THE SURGICAL CASE ASSIGNMENT AND SEQUENCING PROBLEM

Transcription

1 THE SURGICAL CASE ASSIGNMENT AND SEQUENCING PROBLEM A CASE STUDY Word count: Anna Macken Student number : Supervisor: Prof. dr. Broos Maenhout Master s Dissertation submitted to obtain the degree of: Master of Science in Business Engineering Academic year:

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3 THE SURGICAL CASE ASSIGNMENT AND SEQUENCING PROBLEM A CASE STUDY Word count: Anna Macken Student number : Supervisor: Prof. dr. Broos Maenhout Master s Dissertation submitted to obtain the degree of: Master of Science in Business Engineering Academic year:

4 PERMISSION Ondergetekende verklaart dat de inhoud van deze masterproef mag geraadpleegd en/of gereproduceerd worden, mits bronvermelding. I declare that the content of this Master s Dissertation may be consulted and/or reproduced, provided that the source is referenced. Gent, juni 2017 Anna Macken

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6 PREFACE This master thesis completes my five-year trajectory in Business Engineering, Operations Management at Ghent University. This work provided me valuable insights in the field of Operations Management which will be useful in my future career as a business engineer. Furthermore, the case study enhances my understanding of operations management in a practical environment. The writing of this master thesis was a challenging experience which has been supported by a number of people. First, I would like to thank Professor Broos Maenhout for suggesting this interesting topic and providing me the opportunity to write a master thesis in this field of research. I also would like to thank him for his availability, valuable insights and constructive feedback. Second, I would like to thank the AZ Herentals, Doctor Stefaan Verfaillie in particular for the insights he gave me in the working of the operation theatre. I also would like to thank Katrijn Vanlommel, for providing me access to the data files of the AZ Herentals. Finally, I would like to thank my family and friends for their support. Not only during the writing of this master thesis, but during my whole education at Ghent University. My parents and sisters for their everlasting support and the pleasant working environment. Robbe, in particular, for the reading and fine-tuning of my master thesis but more importantly, his support, understanding and patience. I

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8 TABLE OF CONTENTS 1 CHAPTER 1: INTRODUCTION CHAPTER 2: THE HOSPITAL ENVIRONMENT: PROVIDING A CONTEXT TYPES OF HOSPITALS TRENDS IN THE HOSPITAL ENVIRONMENT THE OPERATING THEATRE CHAPTER 3: THE PLANNING AND SCHEDULING OF OPERATING ROOMS DECISION LEVELS Strategic planning level: Case Mix Planning Tactic planning level: Master Surgery Schedule Block scheduling strategy Modified block scheduling strategy Open scheduling strategy Implementation Operational planning level: the surgical case assignment and sequencing problem An example PROBLEM CHARACTERISTICS Task characteristics Types of procedures Types of patients Duration of procedures Departments Uncertainty in operating room scheduling Resource requirements Operating rooms Equipment Staff Recovery room beds III

9 ICU Suite of regular wards Occurrence of the different resources Objectives used in the surgical case assignment and sequencing problem Operating room utilization Overtime Minimizing makespan Patient waiting time Patient priority Financial Occurrence of the different objectives Overview of the different approaches used to address the surgical case assignment and sequencing problem Assignment phase (advance scheduling) Sequencing phase (allocation scheduling) The assignment and sequencing problem addressed by two sub-problems The assignment and sequencing problem using an integrated approach CHAPTER 4: CASE STUDY GENERAL INTRODUCTION TO THE ALGEMEEN ZIEKENHUIS HERENTALS CURRENT SITUATION AT THE AZH The different actors in the operating theatre of the AZH The nursing staff Anesthesiologists Surgeons Patients Categories of surgical cases THE DIFFERENT DECISION LEVELS IN THE AZH Strategic planning level IV

10 4.3.2 Tactic planning level Operational planning level Planning systems Surgical case assignment at the AZ Herentals Sequencing of individual cases at the AZ Herentals CHAPTER 5: MATHEMATICAL MODEL DATA DECISION VARIABLE VARIABLES MODEL DESCRIPTION OF THE OBJECTIVE FUNCTION DESCRIPTION OF THE CONSTRAINTS ASSUMPTIONS CHAPTER 6: ANALYSIS DATA TREATMENT Data on surgeries performed Data on standard durations DESCRIPTION OF THE CURRENT SITUATION Standard versus realized utilization rate Utilization rate of planned durations Utilization rate of realized durations Comparison of the standard versus the realized durations Average patient waiting times Overview of characteristics ANALYSIS BASED ON THE MODEL Performance measures The consequences of using a discrete time representation Hospital perspective V

