schedule.

Surgical specialties are assigned in order of non-increasing surgical duration variance

to the surgical blocks. The patients of the first specialty with the highest surgical

duration variance are first exhausted before assigning patients from another specialty.

After each patient is assigned, the deviation value of the surgery block being filled is

calculated. If the surgery block’s capacity has not been filled (i.e. there is no

overtime), the assignment is accepted and the process continues. If the surgery block

is overfilled, then the deviation value for the current assignment is compared with the

deviation value for one fewer patient. If the current deviation value is less than the

deviation for one fewer patient, then the assignment is accepted. Otherwise, the

assignment is rejected and the patient is assigned to the next surgery block. This

process is repeated until all the patients of the specialty have been assigned.

Once all the patients of the first specialty are assigned, the assignment algorithm

moves onto the next specialty beginning with the first surgery block. By starting

with the first surgery block rather than the next empty block, the process tries to

minimise the number of surgery blocks actually used. The assignment process

continues until all the required patients for each specialty are assigned.

102

Initial assignment algorithm

Inputs

jC,J,I

j,i ,0ijX ∀=

j ,0jD,jE,jT ∀=

iR

I to 1i For =

{

1j

1k

==

∑=

<J

1jiRijX While

{

kijX =

)0,jCjUmax(jT −=

)0,jUjCmax(jE −=

jTjEjD +=

If 0jT ==

1kk +=

Else

{

1ijx'ijx −=

)0,jC'jUmax('

jT −=

)0,'jUCmax('

jE −=

'jT'

jE'jD +=

If 'jDjD <

{

ijx'ijx =

jT'jT =

jE'jE =

103

jD'jD =

1kk +=

}

Else

{

'ijxijx =

'jTjT =

'jEjE =

'jDjD =

1jj +=

1k =

}

}

}

1ii +=

}

Following the assignment of the surgeries to the surgery blocks, a neighbourhood

search heuristic is implemented. This algorithm comprises several sequential

algorithms. The first of these is the max-min algorithm.

The max-min algorithm works out which blocks have the current highest and lowest

earliness and tardiness values. The highest deviation value for the surgery blocks

(whether it be over or under used) is set as an upper-bound, UB. The remaining

algorithms then work to reduce this maximum deviation value (UB) by

systematically removing patients from the surgery block with the highest tardiness

value and moving them to the block with the largest earliness value. If the maximum

deviation (i.e. UB) can be reduced, the max-min algorithm is re-called to update the

new maximum earliness and tardiness values and UB. The search process continues

until the stopping criteria are met.

104

4.7.2 Solution method for the flexible robust surgery assignment model with


The heuristics used to solve the flexible model are the same as those discussed in

5.6.1 with the addition of a second assignment algorithm. This algorithm is called

after the initial assignment of the required number of patients. This ensures the same

number of surgery blocks is used as for the non-flexible model; however, the

deviation may be reduced even further allowing for more efficient planned use of the

theatre capacity.

The second assignment algorithm systematically looks to assign more patients to

surgery blocks that have additional unused capacity. Initially, the heuristic tries to

assign patients of the same specialty as those already in a surgery block; again in an

attempt to exploit the effects of clustering surgeries of the same surgical duration

variance together. If the deviation of the surgery block can be reduced, then the

assignment is accepted. Following this, the algorithm searches for any assignment

regardless of specialty to reduce the deviation.

After the second assignment is implemented, the local search heuristics are again

employed for a final attempt at reducing the deviation of the surgery blocks.

4.8 Robust schedule results

For each of the 100 examples the required number of patients for each specialty is

randomly generated. The specialties used and the distribution parameters associated

with a specialty are given in Section 4.1. The assignment and neighbourhood search

heuristics are implemented using the C# programming language in Microsoft Visual

Studio.

The number of surgery blocks required to robustly assign the patients is unknown

and the constructive and search heuristics discussed in Section 4.7 reflect this by

allowing flexibility in and minimising the number of surgery blocks actually used.

The reason for this is due to the calculations of total treatment time assigned for a set

of patients. Different combinations of patients to the time blocks will result in

105

varying amounts of total slack time required. By assigning patients of the same

specialty and therefore the same surgical duration variance, the amount of slack can

be reduced. This is due to the portfolio effect, mentioned by Hans et al. (2008).

Different constructive heuristics will produce different results for the number of time

blocks required. For each constructive heuristic, the assignment methodology aims

to minimise the number of blocks used and schedule patients of the same specialty

together. Any unused time capacity is dealt with more efficiently by the flexible

assignment model.

4.8.1 Results for the robust surgery assignment model with lognormally

distributed surgical durations

The results of the assignments are presented in Table 7. The performance measures

discussed are maximum, minimum and total time block deviation (in minutes), the

number of time blocks used to generate the schedule and the average number of

patients assigned per time block. Each surgery block has 8 hours capacity. Further

work (not discussed here) could include the analysis of changing this capacity.

Table 7. Schedule results comparing LDSD and NDSD models

Results Lognormal Results Normal Results

Maximum Deviation (mins) 47.24 42.23

Minimum Deviation (mins) 23.37 18.59

Average Total Deviation (mins) 175.75 163.93

Patients per block 6.07 5.50

Number blocks used 4.98 5.50

Total Patients 3014 3014

Number of blocks used 498 550

From the results presented in Table 7 it is evident that the average, maximum and

minimum deviation measures are actually very close between the two different

models. In fact, for the lognormal model, the average deviation values are actually

slightly higher than for the normal model. The similarity between the models

106

suggests the logistics of the heuristics used to generate solutions perform equally

well for surgical durations modelled with either the lognormal or normal distribution.

In both cases, for each individual example, the average maximum deviation is less

than one hour and the average total deviation for the surgery blocks is less than 3

hours. In the worst case, the deviation value for a single surgery block is almost 2

hours, however in the best case it is as little as 5.5 minutes. Considering each

surgery block has 8 hours capacity, in some instances these results represent a

significant amount of either over or under used capacity. One proposal for the reason

for these large deviation values, is due to the inflexibility in patient assignment and

that increasing the flexibility in patient assignment may possibly reduce these. In

many operating theatre departments, surgeons are responsible for selecting and

sequencing their own patients. The flexible assignment model, that incorporates the

surgeon’s requirements with a flexible approach to selecting patients from a waiting

list, addresses this issue.

Taking these deviation results alone, one might think that the normally distributed

surgical duration (NDSD) models perform better than the lognormally distributed

surgical duration (LDSD) models but this is not the case. By looking at the

schedules individually, the deviation can be analysed a little closer. Because

deviation considers both earliness and tardiness, deviation values do not indicate

whether the surgery blocks are over or under used. It turns out that for the LDSD

and NDSD models the proportion of tardy blocks planned is 7.03% and 22.73%

respectively. While both models plan more underused surgery blocks than tardy,

there is a substantial difference between the two models. In addition, when the

NDSD model produces schedules with lower deviation values than the LDSD model,

98% of the time the NDSD model uses not only more blocks, but more tardy blocks,

than the LDSD model to assign the same number of patients.

Although the two models produced very similar ‘deviation value’ results, the other

performance measures illustrate why the robust model using surgery durations drawn

from a lognormal distribution is a better choice than the normal distribution. Before

explaining the details of these results, it is necessary to note the difference between

average patients per block and the total number of patients divided by total number

of blocks given in Table 7. Computing the averages for each example and then

107

taking the mean of these determines the former of these two values. There is

therefore a slight difference between these values.

The average number of patients per time block is greater for the LDSD model

compared with the NDSD model. This means the amount of time assigned per

patient is greater for the NDSD model than the LDSD model. This is due to the

shape of the fitted distribution curves. The modes of the fitted lognormal probability

density functions are further to the left than the modes of the associated normal

curves. Essentially, this results in the LDSD model requiring less time for a patient

than the NDSD model. As a consequence, the average and total number of surgery

blocks used for the same schedules is less for the LDSD model. The NDSD model

requires 550 surgery blocks for the total 3014 patients scheduled while LDSD only

uses 498. This is a significant saving in surgery time and in quantitative terms, if the

52 surgery blocks were filled, this could represent an additional 315 patients (based

on average patient per block figures).

To summarise, both the LDSD and NDSD models create robust schedules with the

expectation that the number of patients to complete surgery within their allocated

time is around 85%. However, the LDSD model requires fewer surgery blocks to

achieve this. In Section 4.1, it was shown that the lognormal distribution is a better

choice for modelling surgical durations than the normal distribution. As a result, the

proposed method of robust patient assignment under the assumption of lognormally

distributed surgical durations is more efficient and more suitable than approximating

with a normal distribution.

In addition to comparing the LDSD model with the NDSD model, the actual

performance of the models was tested using simulation. For a sample of the

schedules, surgery durations were randomly generated from the estimated lognormal

distributions. For each surgery block, the total simulated treatment time of the

patients was compared with the amount of time assigned to the patients. The results

from this simulation are presented in Table 8. The first thing to note from these

results is that both models produce satisfactory results as they achieve an average

proportion of overtime that is less than the desired level of 15.87%. Secondly, it is

evident from the results that a schedule with a lower proportion of overtime,

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produces higher idle time. This would be expected, as the schedule with more

‘robustness’ or buffer would expect fewer overrun surgeries, but at the expense of

unused capacity.

Comparing the models, one may see that the normal model produces less overtime

than the lognormal model, but a much higher percentage of unused capacity. This is

due to the difference in shape of the distributions and illustrates why it is important

to use an appropriate statistical distribution for modelling surgical durations.

Although both distributions may be used for the robust technique, the lognormal

distribution uses a more efficient amount of time to assign to the patients and

achieves better results.

Table 8. Simulation results comparing robust assignment models

MODELS Number of times simulated schedules

exceed assigned time (%)

Difference between simulated time and

assigned time expressed as idle time (%)

Non-flexible Lognormal 14.89 % 37.94 % Non-flexible Normal 11.54 % 42.32 %

Following the comparison of the LDSD and NDSD models, analytic hierarchy

process (AHP) was used to determine the 10 best assignments from the 100 examples

for the Ophthalmology patients. AHP is a technique used to assist with the decision

making process when there are multiple objectives affecting the decision (Winston,

2004). It provides a method for quantifying decisions particularly when the units of

measurement are very different. For example, for the assignment models, the highest

proportion of patients per block, the lowest maximum deviation value and the lowest

total deviation value were used as objectives for choosing the best schedules. The

proportion of patients per block and the deviation values have different units and

therefore it is difficult to compare these as they are. By using AHP, the decision

maker can choose between the different schedules and these chosen assignments

could be used to simplify patient assignment according to the scheduler’s

requirements.

109

The first step of AHP is to determine the weight or priority of each objective, from a

pairwise comparison matrix. A comparison matrix is derived where the entry in row

i and column j (aij) indicates how much more important objective i is than objective j.

Using the comparison matrix, the weight given to an objective can be determined.

For example, entry a13 = 3, indicates that objective 1 is weakly more important than

objective 3. Therefore a13 = w1/w3 = 3. Four alternative comparison matrices were

developed and an example is given in Table 9. The first of these ranks patients per

block slightly higher than maximum deviation and total deviation (which in turn are

ranked equal). The second ranks all objectives equally. The third ranks patients per

block strongly more important than maximum and total deviation and maximum

deviation is ranked slightly higher than total deviation. The fourth ranks maximum

deviation slightly higher than both patients per block and total deviation. Variations

on these priorities may also be tested but will not be discussed here. The resultant

weights based on the comparison matrices are given in Table 10.

Table 9. Example comparison matrix for alternative objectives.

Comparison Matrix Patients/Block Maximum Deviation Total Deviation

Patients/Block 1 3 3

Maximum Deviation 1/3 1 1

Total Deviation 1/3 1 1

Table 10. AHP Objective weights for alternative priorities

Objective weight

a b c d

Patients/block 0.6 0.3 0.7938 0.2

Maximum deviation 0.2 0.3 0.1394 0.6

Total deviation 0.2 0.3 0.0667 0.2

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The next step in AHP was to compute the scores on each objective for the different

assignments. These scores are computed by generating pair wise comparison

matrices comparing each schedule for each of the objectives. Thus, a 100x100

comparison matrix is computed for each of the three objectives. The ratios of

patients per block comparing two schedules were used to generate the first pair wise

comparison matrix, where the entry in row i and column j (aij) indicates how much

better schedule i is than schedule j. For example, the number of patients per block

for schedules 1 and 2 were 6.80 and 6.40 respectively. Therefore entry

06251406

80612 .

.

.a == . For both of the deviation objectives, the inverse relationship

was used because the minimum values were required. For example, the maximum

deviation for schedules 1 and 2 were 36.35 minutes and 58.57 minutes respectively.

Therefore, pair wise comparison matrix entry 611315758

3536112 .

.

./a == means that

schedule 1 is slightly better than schedule 2 for minimising the maximum deviation.

From the three pair wise comparison matrices, the scores on each objective, for each

of the schedules were computed. To determine the overall score for each schedule,

take the sum of the scores for each objective multiplied by the objective’s weight.

The schedule with the highest overall score is chosen.

Based on the results of AHP, the first 5 schedules chosen for all four alternative

priorities are the same (with slight variations in their order). The 10 best schedules

for each of various weights, are presented in Table 11. The two best schedules are

clearly schedules 90 and 85 for all instances. Schedules 51, 92 and 49 round out the

top five, albeit not necessarily in that order.

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Table 11. AHP Results for alternative priorities

Priority a b c d

Rank Schedule number

1 90 90 90 90

2 85 85 85 85

3 49 49 51 51

4 51 51 49 92

5 92 92 92 49

6 42 42 42 13

7 76 76 72 42

8 13 13 13 76

9 72 72 18 17

10 17 17 10 72

Table 12. Schedule details for top 5 AHP selected schedules

Schedule Number 90 85 51 92 49

Number of Specialty 1 4 7 1 7 2






Patients per block 5.75 6.50 7.25 6.25 7.00

Maximum Deviation (mins) 5.32 6.37 8.68 9.65 11.75

Total Deviation (mins) 8.89 19.39 26.04 23.32 18.52

Table 12 indicates the number of patients of each type of specialty that were assigned

and the performance measures for the top 5 schedules (out of 100, as determined by

AHP). A scheduler may use this information to determine surgery assignments for

the same mix of specialties. For example, schedule 90, which was determined as the

best of the 100 schedules, assigns 4, 4, 0, 5, 5 and 5 patients from specialties 1 – 6

respectively. The total number of surgery blocks used for this schedule is 23/5.75 =

4. The maximum deviation of the 4 surgery blocks is 5.32 minutes and total

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deviation is 8.89 minutes. This would be useful for the real life for saving time on

generating schedules and the approach could be used for other mixes of specialties.

One potential flaw would notably be specific specialties receiving preference (or no

preference at all) if a particular assignment is regularly used. For example, schedule

90 assigns no patients of specialty type 3. This schedule could not possibly be used

when specialty 3 patients are required. By using a variety of schedules, however,

this obstacle could be overcome.

4.8.2 Results for the flexible robust surgery assignment model with


The same 100 examples were used to examine the difference between performances

of the constructive heuristics when assignment of patients is flexible. After the

initial assignment and search heuristics assign the fixed number of patients to the

surgery blocks, the second assignment heuristic is implemented, followed again by

local search heuristics. The effect of this is to fill any unused capacity in the surgery

blocks, with additional patients. The effects on the performance measures are

presented in Table 13.