11 Scenario 1: The proposed model Scenario 2: The impact of overtime Patients perspective Scenario 3: Minimization of the average waiting time per patient Surgeons perspective Scenario 4: Maximization of the number of surgical cases scheduled Scenario 5: Patient priorities Scenario 6: Overall surgeons perspective Scenario 7: Multi-objective function considering all objectives Comparison the current scheduling practices of the AZH with the final model CHAPTER 7: CONCLUSION VI

12 LIST OF ABBREVIATIONS AM AZ AZH CHI EPD GYN ICU MIP MSS NKO OR ORT PACU PM THK URO VAT ante meridiem algemeen ziekenhuis (general hospital) algemeen ziekenhuis Herentals (general hospital of Herentals) general surgery electronic patient dossier gynaecology intensive-care unit mixed integer programming master surgery schedule ear, nose and throat disorders operating room orthopedics post-anesthesia care unit post meridiem oral and maxillofacial surgery urology vascular and thoracic surgery VII

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14 LIST OF FIGURES Figure 1: Division of types of hospitalization in general hospitals in terms of percentage (Zorg en Gezondheid Vlaanderen, 2011a) Figure 2: Possible sequences followed by patients Figure 3: The operating theatre process (Guerriero and Guido, 2011) Figure 4: The operating room planning process Figure 5: Types of patients Figure 6: The share of elective versus non-elective cases in the literature (Demeulemeester et al. 2013) Figure 7: The estimation of surgery durations (Cardoen et al., 2007) Figure 8: Division of the operating time over the different disciplines in terms of percentage Figure 9: Time measures in the AZ Herentals Figure 10: Utilization rate of standard durations AZ Herentals Figure 11: Utilization rate of the realized durations of the AZ Herentals Figure 12: Standard and realized utilization rate Figure 13: Realized and standard durations of surgeon Figure 14: Realized and standard durations of surgeon Figure 15: Average patient waiting time per discipline Figure 16: the impact of wu on the amount of overtime IX

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16 LIST OF TABLES Table 1: Average hospitalization period and average occupation of beds (Zorg en Gezondheid Vlaanderen, 2011b) Table 2: Example of a Master Surgery Schedule Table 3: Example of list of surgeries to be performed Table 4: Example of the assignment of surgical cases Table 5: Example of the sequencing of surgical cases Table 6: Overview of the types of resources considered in the literature on the surgical case assignment and sequencing problem Table 7: Overview of the different objectives in the literature on the surgical case assignment and sequencing problem Table 8: Major objectives and their definitions Table 9: Overview of the characteristics of the assignment phase Table 10: Overview of characteristics of the sequencing phase Table 11: Overview of characteristics of the surgical case assignment and sequencing problem considered as two sub-problems Table 12: Overview of characteristics integrated surgical case assignment and sequencing problem. 41 Table 13: The different disciplines in AZH and the number of surgeons per discipline Table 14: The master surgery schedule of the AZH Table 15: The operating room time per discipline Table 16: Priorities of patients Table 17: Example: Operations to be planned for s = Table 18: Example: calculation of overlap constraint for s = Table 19: Explanation of time measures Table 20: Average surgery duration per discipline Table 21: Characteristics of current planning system AZH Table 22: Hospital perspective: basic model Table 23: Overtime settings and their impact Table 24: Hospital perspective: OR utilization versus overtime Table 25: Patients perspective: OR utilization versus average waiting time per patient Table 26: Patients perspective: total number of surgeries versus average waiting time per patient. 79 Table 27: Patients perspective: average waiting time per patient using preemptive approach with α = Table 28: Patients perspective: Average waiting time per patient for different values of alpha XI

17 Table 29: Approaches used to minimize the average waiting time per patient Table 30: Surgeons' perspective: maximize OR utilization versus maximize the number of surgeries scheduled Table 31: Surgeons' perspective: maximize number of surgeries versus overtime Table 32: Priority of patients Table 33: Surgeons' perspective: OR utilization versus average waiting time per child Table 34: Surgeons' perspective: preemptive approach: average waiting time per child Table 35: Surgeons' perspective: OR utilization versus waiting time for urgencies Table 36: Surgeons' perspective: OR utilization versus average waiting time per day hospital patient Table 37: Surgeons' perspective: preemptive approach: average waiting time per day hospital patient Table 38: Surgeons' perspective: scheduling short surgeries first Table 39: Overall surgeons perspective Table 40: Multi-objective function - final model Table 41: Final model: shifting the emphasis put on the different stakeholder groups Table 42: Comparison of the current scheduling practices of the AZH with the final model Table 43: Comparison of the current scheduling practices of the AZH with final model using the same amount of overtime XII