The first thing to note for the flexible results is the same number of surgery blocks

are used as with the fixed case, however the unused time from the non-flexible case

may now be filled with additional patients using the flexible model. All performance

measures are improved for the flexible case illustrating the benefit of allowing

flexibility in patient assignment. The average maximum, minimum and total

deviations were reduced 24.9 minutes, 17.48 minutes and 109.2 minutes respectively.

As a result in the improvement in planned use of the surgery blocks, an additional

407 patients are assigned across the 498 surgery blocks!

In addition to examining the theoretical differences, simulation was used to examine

the performance of the flexible model compared with the fixed model. Although the

flexible model produced a vast reduction in idle time, there was a significant increase

in the proportion of simulated schedules that used more time than was assigned. This

proportion of schedules using more time than assigned exceeded the desired level of

15.87%. This cannot be explained by the schedule differences because the simulated

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time is compared against the assigned time, not the capacity. This means that

differences in these proportions are most likely due to the nature of simulation and

variability in treatment times.

Table 13. Comparison of fixed and flexible assignment methods

Results Fixed Flexible Maximum deviation

(mins) 47.24 22.34

Minimum deviation (mins)

23.37 5.89

Total deviation (mins) 175.75 66.55

Average

Patients per block 6.07 6.88 Number blocks used 498 498 Total

Patients 3014 3421 Number of times

simulated schedules exceed assigned time

14.89% 23.40%

Simulation results Difference between

simulated time and assigned time expressed

as idle time

37.94% 19.87%

Striking an appropriate balance between overtime and theatre idleness would require

further investigation with respect to cost analysis contrasting the cost of overtime and

unused capacity and the profit earned for various types of surgeries. Overall, the

results of Table 13 suggest that the flexible model produces vast theoretical

improvements over the fixed model, however, the simulation showed that this may

be at the expense of an increase in the number of overrun theatres. The usefulness

would then depend on the desirability of overtime against underused capacity.

After establishing the effects of using a flexible assignment approach for robust

scheduling, alternative constructive heuristics were used to sort the patients and

determine the order in which they are assigned to the surgery blocks. The effect of

sequencing patients by the variance of the expected treatment time is examined next

and compared with common approaches, SPT and LPT, which are based on expected

treatment time. The results are presented in Table 14.

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Table 14. Comparison of constructive heuristics

Results HVF LVF SPT LPTAverage maximum deviation (mins) 22.34 18.76 15.44 23.41Average patients per block 6.88 6.90 7.16 6.80Average total deviation (mins) 66.55 54.62 39.73 76.88Total patients assigned 3421 3390 3549 3364Total blocks used 498 493 496 495

The constructive heuristics are compared for average maximum and total deviation,

average patients assigned per block and the total number of patients assigned and the

total number of blocks used. The first step in the solution procedure is to assign the

fixed number of patients to the surgery blocks. To assign the fixed number of

patients, lowest variance first (LVF) uses the fewest number of surgery blocks,

followed by LPT then SPT, whilst highest variance first (HVF) uses the most. After

assigning the fixed number of patients, the remaining theatre capacity is filled by

implementing the secondary assignment algorithm.

By filling in the remaining capacity, SPT goes from the schedule with the lowest

ratio of patients per block to the schedule with the highest. Because SPT used the

most surgery blocks in the original assignment, it had the highest amount of unused

capacity and therefore has the highest potential for adding additional patients. On

top of this, because patients with the lowest treatment time are assigned first, this

means that more patients can be assigned than with any other constructive heuristic.

This can perhaps be explained better numerically. Suppose there are two patients

that require 30 minutes each and a single patient with a 1 hour treatment time. If 1

hour of additional capacity needs to be filled, this could be filled with the two shorter

duration patients using the SPT rule, or the single patient using the LPT rule. Further

evidence to support this suggested finding comes from the LPT results that show it

produces the lowest ratio of patients per block.

While not performing quite as well as SPT, LVF also performed notably well

compared with HVF and LPT. It produced the second lowest deviation scores and

also the second highest ratio of patients per block. It is also noted, that for the fixed

patient assignment, it used the fewest number of surgery blocks, so when patient

numbers are fixed, LVF is perhaps the best constructive heuristic to use. Due to

115

these results, further exploration was done as to whether a hybrid heuristic using the

SPT logic to fill remaining capacity in the LVF schedule, would improve the

schedule results.

Table 15. LVF results when SPT logic is used to assign additional patients

The results presented in Table 15 for the hybrid heuristic indicate that the average

maximum deviation value may be reduced by over 3 minutes to perform as well as

the SPT alone heuristic and an additional 76 patients are scheduled across the 493

surgery blocks. As a result of this increase, there is an increase in the average

number of patients per block. The average total deviation is also reduced by over 10

minutes. These results indeed suggest that the hybrid heuristic using SPT to assign

additional patients could increase the efficiency of the LVF schedule. There is very

little difference between the SPT and SPT-LVF hybrid heuristic performance

measures. The only measure that really differentiates these two is the number of

surgery blocks used. Because LVF uses fewer blocks to assign the fixed number of

patients than SPT, the SPT-LVF hybrid heuristic uses fewer blocks overall.

4.9 Conclusions

The lognormal distribution was shown to be a better choice over the normal

distribution for modeling surgery durations when planning the elective surgery

schedule. When assigning the patients to the surgery blocks, the variance of

expected surgery duration was addressed using a robust assignment approach. The

proposed method of modelling surgery durations with the lognormal distribution was

compared with using the normal distribution. Surgery block deviation was also

introduced and used as a measure of surgical block utilization that incorporates both

earliness and tardiness.

Results LVF original LVF plus SPTAverage maximum deviation (mins) 18.76 15.52Average minimum deviation (mins) 4.93 3.34Average total deviation (mins) 54.62 41.04Average patients per block 6.90 7.06Total patients assigned 3390 3466

116

Results showed that while it was possible with either model to assign patients to the

surgery blocks using the proposed heuristics to minimise the maximum deviation, the

case with lognormally distributed surgical durations required fewer surgery blocks to

achieve this. In the long term, using fewer surgery blocks to assign the same number

of patients would be mean more surgeries could be planned in a year just by

choosing an appropriate statistical distribution to model surgical durations. As a

result, the proposed method of robust patient assignment under the assumption of

surgical durations modelled with the lognormal distribution is more efficient and

more suitable than approximating with a normal distribution.

The NDSD and LDSD models were further compared using simulation, which

showed that both models resulted in overtime below the desired proportion. The

NDSD model however, resulted in a higher level of theatre idle time due to the

differences in shape to the distributions.

One of the limitations of the robust scheduling model is the assumption that a

lognormal distribution may be used to approximate the sum of independent but not

necessarily identical lognormal random variables. This assumption is made to

simplify the calculation of treatment time for the surgeries assigned to a surgical

block. Future work could include investigating the use of a more accurate

distribution for approximating this sum.

Analytic hierarchy process (AHP) was used to select the best 10 schedules from the

100 results based on the number of patients assigned per block, maximum deviation

and total deviation performance measures. Variations in preference for the three

objectives resulted in the same selection of schedules. These chosen schedules could

be used to simplify patient assignment according to the scheduler’s requirements.

Rather than running the robust scheduling model every time a schedule is created a

combination of the selected schedules could be used.

The findings of the fixed robust assignment models also demonstrated the possibility

for improvements in deviation value by increasing the flexibility of the model by

allowing for the assignment of more patients to the surgery blocks in addition to the

117

fixed number of patients. This was supported by the results of the flexible

assignment model. All measures of performance were improved when patient

assignment was not fixed. This demonstrated the potential improvements in the

number of patients that can be scheduled as well as the reductions in deviation

between planned and available time that could be made if such an assignment model

was used for the real-life setting.

Contrasting the fixed and flexible models using simulation showed that the flexible

model now produced a higher proportion of overtime (above the desired level) but a

vastly reduced amount of theatre idleness.

When the constructive heuristics were initially compared, SPT was the best

constructive heuristic followed by LVF. When compared for the number of patients

assigned per block and total and maximum deviation it performed better than all

other constructive heuristics. It was also noted that SPT could possibly be used for

the second assignment algorithm regardless of which constructive heuristic was

originally applied for the initial assignment. The reasoning behind this was to assign

the shorter surgeries to fill the remaining theatre capacity, to increase the number of

patients assigned per surgery block. This idea was applied to the LVF schedule, as it

was the best constructive heuristic for the fixed number of patients, creating an SPT-

LVF hybrid heuristic. The hybrid heuristic greatly improved the results of the LVF

schedule that matched the results of the SPT schedules. These results supported the

idea of using SPT to assign additional patients to increase the efficiency of the

constructive heuristics.

118

Chapter 5. Reactive Scheduling

Robust scheduling was used in the previous chapter to assign patients to the surgery

blocks, generating offline schedules that take into consideration the variability of

patient treatment time. The motivation behind robust schedules is to reduce the

effect of disruptions during implementation in the real life. However, offline models

do not indicate how to deal with disruptions that cannot be handled by a robust

schedule alone. For example, decisions must be made regarding the arrival of

emergency patients that are not in the original schedule.

In the manufacturing scheduling environment, disruptions result in an interruption of

the job being processed by the machine. Usually these occur due to machine

breakdowns or job preemptions, where a higher priority job must be processed. In

the operating theatre, we address job disruptions that necessitate the adjustment of

the original schedule. Two types of disruptions are defined; a theatre (machine)

disruption and patient (job) disruption. Theatre disruptions occur when a theatre

becomes unavailable for some period of time. Examples include equipment failure

or staff shortage/unavailability or the arrival of a high priority emergency patient that

requires the use of an elective theatre. This type of disruption results in the patients

that were initially scheduled for that theatre (and have not yet been treated) to be

delayed and the schedule must be updated to take into account such changes. Patient

disruptions on the other hand occur when treatment times are less than or greater

than the assigned surgery time. If surgery duration is shorter than expected, this

generally means the schedule can be moved forwards without much alteration. Also,

there may be time left at the end of the schedule for adding on emergency cases, or

the additional time may be spent on a patient that exceeds its expected duration. If

patients exceed their expected duration however, this can cause delays in surgery

start time for remaining patients or may even necessitate cancellation of remaining

surgeries to prevent overtime of the theatre.

Two reactive modelling approaches are described. The first model developed

addresses patient assignment but does not consider patient sequence. This model is

formulated similar to the robust scheduling models, and uses a post disruptive

119

management policy, (discussed in Section 2.7) combined with a repair of the existing

schedule. The objective is to minimise the cancellations of electives and maximise

the number of emergency patients added to the schedule. This is achieved by

keeping track of the immediately preceding schedule, before the completion of an

operation. The second reactive scheduling model considers both assignment and

sequencing and also uses a post disruptive management policy. The model is

formulated as a single machine scheduling problem and a full rescheduling approach

is used. Two approaches to modelling surgical durations are discussed. The first

method assumes surgical durations are based on the expected value (from the

appropriate lognormal distribution), whereas the second uses the robust approach,

and assigns a buffer with each patient.

5.1 The robust reactive assignment model

For the reactive assignment model, the number of patients assigned to a single

theatre in the online environment is considered. At each surgery completion, the

reactive model is resolved taking into account any online disruptions that may have

occurred, including any emergency patients that need to be added to the schedule.

Although the model does not directly take into account machine disruptions, they can

be handled indirectly using patient disruptions. For example, say the first patient is

delayed half an hour due to a late start, then the half hour is incorporated into its

treatment time.

Schedules determined in the offline environment using the flexible assignment

model, (results presented in Section 4.8.2) are used as baseline schedules for testing

the reactive assignment model. Changes to the offline schedule during project

implementation are minimised using an online scheduling model that operates in

real-time. The model aims to minimise cancellations of pre-scheduled elective

patients whilst also allowing for additional scheduling of emergency cases, (time

permitting), which may arise during the schedule’s implementation. Surgical

durations are modelled with a lognormal distribution and the single theatre case is

solved.

120

The robust reactive assignment model (RRAM) is implemented at the completion of

each patient’s surgery, which may or may not differ from the expected completion

time. Patient priorities are taken into account allowing for pre-emption by an

emergency patient when necessary. This model is similar to the robust assignment

model in that patients are grouped by specialty. However, as the model is developed

for a single operating theatre, one should note the change in indices. In addition,

patient priorities and type (elective or emergency) are now included in the model.

The model

Indices

: Specialties considered within a surgical category, {1,.., }i i I∈

: Priority, {1,.., }j j J∈

: Type of patient (elective or emergency) , {1,.., }k k K∈

Parameters

idi specialty ofduration surgical Expected :

2 : Expected surgical duration variance of specialty is i

iii specialty for parameterson distributi Lognormal:,σµ

kjC jk type,priority ofbenefit Cost / :

Decision Variables

: The sum of the expected surgical durations assignedM

: The variance of the sum of the expected surgical durations V

121

: Time remaining in the theatreT

: The number of available patients of specialty priority and type

E i, j kijk

: The number of initially assigned patients of specialty priority and type

X i, j kijk

' : The number of assigned patients of specialty priority and type following

the reschedule.

X i, j kijk

Objective function

The objective of the reactive schedule is to minimise the costs of the new schedule.

These costs include the cost of cancelling a patient, the profit (or negative cost)

earned for adding a patient and a penalty for deviating from the original schedule.

The penalty incurred for cancelling a patient (or profit earned for adding one)

depends on their priority level j and type k. This is represented by ( )'jk ijk ijkC X X− .

If ( )'ijk ijkX X− is positive, then at least one patient has been cancelled and a penalty,

jkC is incurred for each cancellation. If ( )'ijk ijkX X− is negative, then at least one

patient of specialty i, priority j and type k has been added to the schedule and a profit

jkC per patient is subtracted from the objective function. The second aspect of the

objective function, 'ijk ijkX X− , prevents the addition of patients at the expense of

cancelling another. For example, one patient with penalty of 10 units could be

cancelled and replaced with two patients, each with a profit of 5 units. In practice,

this would generally be considered both impractical and destructive to the efficiency

of the schedule.

( )' '

1 1 1

I J K

jk ijk ijk ijk ijki j k

C X X X X= = =

− + −∑ ∑ ∑ (5-1)


122

The number of patients assigned in the new schedule cannot exceed the number

available. This allows for a cancelled elective patient to be re-assigned at a later

event.

' , , ,ijk ijkX E i j k≤ ∀ (5-2)

For model simplification surgical duration mean and variance of mean estimates are

assumed independent of priority level j and whether the patient is an emergency or

elective.

2

2i

ii ed

σµ +

= (5-3)

2222 1 iiii ees

σµσ +

−= (5-4)

The sum of the expected durations and the variance of the patients assigned to the

theatre are given respectively by

∑

∑ ∑=

= = =

I

i

iiJ

j

K

kijk eeXM

1

2

1 1

' σµ (5-5)

∑

−

∑ ∑=

=

+

= =

I

i

iiiJ

j

K

kijk eeXV

1

222

1 1

' 1σµσ

(5-6)

The amount of time that is planned for the patients assigned to the theatre depends on

the level of accuracy desired by the decision maker. The level of accuracy used for

the model is 15.87%, i.e. the probability that surgeries run overtime is less than

15.87%. Equation 5-7 ensures the time available in the theatre is sufficient to

complete all the assigned surgeries, based on this level of accuracy.

123

+

+≥

2

2

ln

2

2 M

MV

eMV

MT (5-7)

For a detailed explanation on the calculation of Equation 5-7, refer to Section 4.3.

Finally, the number of patients assigned to each theatre is a positive integer

' , , , ijkX Z i j k+∈ ∀ (5-8)

5.1.1 Solution approach and results

The RRAM is a combinatorial optimisation problem and is a variation of a bounded

knapsack problem in which the patients are the objects to be assigned to a theatre

(knapsack) with limited capacity. The number of patients that can be assigned of a

particular specialty, priority and type is bounded by variable, ijkE . Each patient takes

up a portion of the limited capacity and has a benefit, jkC associated with its

assignment. However, in this case, the amount of time assigned for a given patient

specialty is not fixed, but depends on the existing assignment of patients.

The generalised bounded knapsack problem maximises the benefit associated with

assigning the packed items. In our model, this is analogous to minimising

'

1 1 1

I J K

jk ijki j k

C X= = =

−∑ ∑ ∑

. In order to minimise the costs of our objective, one must also

solve the bounded knapsack problem, however, we have the added consideration of

minimising changes from the original assignment of patients. Therefore our problem

is just as computationally difficult as the bounded knapsack problem.

Knapsack problems have received much attention in the literature and are known to

be NP hard in the ordinary sense and may be solved in pseudo-polynomial time. An

algorithm that runs in pseudo-polynomial time has a running time that is polynomial

in the numeric value of the input. Thus, as the number of patients increases, so does

124

the computation time. However, in the case of the single operating theatre, the

number of patients is relatively small and therefore can be solved without the use of

metaheuristics.

Various exact solution methods exist including dynamic programming, branch and

bound and exhaustive methods. Dynamic programming requires the problem to be

broken up into stages that can individually be solved. Due to the robust calculations

of treatment time, dynamic programming cannot be applied to the RRAM. Branch

and bound, commercial software or an enumerative approach could be used for the

RRAM. The benefit of an enumerative algorithm over commercial software and

branch and bound, is that the code can easily be adapted to incorporate other

requirements of the scheduler, if they are deemed necessary. For example, if an

elective is cancelled relatively towards the beginning of a schedule, then the user

may opt not to fill in the remaining capacity with available emergencies at that point

in time. It may be better to wait until further surgeries are completed, because if they

complete early, this may allow the cancelled elective to be re-scheduled. If however

an emergency patient were allowed to take up the available capacity early on, there

would be no room for the cancelled elective. For this reason, an enumerative

algorithm is proposed (coding is provided in Appendix D) and implemented using

Visual Basic.

Implementing the model in Visual Basic has the additional benefit of a user-friendly

interface, designed for use within the practical setting. The model is run after the

completion of each operation. The user is prompted for information on the duration

of the completed surgery, the type of surgery (elective or emergency), and the

specialty and priority of the patient treated. In addition, information on emergency

patient arrivals can be added at any time.

The enumerative algorithm (detailed below) is used to search for the best schedule

based on the information supplied. The objective is to minimise changes from the

original schedule whilst also searching for the schedule that produces the best

objective value. This is important because it is not practical from a resource

viewpoint to make large changes to the original schedule. The algorithm fills any

available capacity with emergency patients where possible. Therefore at this stage,

125

the idea of postponing the assignment of an emergency because of the possibility of

reassigning an elective is not considered.

Reactive assignment algorithm

1. Remove the completed patient from the current schedule.

2. If the total amount of time used by the completed patients does not exceed the

total capacity

Go to step 3.

Else

End.

3. If the amount of time required for the remaining patients in the schedule exceeds

the total remaining capacity

Go to step 4.

Else

Go to step 5.

4. Do while the amount of time required for the patients in the schedule exceeds the

total remaining capacity

Remove the patient from the schedule that minimises the penalty imposed on

the objective function.

Loop.

Go to Step 5.

5. Do while the amount of time required for the patients in the schedule is less than

the total remaining capacity

Add the patient from the unscheduled list of available patients, to the

schedule that maximises the improvement in the objective function, whilst

satisfying capacity constraints. If no such patient exists, Exit loop.

Loop.

End.

Results of the robust assignment model developed in Section 4.7.2 were used for

testing the RRAM. Specifically, the flexible assignments generated using the

hybridized constructive heuristic LVF-SPT were used. Surgical durations of the

patients are assumed to be from a lognormal distribution (determined by analysis of

historical data discussed in Section 4.1). These were randomly generated and

126

entered into the RRAM. For each example, the model assumes a single patient of

each type of emergency patient is available. In the real life case however, obviously

only the available emergency patients would be entered.

The results of 100 test cases are given in Table 16. The performance measures

presented are the number of elective patients originally scheduled that were

cancelled, the number of emergency cases performed, the number of electives

cancelled but later re-scheduled and the number of emergencies added to the

schedule that had to later be cancelled.

Table 16. Results of reactive schedules

Results indicate that across the 100 examples there were a total of 126 elective

cancellations and the RRAM was able to re-schedule 71 electives. The ability to re-

schedule patients, when an already completed patient uses less time than it was

allocated, illustrates the benefit of the robustness built into the model. In addition, by

allowing a calculated amount of extra time for each surgery based on a percentage

determined by the decision maker, the number of cancellations is kept low and

allows for additional emergency patients to be seen. In this case, the model saw 178

additional emergencies performed across the 100 test cases. This ability to schedule

additional patients ensures theatre capacity is used efficiently rather than being left

unused. In 34 cases, emergency cases that had been tentatively scheduled later had

to be cancelled due to lack of time. Allowing for their cancellation also helps to

maintain the efficiency of the theatre utilisation by preventing overruns. For the 100

test cases, only 16% resulted in over-run theatres.

The objective of the model is to minimise the costs of the new schedule by

minimising the number of elective cancellations and maximise the number of

electives that may be added to the schedule. Because the model is re-run after each

Results TOTALNumber Electives Initially Assigned 741Total Patients Completed 793Number Emergencies Completed 178Number Electives Cancelled 126Re-scheduled Electives 71Cancelled Emergencies 34

127

patient’s completion it does not keep track of the preceding objective function

values. The resulting schedule may be different if the objective function values were

carried through for each schedule. For example, the number of elective cancellations

may possibly be reduced with an accompanied decrease in emergencies added to the

schedule.

Other changes to the schedule results could be induced by changes in the scheduler’s

objectives. For example, as mentioned earlier in the chapter, if an elective is

cancelled relatively towards the beginning of a schedule, then the user may opt not to

fill in the remaining capacity with available emergencies at that point in time and

wait until later in the schedule.

In addition to measuring performance indicators, changes in the schedule may be

presented in Gantt charts. Figure 26 illustrates the Gantt chart for one of the

schedules. One might notice in the Gantt chart the variation in treatment times

assigned for patients from one rescheduling step to the next. This is due to the buffer

used for calculating the amount of time that is assigned to the patients in a list. The

way in which individual treatment times are calculated, is as follows. For a schedule

with 8 patients, the total treatment time including the buffer is calculated (let this

time be P8). To calculate the time for the first patient, the total treatment time for the

other 7 patients is calculated (let this time be P7), and subtracted from the time for

the 8 patients (i.e P8 – P7). Overtime, the amount of time assigned to each patient

will reduce.

128

Patient12345678912345679

102345679

10345679

1045679

105679

10679

1079

109

10

10 Final RunTime (mins)

Eighth Run

446236 291 332 4080 75 120 149 189

Initial schedule

First Run

Second Run

480

Third Run

Fourth Run

Fifth Run

Sixth Run

Seventh Run

Figure 26. Gantt Chart illustrating results for one schedule

In the initial schedule, 9 elective patients are assigned to the theatre, each of which is

illustrated with a different colour. After each surgery is completed, the RRAM is

run and an updated schedule is generated. Following the late completion of the first

patient (in yellow), the RRAM shifts the original schedule to the right and post-pones

one elective patient (patient 8, in green) and fills the remaining capacity with an

emergency patient (patient 10, in orange). The next seven procedures (patients 2 – 7

and 9) finish either early or on time. No more emergency patients are added to the

schedule and patient 8 is not re-scheduled. After each of these procedures, the

RRAM is implemented and there is either a ‘left’ shift in the schedule (for an early

completion) or the schedule is unchanged (for on-time completions). The final

patient (patient 10, in orange) requires an additional 5 minutes of surgery time. It is

evident from the Gantt chart that this particular schedule completes ‘on-time’

meaning that it does not exceed the capacity of 480 minutes.

129

This Gantt chart highlights an important flaw with the robust assignment model.

Early on in the simulation, the elective patient in position eight is cancelled and an

emergency patient is added to fill the remaining capacity. The subsequent elective

procedures complete either early or on time, and the elective that was cancelled

could have been re-instated if the emergency patient were not added. This issue, and

the issue of patient sequencing, will be addressed with the bi-criteria reactive

scheduling model. In addition, only 5 minutes of overtime are used to schedule the

emergency patient. This model does not make decisions regarding whether to

schedule a patient that is predicted to be late, particularly towards the end of the

schedule. For example, in the case of the example illustrated, it may have been

possible to complete the elective patient that was cancelled. These decisions require

cost analysis comparing the benefit of performing a surgery against the cost of

overtime, and is left for future research.

5.2 The bi-criteria reactive scheduling models

The RRAM only considers the assignment of patients to the theatre, however does

not consider sequence of the patients. The reactive scheduling model applies single

machine scheduling techniques to the assignment and sequencing of patients in the

online environment. The weighted number of expected tardy jobs (patients

scheduled to complete outside theatre capacity) is minimised and the maximum

tardiness of the late jobs is minimised as a secondary constraint. Jobs have a

common due date and the expected treatment time is taken from the lognormal

distribution.

There are many reasons for considering dual criteria for this problem. In many

practical settings and particularly in the operating theatre, it is important to maximise

the number of jobs that meet their deadline. In the operating theatre, a tardy job may

be cancelled, which is not only expensive but causes dissatisfaction of the patient.

Thus minimising the number of tardy jobs should be a primary concern. It is also

important however, to minimise tardiness so that in the case of extending theatre

capacity (utilising overtime), the amount of overtime is minimised. In addition, a job

that requires only 10 minutes of overtime has a higher chance of being scheduled

than one that requires more time.

130

In addition to considering patient sequence, this model has improvements over the

first in a practical way. In many situations, although a disruptive event may occur, it

may not be necessary to do a full reschedule of tasks. For example, if a procedure

completes earlier than expected, then the remaining schedule can be left shifted’; i.e.

bringing the starting times forward in time, without any change in the sequence.

Several levels of disruptions are defined, following an event;

1. The original schedule, (or the left or right shifted schedule), is no longer

optimal or feasible. A full reschedule is required that satisfies a new set of

constraints.

2. The original schedule, (or the left or right shifted schedule), is no longer

optimal but is still feasible.

3. The original schedule, (or the left or right shifted schedule), is still optimal and

feasible.

The level of effect on the original schedule will affect the approach used to

‘reschedule’. Following a disruptive event, two steps must be followed; i) check for

feasibility; and 2) check for optimality. The first step is simple because this checks

whether the current schedule completes within the allocated time (available

capacity). The check for optimality is a more difficult step because we now may

have additional jobs (emergencies) to consider.

Because of the nature of the operating theatre scheduling problem it is not straight

forward to determine whether the schedule remains optimal. Not only do we have a

NP hard problem and additional patients to consider, but treatment times are

modelled with a lognormal distribution and buffers are calculated around the

treatment times. In fact, the problem would need to be resolved and the existing

solution would need to be compared with the optimal solution.

5.2.1 The models

For this problem, jobs have a common due date (the remaining theatre capacity) and

the expected treatment time is taken from the lognormal distribution. Two versions

131

of the problem will be considered that are analogous to a knapsack problem. The

theatre’s available operating time is the knapsack of given capacity and the patients

are the items to be loaded into the knapsack. Each patient has a priority represented

as a weight and the weighted number of tardy patients must be minimised, to ensure

the highest priority patients are scheduled on time. Patients that are tardy may or

may not be processed. This decision is made in the online environment by the

scheduler depending on many subjective factors including maximum overtime

allowance and expected duration of the procedure.

The results from the robust assignment models are used as baseline schedules for the

reactive models. The reactive model is used to sequence the initial assignment of

patients. As patients are treated in the online environment, emergency patients may

arrive. At the completion of each procedure, the remaining elective and new

emergency arrivals must be re-scheduled. Figure 27 illustrates this process.

This model differs from the robust and reactive assignment models in its formulation

and notations. Patients are no longer grouped by specialty, but are considered

individually. The priority of the patient is not included as an index, but is indicated

by a penalty. The decision variables have also changed and are now binary integers.

This model is more specific in its formulation, indicating the position in the sequence

of each patient. These changes in the model structure, particularly when compared

with the robust reactive assignment model in section 5.1, will improve the ability to

implement the model in the real life.

132

Time 0

List of patients (not sequenced) assigned to theatre using robust assignment model Initial reactive scheduling model sequences patients Emergency patients arrive during processing of Job 4. Run Reactive model at completion of Job 4

Time 0

1 2 3 4 5

1 2 3 4 5

1 2 3 5 6 7

1 2 3 5 6 7

Figure 27. Process chart for reactive scheduling model

In the RRAM, essentially, the model considers the reactive assignment problem as a

knapsack, in which the patients are fitted subject to the available capacity and to

minimise changes from the original schedule. The benefit of the updated model is

that these same issues are still considered, however they are approached differently.

The new objective is related with the tardiness of the late jobs. In this way, the

model can help the decision maker with choices regarding patients at the end of a

schedule. Secondly, all patients are given a position in the sequence regardless of

whether they are expected to be tardy or not. This handles the ‘nervousness’ that

was seen with the assignment model in 5.1, where patients may be ‘cancelled’ and

rescheduled over the course of the schedule. Rather than ‘cancelling’ a patient, the

procedure is marked as ‘tardy’, letting the decision maker know which of the jobs

133

may possibly need to be rescheduled at a later time. Thirdly, the solution approach

helps to minimise the changes to the original schedule, as opposed to making it an

objective, as will be seen in 5.2.2.

5.2.1.1 Fixed treatment time model

The first case assumes expected treatment times are taken from the lognormal

distribution based on historical data analysis.

Indices

numberPatient :i

This is the list of the patients to sequence. Any new arrivals (emergencies) are added

to the end of the list.

patient of sequencein Position :j

New patients are added to the end of the sequence. The position in the sequence

does not guarantee that there is sufficient capacity for them to be operated on.

Parameters

i:di patient ofduration surgical Expected

theatrein the remainingCapacity :C

tardyif patient ofPenalty i:Pi

Decision Variables

patients all of timesprocessing theof sum The :U

134

=Otherwise 0

position toassigned patient If 1 jiX ij

tardyis position in patient theIf 0

treatedis position in patient theIf 1

=j

jZ j

j:T j position in patient of Tardiness

: Total tardiness of the patientsT

: Minimum sum of weighted tardy patientsk

Objective function

The objective of the schedule is to minimise the total tardiness of the expected late

jobs.

1

J

jj

Min T T=

= ∑ (5-9)


The objective of the model is subject to the minimisation of the weighted number of

expected tardy patients. This is given by the following constraint where

∑≤≤=

I

iiPk

10 and k is equal to the minimum of the sum of the weighted tardy

patients.

( )1 1

1I J

i ij ji j

PX Z k= =

− =∑∑ (5-10)

Each patient must have a position in the sequence regardless of whether or not it is

treated.