18 1 INTRODUCTION CHAPTER 1 During the last years, the number of surgical cases performed in Flemish hospitals increased rapidly, due to the impact of the ageing population and technological advances that have broadened the scope of surgeries. As a consequence, about 59% of the hospital in Flanders indicate to increase their operating room capacity because the current capacity is lacking or because they want to prepare for the future demand (Cardoen, Demeulemeester, & Van der Hoeven, 2007). The acquirement of a new operating room is very expensive, consequently it is more appropriate to increase the efficiency of the planning and scheduling of operating rooms (Pham & Klinkert, 2008). The operating room planning and scheduling problem aims at scheduling the surgical cases of surgeons over the available operating rooms, ensuring the resources are utilized in an efficient way (Jebali, Hadj Alouane, & Ladet, 2006; Marjamaa, Vakkuri, & Kirvelo, 2008; Vijayakumar, Parikh, Scott, Barnes, & Gallimore, 2013). The need for a descent hospital organization and operating room planning is mainly determined by the divergent preferences of the major stakeholders, i.e. the patients, surgeons, nurses and management. They all increase the need for efficiency in the planning of operating rooms (Guerriero & Guido, 2011). Doctors do have different objectives compared to the management of the hospital. They want to perform as many operations as possible in their available time and they want to be assigned as much operating time as possible. On the other hand, they do have personal preferences regarding the timing of their surgical cases (Ozkarahan, 2000; Cardoen, Demeulemeester, & Beliën, 2009a). The management of the hospital wants to maximize its revenues, which is achieved by an optimal allocation of the available capacity. They want patients to stay in the hospital for a long period and they aim at maximizing the patient flow, since this increases their revenues. This preference may not be in line with the preferences of the surgeons, since they are not able to perform a surgery when there are no beds available for their patients (Jebali et al., 2006). Besides the doctors and the management, the patient has right on good-quality care and minimal waiting time. Often the patient wants to leave the hospital as soon as possible, which may not be in line with the preferences of the management. Operating room planning and scheduling efficiency mainly determines the waiting time for patients before a surgery. An adequate scheduling should 1

19 aim at minimizing this waiting time taking into account the patient s characteristics (Riise & Burke, 2011). Also the working requirements and availability of the nurses and administrative personnel need to be taken into account when scheduling. In Belgium it is stipulated by law that there always should be three nurses and one anesthesiologist available during each surgical procedure. Personnel shortage and working legislation make this requirement very difficult to fulfil and this may lead to a resources availability problem (Roland, Di Martinelly, & Riane, 2006). The priorities of a hospital are mainly determined by the relative power each of the above stakeholders has compared to the other stakeholders (Gemmel & Van Dierdonck, 1999). Those different stakeholders make the operating room planning a very complex task, since they do have conflicting preferences and objectives (Guerriero and Guido, 2011; Riise & Burke, 2011)). The operating room planning and scheduling problem encompasses different levels and time horizons. The strategic planning level in the long term, also referred to as the case mix planning, determines the available capacity (Riise & Burke, 2011). The tactic planning level divides the available operating room time among the different disciplines, which leads to the master surgery schedule (Santibanez, Begen, & Atkins, 2007). The operational planning level on the short term, also called the surgical case assignment and sequencing problem, decides on the assignment of the surgical cases to the operating time blocks and the sequencing of the surgical cases within these time blocks (Guerriero and Guido, 2011; Doulabi, Rousseau & Pesant, 2014). The objectives pursued at this stage aim at maximizing the utilization of the hospital by assigning the surgical cases to the right operating rooms and time slots (Vijayakuram et al., 2013). The operational planning level will be the focus of this master thesis. The emphasis in this work lies on the integration of the two stages of the operational planning level, i.e. the allocation and sequencing of surgical cases. The integration of the surgical case assignment and sequencing problem is a less examined topic in the literature of operating room planning and scheduling (Doulabi et al., 2014). This master thesis is performed in collaboration with the AZ Herentals, a regional hospital in the province of Antwerp, Belgium. Based on interviews the current scheduling practices at the AZ Herentals are examined. This enables us to identify the conflicting objectives of the various stakeholder groups. Based on this information and the conclusions that can be drawn from the literature review, a mathematical model will be developed that tackles the integrated surgical case assignment and sequencing problem. Next, a multi-objective function approach will be used in order 2