135

iXJ

jij ,1

1∀=∑

= (5-11)

Each position can only be filled with one patient.

jXI

iij ,1

1∀=∑

= (5-12)

The completion time of the patient in the j th position is given as

1 1

jI

im ii m

X d= =∑∑ (5-13)

And the total amount of time required for all patients in the list is given by

1 1

I J

ij ii j

X d U= =

=∑∑ (5-14)

The number of patients that can be treated depends on whether there is sufficient

capacity. If the amount of time required for the first j patients, is less than the

available capacity, then those j patients will be treated. If however there is

insufficient capacity for patient in position j + 1, then patient in position j + 1 will

not be treated. For any patient in the j th position, that is not treated (i.e. is tardy) then

jZ will be 0. This is shown mathematically with the following constraint.

1 1

(1 )jI

im i mi m

X d C U Z= =

− ≤ −∑∑ (5-15)

The above constraint ensures that when jZ = 1, the patient in the j th position is not

tardy. When jZ = 0, the left hand side of Equation 5-15 may or may not be positive.

This means that used time may or may not exceed capacity. However, due to the

constraint given by Equation 5-10, the weighted sum of tardy patients must be

136

minimised. Therefore, the model would select jZ = 1 over jZ = 0. On the other

hand, it is impossible for used time to exceed capacity with jZ = 1.

The next constraint links tardiness variable jT with variable jZ as well as calculates

jT .

1 1(1 ),

jI

j im i mi m

T X d C Z m= =

= − − ∀∑ ∑

(5-16)

When the used time exceeds capacity, 1 1

jI

im ii m

X d C= =

−∑∑ > 0 and jZ = 0. Hence

1 1

jI

j im ii m

T X d C= =

= −∑∑ . When capacity exceeds used time, 1 1

jI

im ii m

X d C= =

−∑∑ < 0, jZ =

1 and therefore 0jT = .

Finally, there are binary integer constraints on variables ijx and jz .

0 or 1, ,ijX i j= ∀ (5-17)

0 or 1, jZ j= ∀ (5-18)

5.2.1.2 Robust model

The robust case is dynamic and complicated by calculations of treatment time of the

sequenced patients. The calculations of assigned surgery time include an amount of

slack for each patient and are not simply an addition of expected duration and

variance. In addition, one cannot simply divide the amount of buffer by the number

of patients and assign to each equally. Buffers are not equally distributed like this

because of variance differences between different specialties. The following

numerical example illustrates this problem and the approach that is used to assign the

buffer (and by extension, the total amount of treatment time) to each patient.

Three patients a, b and c with lognormal distribution parameters

,0.1 =aµ 5.1=bµ and 0.2=cµ and ,1.02 =aσ 15.02 =bσ and 2.02 =cσ , must be

137

sequenced on a single theatre. If each individual patient were assigned to their own

operating theatre, the amount of time required for each could be calculated using

Equations 5-3 to 5-7. This would give 3.72, 6.60 and 11.56 time units for each

patient respectively. If the sequence a, b, c is assigned to a single operating theatre,

then 20.06 time units would be required for patients a, b and c. For patients a and b

alone, 9.74 time units would be required. These calculations may be used to

determine the amount of time for each individual patient. Patient a, would require

3.72 time units, patient b would require 9.74 – 3.72 = 6.02 time units and patient c

needs 20.06 – 9.74 = 10.32 units. This illustrates the reduction in buffer required by

assigning the patients to a single operating theatre, over 3 individual theatres, and the

reduction in treatment time for the individual patients.

By calculating assigned treatment times in this manner, a further problem is

introduced. Patient sequence also affects calculations of individual assigned time.

Using the same numerical example as above, the alternative sequence a, c, b would

yield different treatment times for the individual patients. Patient a would require

3.72 time units, 10.96 time units for patient c and 5.37 time units for patient b.

Clearly, calculating assigned time per patient using this approach is affected by

patient sequence.

A number of additions and adaptations to the parameters, variables and constraints

are used for the robust version. The objective function Equation 5-9, constraints 5-

10 to 5-12 and binary constraints 5-17 and 5-18 remain the same.

Changed parameters

2is : Expected surgical duration variance of job i

jd : Amount of time assigned to the job in position j, given the current assignment of

jobs in positions 1 to j-1.

ii ,σµ : Lognormal distribution parameters for job i

138

Changed Variables

M : The sum of the expected surgical durations of all jobs in the list.

jsM : The sum of the expected surgical durations of jobs in sequence 1 to j.

V : The sum of the variances of the expected surgical durations of all the jobs in the

list.

jsV : The sum of the variances of the expected surgical durations of jobs in sequence

1 to j.

js : The current sequence of jobs in positions 1 to j.

jt : The amount of time assigned for jobs in positions 1 to j.

Changed Constraints

Equations 5-13 to 5-16 are replaced with all of the following equations.

The expected surgical duration, id and the variance, 2is of patient i are respectively

given by

2

2i

ii ed

σµ +

= (5-19)

2222 1 iiii ees

σµσ +

−= (5-20)

Although these are the same as Equations 4.3 and 4.4 given in Chapter 4, the index i

in this case represents patient number not patient specialty. iµ and iσ are lognormal

random variable parameters for patient i, determined by analysis of historical data.

139

The sum of the expected durations and the variances of all patients in the sequence

(regardless of whether they are tardy or not) are given respectively by

2

1

in

i

i eeMσµ

∑==

(5-21)

∑

−=

=

+n

i

iii eeV1

2221

σµσ (5-22)

The total amount of time required for all patients in the list is given by

UeMV

M M

MV

=

+

+2

2ln

2

2 (5-23)

The above equation is formulated such that the probability that surgeries run

overtime is less than 15.87%, assuming x = 1 (see Section 0).

As discussed at the beginning of this chapter, the time assigned to each patient is not

equal to its expected treatment time, but includes a ‘buffer’ component. The

following four constraints are used to calculate the amount of time assigned to each

patient, which was also noted to depend on patient sequence. The sum of the means

of the jobs in sequence 1 to j is

2

1 1,

j ni i

s ijjj i

M X e e jσµ

= == ∀∑ ∑ (5-24)

The sum of the variances of the means of the jobs in sequence 1 to j is

2 22

1 11 ,

jnii i

s ijji j

V X e e jσ µ σ+

= =

= − ∀∑ ∑ (5-25)

The completion time of the jobs in the sequence 1 to j is

140

2

ln22

2,

V Msj sj

Ms sj j

j

s sj j

Mt e j

V M

+

= ∀

+

(5-26)

The amount of time assigned to the job in position j, given the current assignment of

jobs in positions 1 to j - 1 is

1, j j jd t t j−= − ∀ (5-27)

The number of patients that can be treated depends on whether there is sufficient

capacity. If the amount of time required for the first j jobs, is less than the available

capacity, then those j jobs will be treated. If however there is insufficient capacity

for job in position j + 1, then job in position j + 1 will not be treated. For any job in

the j th position, that is not treated (i.e. is tardy) then jZ will be 0. This is shown

mathematically with the following constraint.

(1 ), j jt C U Z j− ≤ − ∀ (5-28)

The next constraint links tardiness variable jT with variable jZ as well as calculates

jT .

( )( )1 , j j jT t C Z j= − − ∀ (5-29)

5.2.2 Solution approach

The problem being solved has two aspects. The first is to find the assignment of

patients that minimises the weighted number of tardy jobs. Knapsack problems

however are usually formulated as a maximisation problem, therefore this problem

can be restated as maximising the number of weighted jobs that complete on time.

141

This can be shown using Equation 5-10, where k is the sum of the number of

weighted tardy patients and Zj = 0 if patient j is tardy, 1 otherwise.

( )∑ ∑ =−= =

I

i

J

jjiji kZXP

1 11 (5-10)

Since k is the minimum sum of the number of weighted tardy patients, then the

maximum number of weighted patients that complete on time may be given by

rearranging 5-10 as follows

∑=

∑=

∑=

∑=

−=I

i

J

j

I

i

J

jijijiji kXPZXP

1 1 1 1. (5-30)

The LHS of Equation 5-30 is maximised when k is minimised and ∑=

J

jjZ

1is

maximized, i.e. as many patients are finished on time as possible.

Solving the minimum weighted tardy patients problem is therefore equivalent to a

0-1 (binary) knapsack problem, in which the total benefit of the assigned items is

maximised (in this case, the sum of the weighted tardy patients). Therefore the

solution to the maximisation problem also provides the solution to the associated

minimisation problem and vice versa.

To solve the first stage knapsack problem, the models illustrated in 5.2.1.1 and

5.2.1.2 need only a slight adjustment. The objective function of these models is

replaced with constraint 5-10. Therefore, the objective is no longer to minimise the

sum of the tardiness of the late patients, but to minimise the weighted tardy patients.

The 0-1 knapsack problem has been shown to be NP hard and may be solved in

pseudo-polynomial time using dynamic programming. Other solution approaches

include branch and bound, greedy algorithms and various approximation algorithms

(Martello & Toth, 1990).

142

The bi-criteria reactive scheduling problem is relatively small and usually involves

less than 10 surgeries to be scheduled. This problem can therefore be solved using

exact solution approaches in a reasonable amount of time. Dynamic programming

would seem a good choice for this problem, and for the initial model this is true.

However, for the robust version of the problem, stages cannot be separated because

the amount of time assigned for each additional patient depends on the surgeries

already assigned. Therefore, the key element for using dynamic programming (i.e.

separating the problem into stages) cannot be performed. For this reason, branch and

bound has been selected as the solution approach. When discussing the solution

approach for the reactive sequencing problem, attention is focused on the robust

version of the problem. The model with fixed treatment times may be solved using

the original algorithms on which our developed methods are based.

The Horowitz-Sahni (H-S) algorithm is a branch and bound algorithm for solving 0-1

knapsack problems (Martello & Toth, 1990). The Stuart-Kozan (S-K) algorithm is

developed for the robust version of the bi-criteria reactive scheduling model, based

on the idea of the H-S algorithm. In this section, proof that the S-K algorithm

provides the optimal solution is given as well as the algorithm itself.

The first element of proving optimality for the S-K algorithm lies with the ratios of

item benefit to item weight. The following theorem and proof are adapted from

Martello and Toth (1990) for the robust problem. The original proof applies to the

non-robust version.

The knapsack problem (KP), may be relaxed by removing the binary constraints on

the decision variables, ix , and replacing them with 10 ≤≤ ix , to give the linear

relaxation problem C(KP). The items are then inserted into the knapsack according

to decreasing ratio of penalty of tardiness of item i to duration of item i, if it is in the

j th position according to the current sequence 1,..,j, ik

i

d

p. (Therefore, the items that

provide the highest penalty per unit of treatment time are inserted first). The

following algorithm is used to sequence the patients.

143

Procedure SK – Sort:

The following new notations are required:

ijM : The sum of the expected surgical durations of the first j jobs if job i is in

position j.

ijV : The sum of the variances of the expected surgical durations of the first j jobs if

job i is in position j.

ijd : The treatment time of patient i in position j, given the current sequence 1 to j -

1.

( ) { }nN ,..,2,1=

( ) { }φ=js

Input: ( ) ( ) iiij p,,,n,N,s σµ

Output: ( )ijX

Begin

6. [initialise]

For i = 1 to n do

2

1ii

i eeMσµ=

222

1 1 iiii eeV

σµσ +

−=

+

+=

21

211

ln

211

21

1iM

iMiV

ii

ii e

MV

Md

End do

144

Find ( )Nqd

pq

q

q ∈

= ,max1

11 =qx

( ) ( ) { }qNN −=

( ) ( ) { }qss jj +=

1 0,iX i q= ∀ ≠

11 qdd =

11 qMM =

11 qVV =

7. [sort]

Do while nj to2=

( )Ni,eeMM iijij ∉∀+= σµ−

2

1

( )Ni,eeVV iiijij ∉∀

−+= σ+µσ

−222

1 1

( )Ni,deMV

Md

j

kk

ijM

ijMijVln

ijij

ijij ∉∀−

+= ∑

−

=

+

1

1

2

2

2

2

Find ( )Nqd

pq

q

q ∈

= ,max1

1qjX =

( ) ( ) { }qNN −=

( ) ( ) { }qss jj +=

0,ijX i q= ∀ ≠

qjj dd =

qjj MM =

qjj VV =

1+= jj

145

Loop

3. Output sequence (js ), ( )ijX , ( )jd

End.

The first step of the sorting algorithm determines the patient with the highest penalty

per unit treatment time. For each patient i, determine the duration of the patient if it

were scheduled in the first position. Calculate the ratio of penalty per unit treatment

time for each patient i if it were in position 1. The patient with the highest ratio is set

to variable q and is removed from the list of patients to schedule (N) and added to the

schedule list ( )js . For all other patients, variable 1ix is set to naught and variables

1d , 1M and 1V are determined.

The second step in the process determines the order of the remaining patients starting

from position j = 2 to n. For each patient still on list (N), its treatment time if it were

in position j is calculated. This is equal to the total time allocated to the jobs in

positions 1 to j, less the duration of jobs in positions 1 to j-1 (which have already

been determined). The maximum ratio of penalty per unit treatment time for each

patient i, given it is in position j, determines the next patient in the sequence. This

process continues until all patients are sequenced.

Using the sequence (js ), generated by the SK-Sort algorithm, the first item that does

not fit into the knapsack is called the critical item, s, i.e.

{ }min : js j t c= > (5-31)

(Where tj is the treatment time of jobs 1 to j, given by Equation 5-27)

Using the information on the critical item, the following theorem demonstrates the

optimal solution to the knapsack problem (KP).

146

Theorem III

The optimal solution X of C(KP) is

, for 1,.., 1 and 1,..,ij ijX X j s i n= = − = from SK-sort (5-32)

0, for 1,.., and 1,..,ijX j s n i n= = + = (5-33)

Find : 1 isq i X= = from SK-sort (5-34)

∑−=−

=

1

1

s

jjdcc (5-35)

cqs

s

Xd

= (5-36)

0,isX i q= ∀ ≠ (5-37)

Proof

Any optimal solution X of C(KP) must be maximal and 1

n

i ij

d X c=

=∑ . Assume that

1

1

+

+>j

j

j

j

d

p

d

p for all j and let some *X be the optimal solution of C(KP). Suppose for

some k < s that * 1kX < and so there must be some * qqX X> for at least one item

sq ≥ . Given a sufficiently small 0>ε , the value of *kX could be increased by ε

and *qX decreased by

q

k

d

dε . The objective function would thus be increased by

0>

−

q

kqk d

dppε , since

q

q

k

k

d

p

d

p> , which is a contradiction. Similarly, it can be

shown that * 0kX > for k > s is impossible. Therefore ss

cX

d= in the optimal

solution.

The second aspect of proving optimality of these algorithm lies with the branching

scheme used. From the optimal solution of C(KP), two branches or sub-problems

may be formed; one with 0qsX = the other with 1qsX = . When two new sub-

147

problems are formed by branching on any sub-problem, the left branch takes the

value 0. Also, because of the sorting of patients (jobs) in non-increasing order of

benefit per unit of treatment time, any solution to a sub-problem cannot exceed the

best solution of the branch from which it was created.

A variation of a deterministic 0-1 knapsack algorithm, called the Horowitz-Sahni

algorithm (Martello & Toth, 1990), is derived for the robust version of the problem.

The new algorithm is called the Stuart-Kozan algorithm. The robust version of the

bi-criteria reactive scheduling model is dynamic. Calculations of assigned time for

each patient depend on the number and type of patients already assigned to the

theatre. Because of this, after the assignment of each patient, the ratio of benefit to

weight (i.e. penalty to treatment time) may change. This means that after each

assignment, the sort order must be updated. The Stuart-Kozan algorithm is presented

below and the Visual Basic code is also provided in Appendix D.

The Stuart-Kozan algorithm

Assume the patients are sorted according to the SK-Sort algorithm. Patients are

therefore listed in non-increasing order of penalty per unit of expected treatment

time. According to this order, patients are numbered i = 1 to n. The initial sequence

is given by )( ks .

Procedure SK:

Input: )(),(),(,, iii vmpcn , ( )ks

Output: , ( )iz x

Begin

1. [initialise]

1

I

ii

z p=

= ∑

zz ='

01 =+np

1=i

2. [compute lower bound l]

148

find { }cdjr k >= :min

[ ][ ]

−−+∑=

−−

−

= 11

1

rr

rr

r

ikk dd

Pdcpl

If lzz −≤ ' then go to 5.

3. [perform a forward step]

While cdi ≤ do

ipzz −= ''

' 1iX =

1+= ii

loop

If ni ≤ then

0' =ix

1+= ii

End if

If ni < then go to 2.

End if

If ni = then go to 3.

End if

4. [update the best solution so far]

If zz <' then

'zz =

for nk to1= do 'kk xx =

End if

ni =

If 1' =nx then

npzz += ''

0' =nx

End if

5. [backtrack]

149

find { }1:max ' =<= kxikj

If no such j then return;

jpzz += ''

0' =jx

1+= ji

go to 2.

End.

The second aspect to the bi-criteria problem is to find the schedule that minimises

total tardiness of the tardy jobs. Using the solution to the knapsack problem, the

sequence will be determined using the SPT rule, i.e. in order of non-decreasing

treatment time. A proof of how SPT minimises the total tardiness is also given

below.

Theorem IV

SPT minimises sum of completion times, ∑=

n

jjC

1.

Proof

Take the sequence of n jobs sequenced by SPT and indexed such that ji pp < and

ji < . The completion time of any job in the sequence, k, is given by ∑==

k

iik pC

1

and nnn

jj pppnnpC +++−+=∑ −

=121

12...)1(

jn

jp)jn(∑ +−=

=11

Consider the sequence obtained by interchanging any two jobs, b and c, in the

sequence where cb pp < and cb < .

150

The SPT sequence gives the sum of completion times as

ncbn

jSPTj ppcnpbnpnnpC +++−++−++−+=∑

=...)1()1(...)1( 21

1,

By interchanging jobs b and c the sum of the completion times is

nbcn

jeinterchangj ppcnpbnpnnpC +++−++−++−+=∑

=...)1()1(...)1( 21

1,

To show that SPT minimises the sum of completion times it must be show that

∑>∑==

n

jSPTj

n

jeinterchangj CC

1,

1,

01

,1

, >∑−∑==

n

jSPTj

n

jeinterchangj CC

Thus ( )nbc ppcnpbnpnnp +++−++−++−+ ...)1()1(...)1( 21 -

( )ncb ppcnpbnpnnp +++−++−++−+ ...)1()1(...)1( 21 >0

( ) ( ) 0)1()1()1()1( >+−++−−+−++− cbbc pcnpbnpcnpbn

[ ] [ ] 0)1()1()1()1( >+−−+−++−−+− bc pbncnpcnbn

( ) [ ] 0>−+− bc pcbpbc

( ) [ ] 0>−−− bc pbcpbc

Since bc pp > this is true.

151

Theorem V

For the single machine scheduling problem with a common due date, SPT rule

minimises total tardiness,∑=

n

jjT

1.

Proof

∑ −=∑==

n

jjj

n

jj DCT

11)0,max(

Since only the tardy jobs are considered, tardiness can be simplified to

∑ −=∑==

n

jjj

n

jj DCT

11)(

∑ ∑−== =

n

j

n

jjj DC

1 1

Due dates are equal therefore

jn

jj

n

jj nDCT −

∑=∑== 11

Therefore since SPT minimises∑=

n

jjC

1(Theorem IV), it also minimises ∑

=

n

jjT

1for the

single machine problem with a common due date.

To summarise the solution method for the reactive sequencing model, the patients are

firstly sequenced using SK-Sort. Secondly, the on-time and tardy jobs are

determined using the SK-algorithm and finally, the tardy jobs are sequenced using

SPT.

152

5.2.3 Schedule results and conclusions

This section compares the results between the fixed treatment time and robust

schedules. Schedule comparisons are made for initial and final sequences, number of

patients completed, number of patients that use more time than was allocated,

number of emergencies completed compared with total that arrive during scheduling

period and the number of electives that are postponed.

The branch and bound heuristics and sorting algorithms were coded in Visual Basic.

Actual treatment times were generated using the lognormal distribution and

appropriate distribution parameters according to the surgery performed. The flexible

robust assignment model (FRAM), of Section 4.5.2, was used to determine initial

patient assignments. Sequences were determined for these schedules using the

branch and bound algorithms discussed in Section 5.2.2. These initial sequences are

presented in Table 17.

The FRAM assigns patients to theatres to minimise deviation of the time blocks.

Therefore some initial schedules may require slightly more capacity than what is

available. When determining the initial sequence of patients of these schedules, it is

assumed there is sufficient capacity to schedule all patients, i.e. that capacity equals

required time. Therefore the sequence is generated using the non-increasing ratio of

penalty per unit of treatment time. Differences in the initial sequence between the

fixed treatment time and robust cases are due to the calculations of treatment time.

As was demonstrated earlier, the robust case includes an amount of buffer in the

calculation and hence the sorting order is checked after each patient addition. In

98% of schedules however, the initial sorting order is the same.

153

Table 17. Comparison of initial patient sequences for fixed treatment time and

robust models

Deterministic Sequence Robust Sequence1 {2, 3, 4, 5, 6, 1} {2, 3, 4, 5, 6, 1}2 {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9}3 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}4 {1, 2, 3, 6, 7, 4, 5} {1, 2, 3, 6, 7, 4, 5}5 {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}6 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}7 {1, 2, 3, 4, 5, 6, 9, 7, 8} {1, 2, 3, 4, 5, 6, 9, 7, 8}8 {1, 2, 3, 4, 5, 6, 8, 7} {1, 2, 3, 4, 5, 6, 8, 7}9 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}10 {1, 2, 5, 6, 3, 4} {1, 2, 5, 6, 3, 4}11 {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9}12 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}13 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}14 {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}15 {1, 2, 6, 3, 4, 5} {1, 2, 6, 3, 4, 5}16 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}17 {1, 2, 3, 4, 5, 6, 9, 7, 8} {1, 2, 3, 4, 5, 6, 9, 7, 8}18 {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}19 {1, 2, 3, 4} {1, 2, 3, 4}20 {1, 5, 2, 3, 4} {1, 5, 2, 3, 4}21 {1, 2, 3, 4, 5, 7, 8, 6} {1, 2, 3, 4, 5, 7, 8, 6}22 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}23 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}24 {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}25 {1, 2, 3, 4, 5, 8, 6, 7} {1, 2, 3, 4, 5, 8, 6, 7}26 {1, 2, 3, 4, 5, 8, 6, 7} {1, 2, 3, 4, 5, 8, 6, 7}27 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}28 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}29 {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}30 {1, 2, 3, 4, 5, 6, 8, 7} {1, 2, 3, 4, 5, 6, 8, 7}31 {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9}32 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}33 {1, 2, 3, 4} {1, 2, 3, 4}34 {1, 2, 3, 4, 5, 6, 9, 7, 8} {1, 2, 3, 4, 5, 6, 9, 7, 8}35 {1, 2, 3, 4, 5, 6, 12, 7, 8, 9, 10, 11} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}36 {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9}37 {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9}38 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}39 {1, 2, 6, 3, 4, 5} {1, 2, 6, 3, 4, 5}40 {1, 2, 3, 4, 5, 7, 8, 6} {1, 2, 3, 4, 5, 7, 8, 6}41 {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}42 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}43 {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}44 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}45 {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10} {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10}46 {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}47 {1, 2, 3, 4} {1, 2, 3, 4}48 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}49 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}50 {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}

Schedule Number

Initial Sequence

154

Table 18. Comparison of final patient sequences for fixed treatment time and

robust models

Deterministic Sequence Robust Sequence1 {2, 3, 4, 5, 6, 7, 1} {2, 3, 4, 5, 6, 7, 1}2 {1, 2, 3, 4, 5, 6, 7, 8, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9}3 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15}4 {1, 2, 3, 6, 7, 8, 4, 9} {1, 2, 3, 6, 7, 8, 4, 9}5 {1, 2, 3, 4, 9, 7, 8, 5} {1, 2, 3, 4, 9, 7, 8, 5}6 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}7 {1, 2, 3, 4, 5, 12, 6, 9, 11, 7, 8, 10} {1, 2, 3, 4, 5, 12, 6, 9, 10, 11, 7, 8}8 {1, 2, 3, 4, 5, 6, 8, 7} {1, 2, 3, 4, 5, 6, 8, 7}9 {1, 2, 3, 9, 4, 5, 6, 7, 10} {1, 2, 3, 9, 4, 5, 6, 7, 10}10 {1, 2, 5, 6, 8, 3, 4, 7} {1, 2, 5, 6, 8, 3, 4, 7}11 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}12 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11} {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11}13 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}14 {1, 2, 3, 4, 9, 8, 10} {1, 2, 3, 4, 8, 9, 10}15 {1, 2, 6, 3, 8, 7, 9, 4, 5} {1, 2, 6, 3, 8, 7, 9, 4, 5}16 {1, 2, 3, 4, 5, 6, 14, 7, 8, 9} {1, 2, 3, 4, 5, 6, 14, 7, 8, 9}17 {1, 2, 3, 4, 5, 6, 9, 7, 8} {1, 2, 3, 4, 5, 6, 9, 7, 8}18 {1, 2, 3, 8, 4, 7} {1, 2, 3, 4, 8, 5}19 {1, 2, 5, 3} {1, 2, 5, 3}20 {1, 5, 2, 10, 11, 3, 4} {1, 5, 2, 10, 11, 3, 4}21 {1, 2, 3, 4, 5, 7, 8, 9} {1, 2, 3, 4, 5, 7, 8, 9}22 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}23 {1, 2, 8, 3, 4, 5, 6, 7} {1, 2, 8, 3, 4, 5, 6, 7}24 {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}25 {1, 2, 3, 4, 5, 8, 6, 7} {1, 2, 3, 4, 5, 8, 6, 7}26 {1, 2, 3, 4, 10, 5, 8, 11, 12, 6, 7, 9} {1, 2, 3, 4, 10, 5, 8, 11, 6, 12, 7, 9}27 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}28 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}29 {1, 2, 3, 4, 6, 5} {1, 2, 3, 4, 5, 6}30 {1, 2, 3, 4, 5, 6, 9, 10, 8} {1, 2, 3, 4, 5, 6, 9, 10, 8}31 {1, 2, 3, 4, 5, 6, 7, 8, 10, 9} {1, 2, 3, 4, 5, 6, 7, 8, 10, 9}32 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}33 {1, 2, 7, 8} {1, 2, 7, 8}34 {1, 2, 3, 4, 5, 6, 9, 7, 10} {1, 2, 3, 4, 5, 6, 9, 10, 11, 7}35 {1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 7, 8, 9} {1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 10, 11}36 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7, 8}37 {1, 2, 3, 4, 5, 6, 12, 7, 8, 11, 10} {1, 2, 3, 4, 5, 6, 12, 7, 8, 11, 9}38 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}39 {1, 2, 6, 3, 4, 5, 7, 8} {1, 2, 6, 3, 5, 4, 7, 8}40 {1, 2, 3, 4, 5, 7, 8, 6, 10} {1, 2, 3, 4, 5, 7, 8, 6, 10}41 {1, 2, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6}42 {1, 2, 3, 4, 5, 8, 6, 7} {1, 2, 3, 4, 5, 8, 6, 7}43 {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}44 {1, 2, 3, 4, 5, 10, 6} {1, 2, 3, 4, 5, 10, 6}45 {1, 2, 3, 4, 5, 6, 7, 8, 13, 9, 12, 11, 14} {1, 2, 3, 4, 5, 6, 7, 8, 13, 9, 12, 11, 14}46 {1, 2, 3, 8, 4, 10, 5, 6, 7} {1, 2, 3, 8, 5, 10, 4, 6, 7}47 {1, 2, 3} {1, 2, 3}48 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}49 {1, 2, 3, 4, 5, 6, 7, 14, 8, 9, 10} {1, 2, 3, 4, 5, 6, 7, 14, 8, 10, 12}50 {1, 2, 3, 4, 5, 7, 6, 9, 10} {1, 2, 3, 4, 5, 7, 6, 9, 10}

Final Sequence

Schedule Number

155

Table 18 indicates the final sequence of patients for each of the fixed treatment time

and robust schedules. Of the schedules tested, 78% of the final sequences for the

fixed treatment time model were the same as for the corresponding robust model.

Although the actual sequences were similar in most cases, the robust schedules were

more conservative and more likely to accurately predict overtime than the fixed

treatment time model. This may have practical benefits for making decisions on

whether or not to postpone a patient that has the potential to require overtime.

The schedules were also compared against each other and also against the robust

assignment model of section 5.1 for the number of patients completed and how many

of these were emergencies and how many electives were postponed (see Table 19).

From the results, there was very little variation between the fixed and robust

schedule. On occasions, the fixed schedule processed more emergencies than the

corresponding robust schedule and vice versa. Overall, the total number of

emergencies processed for the test cases was 65 and 62 respectively for the fixed and

robust models. The slight differences between individual cases are due to variations

in the sequencing of the operations.

The number of elective cases postponed for all schedules was 27 and 23 respectively

for the fixed and robust cases. So this indicates that while the fixed schedules

processed slightly more emergency cases than the robust models, this was at the

expense of elective postponements. The reason this occurred can be explained by the

difference in assignment of treatment time between the fixed and robust models.

Because the fixed treatment time model does not assign a buffer with each patient,

when an elective is postponed, it is more likely to be replaced with an emergency

patient. On the other hand, because the robust model assigns more time per patient,

the remaining capacity after a cancellation is less than the fixed treatment time

model, and therefore there is less chance of filling that capacity with an emergency

patient. As the day progresses, if surgeries complete earlier than expected, the

previously postponed elective patient can be re-scheduled. However, in the fixed

treatment time case, if an emergency patient has been scheduled, then there will not

be enough time to reschedule the postponed elective.

156

Both models performed better than the robust assignment model. For the same set of

schedules, a total of 398 patients were assigned for the assignment model compared

with 415 for the sequencing models. In addition, the assignment model assigned 80

emergencies at the expense of 59 elective cancellations. For the sequencing models,

only 27 and 24 electives were cancelled for the fixed treatment time and robust

versions respectively. This illustrates the improvement over the robust assignment

model.

Besides the differences previously discussed, the reason why there is very little

variation between the fixed treatment time and robust results for these performance

measures is most likely due to the fact that both schedules are based on the robust

assignment model results. As a result, the amount of time initially assigned to the

patients in the fixed treatment time model is equal to that assigned to the robust

model. If the fixed treatment time model were based on a fixed treatment time

assignment model, then the performance result would most likely not be so similar.

More initial elective patients assigned overall would be expected and as a result there

would be more elective cancellations.