20 to take into account the various objectives of the stakeholder groups. It will be tried to improve the current scheduling practices at the AZ Herentals for the different stakeholders to the best extent. Overview This master thesis is organized as follows. Chapter 2 provides a context for the operating room planning and scheduling problem and zooms in on the functioning of the operating theatre. Chapter 3 reviews the literature on the different decision levels of the operating room planning problem and provides a detailed overview of the surgical case assignment and sequencing literature. Chapter 4 describes the case study on which the proposed mathematical formulation in Chapter 5 will be based. Chapter 6 provides an analysis of the mathematical model by formulating various objectives that measure the impact of the schedule on different stakeholder groups. Finally, Chapter 7 draws a conclusion. 3

21 2 THE CHAPTER 2 HOSPITAL ENVIRONMENT: PROVIDING A CONTEXT In the following chapter a short overview of the hospital environment in Belgium is given in order to provide a context for the operating room planning and scheduling problem and the case discussed. In paragraph 2.1 the different types of hospitals in Belgium are discussed. Paragraph 2.2 gives an overview of some general trends that appear in the hospital environment. In paragraph 2.3 one specific division of the hospital is discussed more in detail, i.e. the operating theatre. 2.1 TYPES OF HOSPITALS Guinet and Chaabane (2003) define a hospital as a multi-service production system, which is constrained by finite capacity material and human resources. Its objective is to offer the best healthcare at the lowest cost. The hospital environment is a labour-intensive sector. The hospitals are one of the largest employers in Belgium. Also indirectly they employ many people such as suppliers of medical equipment, suppliers of food, etc (Cardoen et al., 2007). Different types of hospitals can be distinguished. First, general hospitals treat patients for all kind of diseases by offering medical and specialized services. In 2016, Flanders and Brussels counted 63 general hospitals that represent beds (Zorg en Gezondheid Vlaanderen, 2016). The number of general hospitals in Belgium is characterized by a declining trend due to reorganizations and mergers in the hospital environment. Second, psychiatric institutions offer specialized treatment for patients with serious mental problems. Third, specialized hospitals are smaller hospitals, for example revalidation hospitals, that offer specialized care for a certain segment of the population. Finally, university hospitals are linked to a university that offers medical science. Belgium possesses seven university hospitals in total, they are coordinated by the counsel of university hospitals ( Raad van Universitaire Ziekenhuizen, RUZB ). According to information of a survey on operating theatre planning in Flanders, 46% of the hospitals in Flanders have a bed capacity between 150 and 300 beds (Cardoen et al., 2007). 4

22 Percentage distribution of types of hospitalization 2.2 TRENDS IN THE HOSPITAL ENVIRONMENT Two main trends appeared in the hospital environment during the last years: o A significant increase in the number of day hospitals from in 1999 to in 2011 took place. Day hospitalization occurs when the patient leaves the hospital at the same day he has been operated and includes both medical and surgical procedures. The increase is mainly due to the increase of medical procedures in day hospitals by seven percentage points. This evolution is visualized in Figure 1 (Zorg en Gezondheid Vlaanderen, 2011a). 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% classic day Figure 1: Division of types of hospitalization in general hospitals in terms of percentage (Zorg en Gezondheid Vlaanderen, 2011a). o A decrease in the average hospitalization period and an increase in the bed utilization rate, which is related to an increase in the number of staying patients at the hospital. The average hospitalization period can be defined as the number of nursing days divided by the total number of stays in the hospital. The utilization rate concerns the occupation of beds in the hospital (Zorg en Gezondheid Vlaanderen, 2011b) Number of staying patients Average hospitalization period 8,32 days 8,04 days 7,49 days Average bed occupation 78,43% 79,21% 82,40% Table 1: Average hospitalization period and average occupation of beds (Zorg en Gezondheid Vlaanderen, 2011b). The increase in the number of day hospitalizations, the increase of the average bed occupation and the increase in the number of staying patient indicate that there is a strong pressure on the management of hospitals. This emphasizes the need for adequate Operations Research techniques to 5