The last issue to address is whether the fixed treatment time rather than the robust

version of the reactive model could be used in conjunction with the flexible robust

assignment model. There is a slight disadvantage already seen and discussed with

this idea – the slight increase in emergency cases at the expense of elective

postponements. The concept was further investigated by comparing the assigned

treatment time for the fixed treatment time and robust models with the actual amount

of time used in the simulations. From the results presented in Table 20 it is clear that

the fixed treatment time case underestimates actual treatment time for 74% of the

patients compared with 30% for the robust model. This again illustrates the benefit

of using the robust approach as the fixed treatment time model consistently

underestimates the surgical duration.

157

Table 19. Comparison of performance measures for fixed and robust schedules

against robust assignment model

# completed # emergencies completed # electives postponed1 7, 7, 6 1, 1, 0 0, 0, 02 9, 9, 9 0, 0, 1 0, 0, 13 11, 11, 9 1, 1, 0 1, 1, 24 8, 8, 6 2, 2, 0 1, 1, 15 8, 8, 5 3, 3, 0 1, 1, 16 10, 10, 7 0, 0, 0 0, 0, 37 12, 12, 9 3, 3, 0 0, 0, 08 8, 8, 9 0, 0, 3 0, 0, 29 9, 9, 6 2, 2, 0 0, 0, 1

10 8, 8, 5 2, 2, 0 0, 0, 111 8, 8, 7 0, 0, 0 1, 1, 212 12, 12, 11 1, 1, 0 0, 0, 013 8, 8, 7 0, 0, 0 0, 0, 114 7, 7, 5 2, 3, 0 1, 2, 115 9, 9, 7 3, 3, 2 0, 0, 116 10, 10, 9 1, 1, 2 1, 1, 317 9, 9, 9 0, 0, 1 0, 0, 118 6, 6, 8 2, 1, 3 2, 1, 119 4, 4, 3 1, 1, 0 1, 1, 120 7, 7, 6 2, 2, 2 0, 0, 121 8, 8, 8 1, 1, 1 1, 1, 122 8, 8, 10 0, 0, 3 0, 0, 123 8, 8, 11 1, 1, 5 0, 0, 124 6, 6, 6 0, 0, 2 0, 0, 225 8, 8, 8 0, 0, 1 0, 0, 126 12, 12, 11 4, 4, 3 0, 0, 027 8, 8, 10 1, 1, 3 0, 0, 028 8, 8, 10 1, 1, 4 0, 0, 129 6, 6, 8 2, 1, 4 1, 0, 130 9, 9, 5 2, 2, 0 1, 1, 331 10, 10, 10 1, 1, 2 0, 0, 132 7, 7, 8 0, 0, 2 1, 1, 233 4, 4, 5 2, 2, 3 2, 2, 234 9, 10, 9 1, 2, 2 1, 1, 235 13, 12, 11 3, 1, 0 2, 1, 136 8, 8, 11 0, 0, 4 1, 1, 237 11, 11, 10 3, 2, 2 1, 0, 138 7, 7, 6 0, 0, 0 0, 0, 139 8, 8, 7 2, 2, 1 0, 0, 040 9, 9, 9 1, 1, 3 0, 0, 241 6, 6, 10 1, 1, 5 0, 0, 042 8, 8, 9 1, 1, 2 0, 0, 043 5, 5, 6 0, 0, 3 0, 0, 244 7, 7, 6 1, 1, 0 1, 1, 145 13, 13, 12 3, 3, 1 1, 1, 046 9, 9, 10 3, 3, 4 0, 0, 047 3, 3, 3 0, 0, 0 1, 1, 148 7, 7, 7 0, 0, 1 1, 1, 249 11, 11, 9 1, 1, 0 3, 3, 450 9, 9, 10 4, 4, 5 0, 0, 0

Deterministic, Robust, AssignmentSchedule Number

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Table 20. Percentage of patients requiring more time than assigned

Schedule Number Deterministic Robust1 0.83 0.332 0.89 0.113 0.91 0.274 0.63 0.255 0.88 0.256 0.80 0.207 0.83 0.178 0.63 0.389 0.63 0.3810 0.63 0.0011 0.56 0.5612 0.75 0.3313 0.75 0.2514 0.86 0.4315 0.44 0.2216 0.90 0.5017 0.89 0.4418 0.67 0.3319 0.80 0.4020 0.86 0.1421 0.75 0.2522 0.75 0.3823 0.50 0.2524 0.67 0.5025 1.00 0.2526 0.67 0.2527 0.75 0.0028 0.50 0.1329 0.50 0.3330 1.00 0.7831 0.60 0.4032 0.86 0.5733 1.00 0.5034 0.80 0.4035 0.92 0.3336 0.78 0.4437 0.73 0.1838 0.86 0.4339 0.88 0.2540 0.67 0.2241 0.33 0.1742 1.00 0.1343 1.00 0.4044 0.86 0.4345 0.77 0.0846 0.56 0.2247 1.00 0.3348 1.00 0.2949 1.00 0.4550 0.11 0.00

Total 74% 30%

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5.3 Conclusions

An innovative online assignment model for a single Operating Theatre is developed

and solved. The model is run in real-time following the completion of each

operation and minimises cancellations whilst also allowing for additional scheduling

of emergency cases (time permitting), which may arise during the schedule’s

implementation. The problem is NP hard in the ordinary sense and hence an exact

solution approach was used. The model was developed and implemented using

Visual Basic.

In Section 4.3 it was illustrated how the choice of x affects the proportion of patients

that are expected to be completed on time. When x = 1, the model expects

approximately 85% of surgeries to be completed on time based on the amount of

slack assigned for each patient. This means that in 85% of cases, the actual time will

be less than or equal to the assigned time. The robustness of the model not only

helps to minimise the number of elective patients that are cancelled but also allows

for the addition of emergency cases. The likelihood of fitting in emergency cases

increases with the more surgeries that complete early. However if the proportion

expected to complete on time is too high, then the chance of under filling capacity

increases, especially if there are no emergency cases on a particular day. On the

other hand, if the proportion is too low, then capacity may be exceeded at a higher

rate and no emergencies may be scheduled and elective cancellations may result.

Determination of the optimal proportion depends on the management style of the

hospital, but from a theoretical view, could possibly be investigated using simulation

as a further issue of research.

The RRAM only considered the assignment of patients to the theatre, however did

not consider the sequence of the patients. The reactive scheduling model applied

single machine scheduling techniques to the assignment and sequencing of patients

in the online environment. Two criteria were used for the model; the first was to

minimise the weighted tardy jobs and the secondary constraint was the minimisation

of the tardiness.

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A branch and bound approach was used to solve the problem first assuming fixed

treatment time surgical durations and secondly, applying the more complicated

robust scheduling theory. Both models were able to reschedule the patients

following each completed surgery. The robust reactive approach was better for

predicting overtime whereas the fixed treatment time approach consistently

underestimated treatment times.

A number of areas of possible future work have been identified for the reactive

scheduling problem. Firstly, a cost analysis comparing the cost of running theatres in

overtime, the costs of cancelling patients and the profit earned for procedures could

be performed. This would have practical benefits for assisting decisions on whether

or not to postpone a patient that has the potential to require overtime. At present, this

is a subjective decision made by OT staff based on many factors including the

amount of capacity remaining in the theatre and the expected duration of the

procedure. Secondly, the reactive models developed in this chapter consider only a

single operating theatre. Future work will involve expanding on this work for the

multiple theatre case, which is NP hard in the strong sense.

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Chapter 6. Conclusions and future work

This thesis finishes with a summary of the work conducted, conclusions and possible

future work arising from the simulation and mathematical models.

6.1 Conclusions

Chapter 2 presents a literature review on the most relevant research related to the

problem. Included are discussions on general OT studies focusing on administrative

issues, simulation applied to healthcare, scheduling in the OT and an introduction to

robust and reactive scheduling techniques. This chapter illustrated specific gaps in

the literature (discussed in Section 1.4.1), which are addressed by this thesis (see

Section 1.5 Contribution of the thesis).

A simulation model is presented in Chapter 3 that explores the effects of changing

patient arrivals and elective theatre scheduling disciplines on the operating theatre

department’s performance measures. Simulation was chosen because of its ability to

immediately observe the effects that changes to parameters and inputs have on a

system and also its use as a visual tool for explaining model outputs to decision

makers. Model observations form decision support by predicting the effects changes

in the model’s parameters may have on a system.

Modelling of the OT department was achieved through the use of blocks. Service

distributions and emergency patient inter-arrivals were generated based on historical

patient data. Efficiency measures calculated for the real system were compared with

simulation outputs to validate the model. The model was used to explore the effects

that increasing patient arrivals and alternative elective patient admission disciplines

would have on the performance measures.

The relationships between the performance objectives and changes made on the

system observed during sensitivity analysis were explained by the underlying

assumptions of the system. Emergency patient waiting times and the number of

elective cancellations appeared to increase exponentially as the number of arrivals

was increased. The relationship between elective patient waiting time, elective

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theatre utilisation rate and emergency theatre utilisation rate with changing arrivals

appeared linear and less pronounced. There was no significant impact on elective

theatre utilisation rate for any change in arrival, which suggested that the elective

theatres were running at near maximal activity. This claim may be supported by the

exponential effect seen on the number of cancellations when arrivals increased by

30%, with no accompanied effect on elective theatre utilisation rate.

Constructive heuristics were used to generate schedules and the results of the

alternative methods were compared. Least flexible job (LFJ) first improved patient

waiting times and reduced the number of cancellations. Shortest processing time

(SPT) first improved patient waiting times, had a lesser impact on elective patient

waiting times than LFJ, and did not affect the number of cancellations. Longest

processing time (LPT) first improved emergency patient waiting times but was found

to increase both elective patient waiting times and the number of cancellations. The

improvement in emergency patient waiting times for all disciplines suggested this

was a result of reducing elective scheduling variability. The scheduling disciplines

did not have a significant effect on theatre utilisation rates.

An important result of this study was its potential implementation as decision support

for surgical planners wishing to adopt an alternative scheduling discipline or for

predicting the effects than an increase in patients will have on performance criteria.

Traditionally, theatre utilisation rate has been considered the only important

performance measure for the OT. This study clearly demonstrated, however, the

need to consider the effects on patient waiting times and elective cancellations as an

adjunct measure of performance.

While simulation provides a way to examine scenarios without actually impacting

the real life system, it does not optimise the system. For this reason, theoretical

models based on operation research techniques were also used with respect to

elective scheduling procedures and to develop a new tool for the dynamic online

scheduling problem incorporating emergency patients.

In Chapter 4, the data analysis for the robust and reactive models precedes the robust

assignment models. The first scheduling models that were investigated were the

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offline schedules for elective patients. Surgical durations are known to be variable

and depend on many factors including the type and severity of illness and the

experience of the surgeon performing the operation. Even with detailed statistical

analysis it is very difficult to predict how long a procedure will take. When the

assigned time for a surgery is less than the actual time, this can lead to either delayed

starts for remaining procedures, cancellations or overrun theatres, which is an added

cost to the running of the department. Robust patient assignment was used to address

this issue faced by schedulers.

The robust patient assignment models developed, used buffers to absorb treatment

time variation where the amount of slack planned for each block is based on the

variations in surgical durations of the patients scheduled in the block. The lognormal

distribution was verified as a better distribution choice for modelling surgery

durations than the normal distribution. The summation of lognormal variables

however, is not modelled by a normal distribution, as with the summation of some

other types of random variables. Using the sum of lognormal variables was a point

of innovation for robust patient assignment in the operating theatre.

An alternative performance measure to tardiness, namely deviation, was considered.

The deviation of a surgery block was defined as the difference between used and

available surgery block capacity. Costs are incurred from both over and under use of

the available theatre capacity. For this reason, deviation was introduced because it

considers both earliness and tardiness.

The innovative model that based surgical durations on the lognormal distribution was

compared with the case of normally distributed surgical durations. The resulting

schedules required fewer surgery blocks to assign the required number of patients

when a lognormal distribution was used. This suggested that choosing an

appropriate statistical distribution to model surgical durations would have positive

practical implications for elective surgery scheduling.

The findings of the fixed robust assignment models also suggested that using a

flexible approach to surgery assignment would improve the efficiency of the planned

elective surgery schedules. This was supported by the results of the flexible

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assignment model, which saw improvements in all performance measures when the

fixed and flexible models were compared. In real life, surgical consultants are

generally responsible for selecting and sequencing their own patients. It was shown

that adapting the robust patient assignment model to allow patients to be selected

directly from a waiting list, could improve the resultant schedules. This

demonstrated the potential improvements in the number of patients that can be

scheduled as well as the reductions in deviation between planned and available time

that could be made if such an assignment model was used for the real-life setting.

Innovative constructive heuristics based on surgical duration variance and meta-

heuristics were proposed for solving the model and then compared with traditional

approaches. The traditional scheduling heuristic SPT, followed by lowest variance

first (LVF), performed better than all other constructive heuristics when the number

of patients assigned per block and then the total and maximum deviation values were

compared. It was also noted that SPT could possibly be used for the flexible model

regardless of which constructive heuristic was originally applied for the initial

assignment of patients. The reasoning behind this was to assign the shorter surgeries

to fill the remaining theatre capacity, to increase the number of patients assigned per

surgery block. Using this idea, an innovative hybrid constructive heuristic, SPT-

HVF was developed, which performed as well as the SPT heuristic and used fewer

surgery time blocks overall to achieve the result. These results supported the idea of

using SPT to assign additional patients to increase the efficiency of the constructive

heuristics.

Analytic hierarchy process (AHP) was used to select the best schedules from the

results based on the number of patients assigned per block, maximum deviation and

total deviation performance measures. Variations in preference for the three

objectives resulted in the same selection of schedules. These chosen schedules could

be used to simplify patient assignment according to the scheduler’s requirements.

Rather than running the robust scheduling model every time a schedule is created, a

combination of the selected schedules could be used.

An important aspect of the robust models developed is that they are applicable

irrespective of whether patient assignment is surgeon specific or not. This means

165

that the method can be applied when theatre capacity is assigned to specific surgical

consultants or surgical teams, or in an open scheduling system where capacity is not

assigned in this manner. As a result, a generalised model may be easily adapted to

other hospital systems and can easily incorporate hospital specific requirements.

Robust scheduling was used to assign patients to the surgery blocks, generating

offline schedules that take into consideration the variability of patient treatment time.

Offline models, however, do not indicate how to deal with disruptions that occur in

the online environment.

Two types of disruptions are defined for operating theatre scheduling; a theatre

(machine) disruption and patient (job) disruption. Theatre disruptions occur when a

theatre becomes unavailable for some period of time. Patient disruptions on the

other hand occur when treatment times are less than, or greater than, the assigned

surgery time.

If a surgery’s duration is shorter than expected, this generally means the schedule can

be moved forwards without much alteration. Also, there may be time left at the end

of the schedule for adding on emergency cases, or the additional time may be spent

on a patient that exceeds their expected duration. If patients exceed their expected

duration however, this can cause delays in surgery start time for remaining patients,

or may even necessitate cancellation of remaining surgeries to prevent overusage of

the theatre.

Two types of models were developed in Chapter 5 that used the results of the offline

robust surgery assignment models as baseline schedules. The first was a reactive

assignment model that determined the number of patients that could be assigned to

the available theatre capacity. The objective was to minimise the number of elective

cancellations, maximise emergency additions, whilst also minimising the deviation

from the original schedule. This model was developed and implemented using

Visual Basic.

Results for the initial assignment model showed it was capable of adapting

appropriately to disruptions in the online environment by delaying, rescheduling or

166

adding additional surgeries according to the available operating time capacity. The

ability to re-schedule patients, when an already completed surgery used less time

than it was allocated, illustrated the benefit of the robustness built into the model. In

addition, by allowing a calculated amount of extra time for each surgery based on a

percentage determined by the decision maker, the number of cancellations was kept

low and additional emergency patients could be treated. This ability to schedule

additional patients ensured theatre capacity is used efficiently rather than being left

unused. Allowing for the cancellation of emergencies also helped to maintain the

efficiency of the theatre utilisation by preventing overruns.

The second reactive approach expanded on the first by addressing patient

sequencing. Two approaches to estimating surgical durations were used. The first

assumed surgical durations were fixed and based on the expected duration. The

second used the robust theory approach and incorporated a portion of slack in the

surgical duration calculations. Both models were developed using single machine

scheduling techniques. Two criteria were used for the models; the first was to

minimise the weighted tardy jobs and the secondary constraint was the minimisation

of the tardiness.

A branch and bound approach was used to solve both models. These models also

responded appropriately to the online disturbances that might occur in the operating

theatre with the added benefit of explicitly stating the sequence of patients. In

addition, the robust reactive approach was better for predicting overusage whereas

the fixed treatment time approach consistently underestimated treatment times.

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6.2 Future work

A number of potential avenues for future work were identified throughout this

research.

In section 4.3, the sum of independent lognormal variables was approximated with a

lognormal distribution. This was used to determine the amount of time (including a

slack component) that is assigned to the surgeries assigned to a theatre. It was

illustrated how the proportion of patients that are expected to be completed on time if

affected by this calculation. Increasing the slack component of this assigned time

lowers the expected overtime. The current model expects that in 85% of schedules,

the actual time will be less than, or equal to, the assigned time. The robustness of the

model not only helps to minimise the number of elective patients that are cancelled

but also allows for the addition of emergency cases. The likelihood of fitting in

emergency cases increases with the number of surgeries that complete early.

However if the proportion expected to complete on time is too high, then the chance

of under filling capacity increases, especially if there are no emergency cases on a

particular day. On the other hand, if the proportion is too low, then capacity may be

exceeded at a higher rate and no emergencies may be scheduled and elective

cancellations may result. Determination of the optimal proportion depends on the

management style of the hospital, but from a theoretical view, could possibly be

investigated using simulation as a further issue of research.

One of the limitations of the robust and reactive scheduling models is the assumption

that a lognormal distribution may be used to approximate the sum of independent but

not necessarily identical lognormal random variables. This assumption is made to

simplify the calculation of treatment time for the surgeries assigned to a surgical

block. Future work could include investigating the use of a more accurate

distribution for approximating this sum.

The objective of the reactive scheduling model (in particular the assignment model)

was to minimise the costs of the new schedule by minimising the number of elective

cancellations and maximise the number of electives that may be added to the

schedule. An exact solution approach was coded in Visual Basic to achieve this,

168

however, because the model is re-run after each patient’s completion it does not keep

track of the preceding objective function values. The resulting schedule may be

different if the objective function values were carried through for each schedule. For

example, the number of elective cancellations may possibly be reduced with an

accompanied decrease in emergencies added to the schedule. Other changes to the

schedule results could be induced by changes in the scheduler’s objectives. For

example, if an elective is cancelled towards the beginning of a schedule, then the

user may opt not to fill in the remaining capacity with available emergencies at that

point in time and wait until later in the schedule. Changes such as these would

depend on the specific end user and would form an interesting future study.

In addition to the above changes to the reactive model, more pertinent work would be

an analysis comparing the cost of running theatres overtime, the costs of cancelling

patients and the profit earned for procedures. This may have practical benefits for

assisting decisions on whether or not to postpone a patient that has the potential to

require overtime. At present, this is a subjective decision made by OT staff based on

many factors including the amount of capacity remaining in the theatre and the

expected duration of the procedure. Secondly, the reactive models developed in this

chapter consider only a single operating theatre. Future work will involve expanding

on this work for the multiple theatre case, which is NP hard in the strong sense.

169

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178

APPENDIX A

Floor Plan PAH OT Department

179

APPENDIX B

Figure 28. BE&T LOS distribution

fitting results

Figure 29. CARD LOS distribution

fitting results

Figure 30. COLO LOS distribution

fitting results

Figure 31. FMAX LOS distribution

fitting results

Figure 32. ENT LOS distribution

fitting results

Figure 33. HPB LOS distribution

fitting results

180

Figure 34. NSUR LOS distribution

fitting results

Figure 35. PLAS LOS distribution

fitting results

Figure 36. OPHT LOS distribution

fitting results

Figure 37. RTPT LOS distribution

fitting results

Figure 38. ORTH LOS distribution

fitting results

Figure 39. UGI LOS distribution fitting results

181

Figure 40. UROL LOS distribution

fitting results

182

APPENDIX C

Algorithms

Robust scheduling models

Initial assignment algorithm Inputs

jC,J,I

j,i ,0ijX ∀=

j ,0jD,jE,jT ∀=

iR

I to 1i For = {

1j

1k

==

∑=

<J

1jiRijX While

{ kijX =

)0,jCjUmax(jT −=

)0,jUjCmax(jE −=

jTjEjD +=

If 0jT ==

1kk += Else {

1ijx'ijx −=

)0,jC'jUmax('

jT −=

)0,'jUCmax('

jE −=

'jT'

jE'jD +=

If 'jDjD <

{

ijx'ijx =

jT'jT =

jE'jE =

jD'jD =

183

1kk += } Else {

'ijxijx =

'jTjT =

'jEjE =

'jDjD =

1jj +=

1k = } }

} 1ii +=

}

Neighbourhood search algorithm

I to 1i For

1i

IM

==

=

{ 0maxijX If =

1ii += Else {

1maxijX'maxijX −=

1minijX'minijX +=

)0,C'maxusedmax('

maxjT −=

)0,C'minusedmax('

minjT −=

)0,'maxusedCmax('

maxjE −=

)0,'minusedCmax('

minjE −=

'maxjT'

maxjE'maxjD +=

'minjT'

minjE'minjD +=

If UB'minjD,maxjDmax <

{

'maxijXmaxijX =

'minijXminijX =

184

'maxjTmaxjT =

'minjTminjT =

'maxjEmaxjE =

'minjEminjE =

'maxjDmaxjD =

'minjDminjD =

Call max-min algorithm Return } Else { For M to 1m = { If im ≠ { 0minmjX If =

1mm += Else {

1maxmjX'maxmjX +=

1minmjX'minmjX −=

)0,C'maxusedmax('

maxjT −=

)0,C'minusedmax('

minjT −=

)0,'maxusedCmax('

maxjE −=

)0,'minusedCmax('

minjE −=

'maxjT'

maxjE'maxjD +=

'minjT'

minjE'minjD +=

If UB'minjD,'

maxjDmax <

{ For mn1 ≤≤ {

'minnjXminnjX =

'maxnjXmaxnjX =

}

'minijXminijX =

'maxijXmaxijX =

'maxjTmaxjT =

'minjTminjT =

185

'maxjEmaxjE =

'minjEminjE =

'maxjDmaxjD =

'minjDminjD =

Call max-min algorithm Return } Else {

'maxjD'

minjE or 'minjD'

maxjT If >>

{ For M to 1mm += { If im ≠ {

0'maxmjX If =

1mm += Else {

1maxmjX'maxmjX −=

1minmjX'minmjX +=

)0,C'maxusedmax('

maxjT −=

)0,C'minusedmax('

minjT −=

)0,'maxusedCmax('

maxjE −=

)0,'minusedCmax('

minjE −=

'maxjT'

maxjE'maxjD +=

'minjT'

minjE'minjD +=

If UB'minjD,'

maxjDmax <

{ For mn1 ≤≤ {

'minnjXminnjX =

'maxnjXmaxnjX =

}

'minijXminijX =

'maxijXmaxijX =

'maxjTmaxjT =

'minjTminjT =

186

'maxjEmaxjE =

'minjEminjE =

'maxjDmaxjD =

'minjDminjD =

Call max-min algorithm Return } Else { 1mm += break } } } } }

Else 'maxjD'

minjT or 'minjD'

maxjE If >>

{ For M to 1mm += { If im ≠ {

0'minmjX If =

1mm += Else {

1maxmjX'maxmjX +=

1minmjX'minmjX −=

)0,C'maxusedmax('

maxjT −=

)0,C'minusedmax('

minjT −=

)0,'maxusedCmax('

maxjE −=

)0,'minusedCmax('

minjE −=

'maxjT'

maxjE'maxjD +=

'minjT'

minjE'minjD +=

If UB'minjD,'

maxjDmax <

{ For mn1 ≤≤ {

'minnjXminnjX =

'maxnjXmaxnjX =

}

187

'minijXminijX =

'maxijXmaxijX =

'maxjTmaxjT =

'minjTminjT =

'maxjEmaxjE =

'minjEminjE =

'maxjDmaxjD =

'minjDminjD =

Call max-min algorithm Return } Else { 1mm += break } } } } } } } } } For In1 ≤≤ {

minnjX'minnjX =

maxnjX'maxnjX =

} } 1i i= + } }

188

APPENDIX D

Visual Basic Code

Robust Reactive Assignment model

Input Emergency Patient Macro

Sub InputEmergency()

Data1 = InputBox("Enter emergency patient specialty")

Data2 = InputBox("Enter emergency patient priority")

ActiveSheet.Next.Select

Range("D23").Select

ActiveCell.Offset((Data1 - 1) * 3 + Data2 - 1, 0).Select

ActiveCell.Value = ActiveCell.Value + 1

ActiveSheet.Previous.Select

End Sub

Re-set Data Macro

Sub Macro5()

Range("D10:D27").Select

Selection.Copy


Range("B5").Select

ActiveSheet.Paste

Range("C5").Select

ActiveSheet.Paste

Range("D5").Select

ActiveSheet.Paste


Range("D29").Select

Range(Selection, Selection.End(xlDown)).Select

Application.CutCopyMode = False

Selection.Copy


Range("B23").Select

ActiveSheet.Paste

Range("C23").Select

ActiveSheet.Paste

Range("D23").Select

ActiveSheet.Paste


Range("E8:IV46").ClearContents

189

End Sub

Run Model Macro

Sub RunModel()

Capacity = Range("B5").Value

mygrab = InputBox("Enter length of last surgery (minutes)", "Grab Time")

Range("C8").Select

Selection.End(xlToRight).Select

CurrentTime = ActiveCell.Value + mygrab

ActiveCell.Offset(0, 1).Select

ActiveCell.Value = CurrentTime

CapacityRemaining = (Capacity - CurrentTime) / 15

Sheets("CHANGE CALCS").Select

Range("H12").Select

ActiveCell.Value = CapacityRemaining

Sheets("INPUT DATA").Select

TypeofPatient = InputBox("What type patient just completed? Enter 1 for Elective,

2 forEmergency.")

specialty = InputBox("What specialty just completed? Enter 1,2,3,4,5 or 6.")

Priority = InputBox("What priority just completed? Enter 1,2 or 3")

If TypeofPatient <> "" Then

If specialty <> "" Then

If Priority <> "" Then

If TypeofPatient = 1 Then

ActiveCell.Offset(2, -1).Select


Selection.Copy


Selection.PasteSpecial Paste:=xlValues, Operation:=xlNone,

SkipBlanks:= _False, Transpose:=False

ActiveCell.Offset((specialty - 1) * 3 + Priority - 1, 0).Select

ActiveCell.Value = ActiveCell.Offset(0, -1).Value - 1

Selection.End(xlDown).Select




Selection.Copy



SkipBlanks:= _False, Transpose:=False


Range("D5").Select


ActiveCell.Value = ActiveCell.Value - 1


Else



Selection.Copy


190


SkipBlanks:= _

False, Transpose:=False



Range("C27").Select


Range(Selection, Selection.End(xlUp)).Select


Selection.Copy



SkipBlanks:= _



Range("D23").Select




End If

If CurrentTime <= Capacity Then

Call ReactiveModel

Else

InputBox ("Time used exceeds capacity. No more patients can be

assigned.")

End If

Else

Range("D8").Select


ActiveCell.Clear

End If

Else

Range("D8").Select


ActiveCell.Clear

End If

Else

Range("D8").Select


ActiveCell.Clear

End If

End Sub

Sub ReactiveModel()

Range("D10").Select



Selection.Copy


Range("B5").Select

Selection.PasteSpecial Paste:=xlValues, Operation:=xlNone, SkipBlanks:= _

191







Selection.Copy


Range("B23").Select



Range("B5").Select


Selection.Copy

Range("C5").Select



If Range("F42").Value = 0 Then

InputBox ("Time used exceeds remaining. Some patients need to be cancelled.

Press enter to continue.")

n = 5

m = 40

Do While n <= m

If Range("C" & n).Value = 0 Then

n = n + 1

Else

Range("C" & n).Select



OBJ = Range("F41").Value

If MaxOBJ = "" Then

MaxOBJ = OBJ

p = n


n = n + 1

Else

If OBJ < MaxOBJ Then

MaxOBJ = OBJ

p = n


n = n + 1

Else


n = n + 1

End If

End If

Else


n = n + 1

End If

End If

192

Loop

Range("C" & p).Select


n = 5

Do While n <= m

If Range("D" & n).Value = 0 Then

n = n + 1

Else

If Range("D" & n).Value > Range("C" & n).Value Then






MaxOBJ = OBJ

q = n


n = n + 1

Else


n = n + 1

End If

Else


n = n + 1

End If

Else

n = n + 1

End If

End If

Loop

If q > 0 Then

Range("C" & q).Select


End If

Else

InputBox ("Time used less than remaining. Look for additional patients. Press

Enter to Continue.")