23 improve the efficiency of hospitals, focusing on the optimal utilization of the resources provided by the hospital (Guerriero & Guido, 2011). 2.3 THE OPERATING THEATRE When surgical cases need to be performed, this takes place in the operating theatre. The operating theatre consists of a bedhold area for waiting and preparation for surgery, different operating rooms, a post-anesthesia care unit (PACU) or recovery room and an Intensive-Care Unit (ICU) (Guinet & Chaabane, 2003; Guerriero & Guido, 2011). The surgical process that occurs in the operating theatre is described in this paragraph. Figure 2 displays the operating theatre and its components. The arrows display the possible sequences followed by patients involved in the operating process. Figure 2: Possible sequences followed by patients. Typically, inpatients come from the suite of regular wards and are transferred to the operating room. Outpatients come from ambulatory surgical centers and emergencies or urgencies come from the emergency department and arrive in the operating room (Jebali et al., 2006). This stage is called the preoperative stage (Jebali et al., 2006; Pham & Klinkert, 2008; Guerriero & Guido, 2011). The next stage is the perioperative stage. The surgical act starts by anesthesia that is being applied by an anesthetic. Afterwards the surgery is performed by one or more surgeons with the assistance of one or several operating room nurses. After surgery the operating room is cleaned by a special cleaning team (Jebali et al., 2006; Pham & Klinkert, 2008; Guerriero & Guido, 2011). Afterwards, the patient is transferred to the recovery room (PACU) where he or she is taken under supervision of nurses until complete awake. Patients that have undergone a deceptive surgery or for example cardiac patients are transferred to the Intensive Care Unit (ICU). After complete awake in the recovery room, inpatients are transferred back to the suite of regular wards. Outpatients are 6

24 transferred to the ambulatory surgical units and when required admitted to the hospital. In most of the cases outpatients are discharged from the hospital on the same day. This last stage is called the postoperative stage (Jebali et al., 2006; Pham & Klinkert, 2008; Guerriero & Guido, 2011). The three different stages identified in the operation process are displayed in Figure 3. Figure 3: The operating theatre process (Guerriero and Guido, 2011). 7

25 3 THE CHAPTER 3 PLANNING AND SCHEDULING OF OPERATING ROOMS The operating room planning and scheduling process involves the allocation of operating room time through to the operational scheduling of individual surgical cases (Santibanez et al., 2007). Surgical cases need to be allocated to operating rooms and resources need to be assigned to individual surgical cases. When creating an operating theatre planning, one should aim at maximizing the use of the available operating room capacity. Besides this, surgery needs to be performed on time, in order to maximize patient satisfaction and to minimize patients waiting time. Additionally, the patient flow should be maximized to ensure the profitability of the hospital and the surgeons (Guerriero & Guido, 2011). The operating room planning and scheduling process results into three distinctive planning phases, namely the strategic planning level, the tactic planning level and the operational planning level. These phases treat the operating planning and scheduling problem on respectively the long term, medium and short term planning horizon. The different decision levels are discussed in paragraph 3.1. The major part of this master thesis will focus on the operational planning level, i.e. the short term surgical case assignment and sequencing problem. This planning level will be discussed in more detail in paragraph DECISION LEVELS The majority of the literature in operating room planning and scheduling consider three distinctive planning phases consisting of the strategic, tactic and operational planning level (Kennedy, 1992; Vissers, Bertrand, & De Vries, 2001; Testi, Tanfani, & Torre, 2007; Wachtel & Dexter, 2008; Guerriero & Guido, 2011). Cardoen, Demeulemeester and Beliën (2010) and Demeulemeester, Beliën, Cardoen and Samudra (2013) follow a different approach using descriptive fields such as patient characteristics and performance measures to analyze the different contributions. The approach followed in this work is based on the first classification, i.e. the strategic, tactic and operational planning phase. The first planning phase is the strategic planning level in the long term. The second phase is the tactic planning level in the mid-long term. This phase is often described as the master surgery schedule. The total available operating time is divided among the different surgeons or groups of surgeons. The 8

26 third and last phase is the operational planning level in the short term. In this phase, also called the surgical case assignment and sequencing problem, the surgical cases that need to be performed are scheduled on the assigned blocks and the sequence of the operations is decided. The total planning process can be summarized as displayed in Figure 4. Figure 4: The operating room planning process Strategic planning level: Case Mix Planning The strategic planning level in the long term, also called the case mix planning, determines the available capacity. This planning level can be considered as a resource allocation problem (Guerriero & Guido, 2011). During the planning of the capacity the resources needed for the accomplishment of the operations are determined. This capacity is determined by the budget available for the operating rooms. There is made an arrangement on the number of operating rooms, the number of surgeons, the number of nurses, material resources etc. Agreements with the different departments are made on the annual patient volumes and the total operating time (van Oostrum et al., 2010). These arrangements lead to the definition of the hospital s supply for surgery on an annual basis (Marques, Captivo, & Pato, 2012). The case mix planning is made for one to five years and is based on historical data and experience (van Oostrum et al., 2010). Several linear, integer linear or linear goal programming models have been defined to solve this strategic planning problem (Kuo et al., 2003; Testi et al., 2007; Blake, Dexter & Donald, 2002). In this phase the objectives mainly pursued aim at maximizing profit or target a budget taking into account the resources, infrastructure and demand (Riise & Burke, 2011) Tactic planning level: Master Surgery Schedule The second phase is the tactic planning level in the mid-long term, typically with a planning horizon of one or several weeks. This phase is often described as the master surgery schedule in the literature. The total available operating time coming from the case mix planning is divided among the 9