MaxOBJ = 0

n = 5

m = 40

Do While n <= m


n = n + 1

Else







193

MaxOBJ = OBJ

p = n


n = n + 1

Else


n = n + 1

End If

Else


n = n + 1

End If

Else

n = n + 1

End If

End If

Loop

If p > 0 Then

Range("C" & p).Select


n = 5

m = 40

Do While n <= m


n = n + 1

Else







MaxOBJ = OBJ

q = n


n = n + 1

Else


n = n + 1

End If

Else


n = n + 1

End If

Else

n = n + 1

End If

End If

Loop

If q > 0 Then

Range("C" & q).Select

194


End If

End If

End If

Range("C5:C22").Select

Selection.Copy


Range("C10").Select




Range("C29").Select



Range("C23:C40").Select


Selection.Copy




End Sub

Robust Reactive Scheduling model Branch and Bound Code

Input Patient Macro

Sub InputNewPatient()

Data1 = InputBox("Enter patient job number (ID)")

Data2 = InputBox("Enter patient specialty")

Data3 = InputBox("Enter patient priority")

Data4 = InputBox("Enter patient type, 1 = Elective, 2 = Emergency")


Range("B4").Select



ActiveCell.Value = Data1

ActiveCell.Offset(0, 1).Value = Data2




ActiveCell.FormulaR1C1 = "=VLOOKUP(RC[-3],R6C14:R11C18,4,FALSE)"







Range(Selection, Selection.End(xlToLeft)).Select

Selection.Copy

Sheets("Initial sequence").Select

Range("B4").Select

195





Range("I5").Select



ActiveCell.FormulaR1C1 = "=EXP(RC[-2]+SQRT(RC[-1]))"


ActiveCell.FormulaR1C1 = "=(RC[-7]/RC[-1])"

Sheets("Results").Select

End Sub

Run Initial Schedule Macro

Sub RunInitialSequence()


Range("f1").Value = 32

Range("B5").Select

Range(Selection, Selection.End(xlToRight)).Select


Selection.Sort Key1:=Range("k5"), Order1:=xlDescending, Header:=xlGuess, _

OrderCustom:=1, MatchCase:=False, Orientation:=xlTopToBottom

Range("D5").Select

i = 1

n = Range("B1").Value

Range("F5").Select

Mean = 0

Variance = 0

Mu = 0

Sigma = 0

Do While i <= n

Mean = Mean + ActiveCell.Offset(i, 0).Value

Variance = Variance + ActiveCell.Offset(i, 1).Value

i = i + 1

Loop

Mu = Math.Log((Mean ^ 2) / (Mean ^ 2 + Variance) ^ (1 / 2))

Sigma = Math.Log((Variance + Mean ^ 2) / (Mean ^ 2))

SumDuration = Math.Exp(Mu + Sigma ^ (1 / 2))

SumDuration1 = SumDuration

capacity = Range("F1").Value

If capacity < SumDuration Then

If capacity > 0 Then

j = 1


z = Range("D1").Value

capacity = Range("f1").Value

capacity_current = capacity

z_current = z

Call FindU

If z_current < z Then

196

z = z_current

Range("l5").Select



Selection.Copy




End If

j = n

Range("l5").Select

ActiveCell.Offset(j, 0).Select

If ActiveCell.Value = 1 Then

capacity_current = capacity_current + ActiveCell.Offset(0, -2).Value

z_current = z_current + ActiveCell.Offset(0, -4).Value

ActiveCell.Value = 0

End If

Call BackTrack

End If

End If

If Range("f1").Value < SumDuration1 Then

Range("B5").Select



Selection.Sort Key1:=Range("m5"), Order1:=xlDescending, Header:=xlGuess, _


Range("B6").Select


Selection.Copy


Range("D9").Select



Else

Range("B6").Select


Selection.Copy


Range("D9").Select



End If

End Sub

FindU Macro

Sub FindU()

Cells.Select

Selection.Copy


Cells.Select

197

Selection.PasteSpecial Paste:=xlFormulas, Operation:=xlNone, SkipBlanks:= _


Range("B5").Select

i = j

r = i

u = 0

Range("F5").Select


If Range("l6").Value = Empty Then

duration = Exp(ActiveCell.Offset(i - 1, 2).Value + Math.Sqr(ActiveCell.Offset(i

- 1, 3).Value))

SumDuration2 = duration

Mu2 = ActiveCell.Offset(i - 1, 2).Value

Sigma2 = ActiveCell.Offset(i - 1, 3).Value

Mean2 = ActiveCell.Offset(i - 1, 0).Value

Variance2 = ActiveCell.Offset(i - 1, 1).Value

Else

Mean2 = Mean + ActiveCell.Offset(i - 1, 0).Value

Variance2 = Variance + ActiveCell.Offset(i - 1, 1).Value

Mu2 = Math.Log((Mean2 ^ 2) / Math.Sqr(Mean2 ^ 2 + Variance2))

Sigma2 = Math.Log((Mean2 ^ 2 + Variance2) / (Mean2 ^ 2))

SumDuration2 = Math.Exp(Mu2 + Math.Sqr(Sigma2))

End If

Do While SumDuration2 <= capacity_current

u = u + ActiveCell.Offset(i - 1, -2).Value

r = i


If r = 1 Then

ActiveCell.Value = ActiveCell.Offset(0, -9).Value

ActiveCell.Offset(0, 1).Value = ActiveCell.Offset(0, -8).Value



ActiveCell.Offset(0, 4).FormulaR1C1 = "=EXP(RC[-2]+SQRT(RC[-1]))"

Else

m = i - 1

ActiveCell.Offset(m, 0).Select

ActiveCell.Value = ActiveCell.Offset(-1, 0).Value + ActiveCell.Offset(0, -

9).Value

ActiveCell.Offset(0, 1).Value = ActiveCell.Offset(-1, 1).Value +

ActiveCell.Offset(0, -8).Value

ActiveCell.Offset(0, 2).FormulaR1C1 = "=LN((RC[-2]^2)/SQRT(RC[-2]^2+RC[-

1]))"

ActiveCell.Offset(0, 3).FormulaR1C1 = "=LN((RC[-3]^2+RC[-2])/RC[-3]^2)"


m = m - 1

End If

i = i + 1

m = i

p = 1

Do While p <= n - m + 1

198

ActiveCell.Offset(p, 0).Value = ActiveCell.Value + ActiveCell.Offset(p, -

9).Value

ActiveCell.Offset(p, 1).Value = ActiveCell.Offset(0, 1).Value +

ActiveCell.Offset(p, -8).Value

ActiveCell.Offset(p, 2).FormulaR1C1 = "=LN((RC[-2]^2)/SQRT(RC[-2]^2+RC[-

1]))"

ActiveCell.Offset(p, 3).FormulaR1C1 = "=LN((RC[-3]^2+RC[-2])/RC[-3]^2)"

ActiveCell.Offset(p, 4).FormulaR1C1 = "=EXP(RC[-2]+SQRT(RC[-1]))"

ActiveCell.Offset(p, -5).Value = ActiveCell.Offset(p, 4).Value -

ActiveCell.Offset(0, 4).Value

ActiveCell.Offset(p, -4).Value = ActiveCell.Offset(p, -11).Value /


p = p + 1

Loop



If ActiveCell.Offset(1, 0).Value <> Empty Then





End If

Range("F6").Select

If i <= n Then

Mean2 = Mean2 + ActiveCell.Offset(i - 1, 0).Value

Variance2 = Variance2 + ActiveCell.Offset(i - 1, 1).Value




Else

SumDuration2 = SumDuration2 + 1000

End If

Loop

If i <= n Then

Mean2 = Mean2 - ActiveCell.Offset(i - 1, 0).Value

Variance2 = Variance2 - ActiveCell.Offset(i - 1, 1).Value

If Mean2 > 0 And Variance2 > 0 Then




u = u + ((capacity_current - SumDuration3) / (SumDuration2 - SumDuration3) *

ActiveCell.Offset(i - 1, -2).Value)

Else

Mu2 = 0

Sigma2 = 0

u = 0

End If

End If

Range("g1").Select

ActiveCell.Value = u


199

ActiveCell.FormulaR1C1 = "=FLOOR(R[-1]C,1)"

u = ActiveCell.Value

Range("F6").Select

If z <= (z_current - u) Then

Call BackTrack

Exit Sub

Else

Call ForwardStep

End If

End Sub

ForwardStep Macro

Sub ForwardStep()


Range("F6").Select

If Range("L6").Value <> Empty Then

Mean = 0

Variance = 0

Range("f6").Select

Do While ActiveCell.Value <> Empty

Mean = Mean + ActiveCell.Value * ActiveCell.Offset(0, 6).Value

Variance = Variance + ActiveCell.Offset(0, 1).Value * ActiveCell.Offset(0,

6).Value


Loop

Range("F6").Select

Mean = Mean + ActiveCell.Offset(j - 1, 0).Value

Variance = Variance + ActiveCell.Offset(j - 1, 1).Value

Mu = Math.Log((Mean ^ 2) / Math.Sqr(Mean ^ 2 + Variance))

Sigma = Math.Log((Mean ^ 2 + Variance) / (Mean ^ 2))

SumDuration = Math.Exp(Mu + Math.Sqr(Sigma))

Else

duration = Exp(ActiveCell.Offset(j - 1, 2).Value + Math.Sqr(ActiveCell.Offset(j

- 1, 3).Value))

SumDuration = duration

Mu = ActiveCell.Offset(j - 1, 2).Value

Sigma = ActiveCell.Offset(j - 1, 3).Value

Mean = ActiveCell.Offset(j - 1, 0).Value

Variance = ActiveCell.Offset(j - 1, 1).Value

End If

Do While SumDuration < capacity_current

If ActiveCell.Offset(j - 1, 0).Value <> Empty Then

duration = SumDuration

z_current = z_current - ActiveCell.Offset(j - 1, -2).Value

r = j

ActiveCell.Offset(j - 1, 6).Value = 1


If r = 1 Then



200




Else

m = j - 1


ActiveCell.Value = ActiveCell.Offset(-1, 0).Value + ActiveCell.Offset(0,

-9).Value

ActiveCell.Offset(0, 1).Value = ActiveCell.Offset(-1, 1).Value +



1]))"



End If

j = j + 1

m = j

p = 1



9).Value




1]))"







p = p + 1

Loop






Selection.Sort Key1:=Range("k5"), Order1:=xlDescending, Header:=xlGuess,

_


End If

Range("F6").Select

Mean = Mean + ActiveCell.Offset(j - 1, 0).Value

Variance = Variance + ActiveCell.Offset(j - 1, 1).Value




Else

Exit Do

End If

201

Loop

If SumDuration <> duration Then

Mean = Mean - ActiveCell.Offset(j - 1, 0).Value

Variance = Variance - ActiveCell.Offset(j - 1, 1).Value




End If

If j <= n Then

ActiveCell.Offset(j - 1, 6).Value = 0

j = j + 1

End If

If j < n Then

Call FindU2

Exit Sub

End If

If j = n Then

Call ForwardStep

Exit Sub

End If


z = z_current

Range("l5").Select



Selection.Copy




End If

End Sub

BackTrack Macro

Sub BackTrack()


i = j - 1

Range("f6").Select

Do While i > 0

If ActiveCell.Offset(i - 1, 6).Value = 1 Then

Mean = Mean - ActiveCell.Offset(i - 1, 0).Value

Variance = Variance - ActiveCell.Offset(i - 1, 1).Value

If Mean > 0 And Variance > 0 Then




Else

Mean = 0

Variance = 0

Mu = 0

Sigma = 0

202

SumDuration = 0

End If

z_current = z_current + ActiveCell.Offset(i - 1, -2)

ActiveCell.Offset(i - 1, 6).Value = 0

j = i + 1

Call FindU2

Exit Sub

Else

i = i - 1

End If

Loop

Exit Sub

End Sub

FindU2 Macro

Sub FindU2()

Cells.Select

Selection.Copy


Cells.Select

Selection.PasteSpecial Paste:=xlFormulas, Operation:=xlNone, SkipBlanks:= _


Range("B5").Select

i = j

r = i

u = 0

Range("F5").Select


If Range("l6").Value = Empty Then

duration = Exp(ActiveCell.Offset(i - 1, 2).Value + Math.Sqr(ActiveCell.Offset(i

- 1, 3).Value))

SumDuration2 = duration

Mu2 = ActiveCell.Offset(i - 1, 2).Value

Sigma2 = ActiveCell.Offset(i - 1, 3).Value

Mean2 = ActiveCell.Offset(i - 1, 0).Value

Variance2 = ActiveCell.Offset(i - 1, 1).Value

Else

Mean2 = Mean + ActiveCell.Offset(i - 1, 0).Value

Variance2 = Variance + ActiveCell.Offset(i - 1, 1).Value




End If

Do While SumDuration2 <= capacity_current

u = u + ActiveCell.Offset(i - 1, -2).Value

r = i


If r = 1 Then



203




Else

m = i - 1


p = 1

Do While ActiveCell.Offset(-p, -3).Value = 0

p = p + 1

Loop

ActiveCell.Offset(0, -3).Value = 1

ActiveCell.Value = ActiveCell.Offset(-p, 0).Value + ActiveCell.Offset(0, -

9).Value

ActiveCell.Offset(0, 1).Value = ActiveCell.Offset(-p, 1).Value +



1]))"



End If

i = i + 1

m = i

p = 1



9).Value




1]))"







p = p + 1

Loop








End If

Range("F6").Select

If i <= n Then

Mean2 = Mean2 + ActiveCell.Offset(i - 1, 0).Value

Variance2 = Variance2 + ActiveCell.Offset(i - 1, 1).Value


204



Else

SumDuration2 = SumDuration2 + 1000

End If

Loop

If i <= n Then

Mean2 = Mean2 - ActiveCell.Offset(i - 1, 0).Value

Variance2 = Variance2 - ActiveCell.Offset(i - 1, 1).Value

If Mean2 > 0 And Variance2 > 0 Then




u = u + ((capacity_current - SumDuration3) / (SumDuration2 - SumDuration3) *

ActiveCell.Offset(i - 1, -2).Value)

Else

Mu2 = 0

Sigma2 = 0

u = 0

End If

End If

Range("g1").Select

ActiveCell.Value = u


ActiveCell.FormulaR1C1 = "=FLOOR(R[-1]C,1)"

u = ActiveCell.Value

Range("F6").Select

If z <= (z_current - u) Then

Call BackTrack

Exit Sub

Else

Call ForwardStep

End If

End Sub

Run Model(2) Macro

Sub RunModel2()


capacity = Range("B5").Value

mygrab = InputBox("Enter length of last surgery (minutes)", "Grab Time")

Range("c8").Select



ActiveCell.Value = mygrab

Dim Data5 As Integer

Data5 = InputBox("For completed patient, enter patient number")


Range("F1").Value = (capacity - mygrab) / 15

Range("l6").Select



205

Selection.ClearContents

Range("B6").Select

Do While ActiveCell.Value <> Data5


Loop



Selection.Delete Shift:=xlUp

Range("J6").Select


ActiveCell.FormulaR1C1 = "=EXP(RC[-2]+SQRT(RC[-1]))"

ActiveCell.Offset(0, 1).FormulaR1C1 = "=(RC[-7]/(RC[-1]))"


Loop

Range("B5").Select






Range("D5").Select

i = 1


Range("F5").Select

Mean = 0

Variance = 0

Mu = 0

Sigma = 0

Do While i <= n

Mean = Mean + ActiveCell.Offset(i, 0).Value

Variance = Variance + ActiveCell.Offset(i, 1).Value

i = i + 1

Loop

Mu = Math.Log((Mean ^ 2) / (Mean ^ 2 + Variance) ^ (1 / 2))

Sigma = Math.Log((Variance + Mean ^ 2) / (Mean ^ 2))

SumDuration = Math.Exp(Mu + Sigma ^ (1 / 2))

SumDuration1 = SumDuration

capacity = Range("F1").Value

If capacity < SumDuration Then

If capacity > 0 Then

j = 1


z = Range("D1").Value

capacity = Range("f1").Value

capacity_current = capacity

z_current = z

Call FindU


z = z_current

Range("H5").Select


206


Selection.Copy




End If

j = n

Range("K5").Select

ActiveCell.Offset(j, 0).Select

If ActiveCell.Value = 1 Then

capacity_current = capacity_current + ActiveCell.Offset(0, -2).Value

z_current = z_current + ActiveCell.Offset(0, -4).Value

ActiveCell.Value = 0

End If

Call BackTrack

End If

End If

If Range("f1").Value < SumDuration1 Then

Range("B5").Select



Selection.Sort Key1:=Range("m5"), Order1:=xlDescending, Header:=xlGuess, _


Range("B6").Select



End If

Selection.Copy


Range("D9").Select



Loop



Else

Range("B6").Select



End If

Selection.Copy


Range("D9").Select



Loop



End If

End Sub

Using Operations Research Methodologies to …Using Operations Research Methodologies to Improve...

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Transcript of Using Operations Research Methodologies to …Using Operations Research Methodologies to Improve...