27 different surgeons or groups of surgeons (Guerriero & Guido, 2011). Historical utilization, hospital costs or gains per discipline can be used as criteria for the division of the operating room time (Testi et al., 2007). Also patient waiting times and actual patient demand for the services of a certain discipline, such as patient waiting lists or appointment requests, can be used as an input for the master surgery schedule (van Oostrum et al., 2010). A lot of research has already been done on this phase. The MSS is adjusted a few times per year depending on changes in hospital budgets, the supply of nurses or changes in the demand (Riise & Burke, 2011). Different approaches have been suggested to address this problem, such as integer or mixed integer linear programming models (Blake et al., 2002; Santibanez et al., 2007; Testi et al. 2007) and heuristic methods (Beliën and Demeulemeester, 2007; Van Oostrum et al., 2008). Different scheduling types are possible depending on the way in which the available operating time is divided, i.e. the block scheduling strategy, the modified block scheduling strategy and the open scheduling strategy (Fei, Chu, & Meskens, 2009). These types are described below Block scheduling strategy In the block scheduling strategy, the total available operating time is divided into time blocks. The duration of one block is usually determined by the staff of the hospital on the base of historical data. Often, blocks of half a day are used. These blocks are subdivided among the surgeons or groups of surgeons, in which they plan their surgeries. In this way a master surgery schedule or operating room block allocation table is created (Guerriero & Guido, 2011). Santibanez et al. (2007) define the most important factors to take into account when creating a MSS as the compatibility between operating room, surgeons availability, block capacity and post-surgical resources. When the blocks are divided among groups of surgeons, for example 15% for the gynaecology department, the blocks still need to be subdivided among the surgeons. The MSS is usually created for a week and then applied cyclically over weeks for a certain period, for example one year (Santibanez et al., 2007). The block system generally results in a higher utilization rate compared to other systems, as a result of a higher utilization of the afternoon operating room time. A MSS approach reduces the variability in the operating room planning and improves the clarity and predictability of the processes. Moreover, doctors experience a lower level of competition and less pressure on the scheduling of cases in advance (Ozkarahan, 2000). A MSS can also be seen as an aggregate production plan that will determine the resource requirements (Santibanez et al, 2007). The major drawback of the block scheduling strategy is the inability to schedule unused time by surgeons among the other surgeons. In this way, expensive operating time is not used efficiently 10

28 when surgeons do not cover the whole block time. An example of a master surgery schedule is presented in Table 2. Monday Tuesday Wednesday Thursday Friday OR AM PM AM PM AM PM AM PM AM PM 1 Ortho Ortho Neuro Ortho Ortho General Ortho General Ortho Neuro 2 General Cardio Cardio Ortho Cardio Ortho Cardio Orho Cardio Ortho 3 General Neuro General Neuro General Neuro Gyn Neuro General Gyn 4 Gyn Gyn Pediatric Gyn Pediatric Gyn Pediatric Pediatric Plastic Plastic Table 2: Example of a Master Surgery Schedule Modified block scheduling strategy In order to cope with the problems of the block scheduling strategy, the modified block scheduling strategy adds some flexibility to the MSS. Certain time blocks are left open in order to incorporate non-elective surgeries or they are made free on a specific release time. In this way, other surgeons can subscribe on the free time blocks on a first come first serve base. Consequently, the unscheduled time for surgeries is subdivided among the other surgeons in order to increase the utilization rate (Guerriero & Guido, 2011). It should be noted that when the release time of unused blocks is too near to the date of surgery, the empty time blocks may remain idle. Besides this, surgeons are often reluctant to discharge unused operating room time (Ozkarahan, 2000). About 94 % of the hospitals in Flanders use the modified block scheduling policy (Cardoen et al., 2007) Open scheduling strategy In the open scheduling strategy, no blocks are assigned to the surgeons and the surgeons are free in asking for available surgery time. This strategy implies a larger amount of flexibility. A schedule is created by filling the empty cells with surgeries following the order of arrival time. This strategy is also called the first-come first-served strategy (Fei, Chu, & Meskens, 2006; Fei et al., 2009). It should be noted that the open scheduling strategy is rarely used when scheduling surgical cases since it results in long waiting times, high cancellation rates and differences between the utilization rates of the different departments (Ozkarahan, 2000). The objectives pursued at the tactic planning level are often aligned with the objectives of the management of the hospital. They aim at maximizing the profitability of the hospital. Additionally, reducing the variability in the bed utilization and maximizing throughput are common used objectives (Cardoen et al., 2010; Demeulemeester et al., 2013). 11

29 The master surgery schedule is the input for the following planning phase, i.e. the operational planning level, where the surgery assignment and sequence will be determined Implementation It should be noted that the organization and inherent culture of a hospital impose implementation issues that may affect the result of the planning process in a negative way. This is one of the main reasons why a master surgery schedule often fails to work in practice or fails to take into account the organizations inherent characteristics. The implementation of the MSS into practice is often seen as one of the most challenging aspects by operation managers (van Oostrum et al., 2010) Operational planning level: the surgical case assignment and sequencing problem The third phase is the operational planning level in the short term. First, the surgical case assignment and sequencing problem is discussed and an example is provided in order to explain clearly the surgical case assignment and sequencing problem. Second, the problems characteristics are discussed. It concerns the task characteristics, resource requirements and performance criteria used in the surgical case assignment and sequencing problem. This part is concluded by providing an overview of the way the different contributions address the surgical case assignment and sequencing problem. The surgical case assignment and and sequencing problem is also called the patient scheduling problem or the elective case scheduling in the literature (Marques et al., 2012). In this last phase, the operations that need to be performed are scheduled on the assigned blocks resulting from the master surgery schedule in the tactic planning level. Critical resources such as surgeons, nurses, operating rooms, bed capacity, medical equipment and operating time need to be assigned to individual surgeries (Riise et al., 2012). Surgery dates and specific start-times are assigned to each surgical case. The operational planning level can further be subdivided into two phases (Guerriero & Guido, 2011). Assignment or advance scheduling step In the first step, also called the advance scheduling, the operations that need to be performed are assigned to the available time blocks for a specific surgeon coming from the master surgery schedule in the tactic planning level. A surgery date and an operating room is assigned to each surgical case (Guerriero & Guido, 2011). 12

30 Sequencing or allocation scheduling step In the second step, also called the allocation scheduling, the sequencing of the surgical cases takes place in order to determine the daily program of each operating room and each time block. This program is known as the Surgical Table Program or Operating Table (Latorre-Núñez, et al., 2016). The surgeries are placed in the right order and time periods, taking into consideration the resource availabilities and priority rules. These priority rules ensure that high priority surgeries are scheduled first before lower priority surgeries. Priorities can be based on different criteria such as out-of-town patients, children, patients using public transportation, patients requiring special tests (Vijayakuram et al., 2013) and moving surgeries that contaminate the operating room towards the end of the day (Latorre-Núñez et al., 2016) An example The assignment and sequencing of surgical cases can be explained by an example, explaining the flow that is followed by patients starting from the first consultation towards leaving the hospital after surgery. This example is based on Riise and Burke (2011) and uses fictitious data. Patients arrive at the hospital for an appointment with a specialist. The necessary treatment is discussed. We use patient a to discuss the example further. The surgery for patient a is registered on the waiting list. This list is displayed in Table 3. Patient a is linked to surgery one which has a duration of one. This duration depends on the type of surgery. The surgery will be performed by Doctor Peeters, who belongs to the Orthopedics department. Patient Surgeries to be performed Duration Discipline Surgeon a surgery 1 1 Ortho Dr. Peeters b surgery 2 3 Ortho Dr. Peeters c surgery 3 4 Ortho Dr. Claeys d surgery 4 2 Ortho Dr. Claeys e surgery 5 5 Ortho Dr. Smets f surgery 6 4 Ortho Dr. Peeters g surgery 7 6 Ortho Dr. Claeys h surgery 8 4 Ortho Dr. Peeters Table 3: Example of list of surgeries to be performed. 13

31 First, the allocation of the surgery takes place, in which a surgery date is decided upon in consultation with the patient. The patient s surgery is allocated to an operating room, a day and a time block. This allocation depends on the availability of operating room time of the surgeon and has been decided upon in the master surgery schedule. If we look at the master surgery schedule as provided in Table 2, we can see the Orthopedics department has access to operating room one on Monday. In this example we assume Dr. Peeters has access to operating room one on Monday in the morning. Dr. Peeters decides to allocate surgery one to operating room one on Monday morning. Surgery eight and two are also assigned to this time block, as shown in Table 4. This admission planning typically takes place several months before hospitalization up to one week before surgery, depending on the urgency of the intervention. Room Surgeon Day Time block Surgeries assigned OR 1 Dr. Peeters Monday AM 1, 8, 2, 6 OR 1 Dr. Claeys Thursday AM 7, 3, 4 OR 2 Dr. Smets Friday PM 5, 10, 9 Table 4: Example of the assignment of surgical cases. A few weeks before hospitalization, the sequencing process takes place in which the specific start time of the surgery is decided upon. The planner communicates with the patients in order to fix some surgeries. In this example, surgery one will be scheduled at start time one and will take one time period. Next, surgery eight is scheduled, which starts at start time two and will take four time periods. This process is repeated for all the assigned surgeries, as displayed in Table 5. The sequencing is often inherent to priority rules that may be specific to the hospital. Often, this surgery sequencing may be reviewed the day before surgery due to the dynamics of the hospital. During the day of surgery the plan can be adapted manually to unforeseen circumstances (Riise & Burke, 2011). 14

32 Room Surgeon Day Time block Surgeries assigned OR1 Dr. Peeters Monday AM 1, 8, 2, 6 Surgery Start time Duration surgery surgery surgery surgery Room Surgeon Day Time block Surgeries assigned OR1 Dr. Claeys Thursday AM 7, 3, 4 Surgery Start time Duration surgery surgery surgery Room Surgeon Day Time block Surgeries assigned OR2 Dr. Smets Friday PM 5, 10, 9 Surgery Start time Duration surgery surgery surgery Table 5: Example of the sequencing of surgical cases. 3.2 PROBLEM CHARACTERISTICS In the following paragraph the characteristics of the surgical case assignment and sequencing problem will be discussed. The allocation and sequencing of surgical cases is dependent on a number of factors. First, task characteristics such as the type of procedure, the type of patient, surgery durations, etc. need to be taken into account. They influence the amount of operating time that needs to be reserved for a certain surgery (Vijayakumar et al., 2013). These task characteristics are discussed in paragraph Second, the resource requirements that influence the ability to schedule surgical cases will be discussed. Critical resources such as surgeons, nurses, operating rooms, recovery room bed capacity, medical equipment and operating time are allocated to individual surgeries (Riise et al., 2012). Each surgery is assigned to a certain surgeon, expected to perform the surgery and an operating room in which the surgery is performed. Additionally, other resources need to be allocated to each surgical 15

33 case, depending on their availability. These resources may be renewable, i.e. medical staff such as nurses, anesthesiologists, etc. Others are non-renewable resources, i.e. equipment, surgical materials, pharmaceuticals, etc. (Latorre-Núñez et al., 2016; Riise et al., 2012; Vijayakuram et al., 2013). Moreover, after surgery each patient requires a bed in the post-anesthesia care unit. This means the capacity of the recovery room beds also needs to be considered when scheduling surgical cases (Latorre-Núñez et al., 2016). Next, the patient is transferred to the suite of regular wards after he has woken up in the recovery room. This capacity is also often seen as a bottleneck (Ma and Demeulemeester, 2010). Third, the performance measures used to evaluate an operating room schedule will be discussed in paragraph The criteria pursued depend on the conflicting needs of the different stakeholders (patients, surgeons, nursing staff and management) of the operating theatre. One or multiple objectives are optimized, satisfying their requirements (Guerriero & Guido, 2011) Task characteristics The objective of the surgical case assignment and sequencing problem is allocating hospital resources to individual surgical cases within a certain time frame. The surgical cases are characterized by a number of factors, i.e. types of procedures, types of patients, duration of procedures, different departments and uncertainty (Guerriero & Guido, 2011; Demeulemeester et al., 2014) Types of procedures In order to enable the scheduling of surgeries, they can be subdivided into three categories based on their frequency of appearance. Category A includes surgeries that occur frequently and can be planned in advance. Category B includes surgeries that can be planned in advance but that do not occur frequently, also called dummy surgeries. Category C consists of all emergency procedures that cannot be planned in advance (van Oostrum et al., 2008) Types of patients Three different classifications of patients exist, i.e. elective and non-elective patients, inpatients and outpatients and add-elective and add-on cases. 16

34 Elective and non-elective patients elective patients non-elective patients emergencies urgencies Figure 5: Types of patients. Each surgical case belongs to a patient. Two types of patients can be considered, namely elective and non-elective patients. Elective patients can be scheduled for surgery in advance. Non-elective patients cannot be scheduled in advance, since surgery appears unexpected for those patients. The latter can be further divided into emergencies and urgencies. Emergencies have to be treated as soon as possible. Urgencies are non-elective patients that do not need a surgery as soon as possible, their treatment can be postponed for a short period of time (Guerriero and Guido, 2011; Demeulemeester et al., 2013). Most research focuses on elective patients and ignores the difficulties caused by emergencies. Besides this, it is often not specified for which type of patients an operating room schedule is developed (Pham & Klinkert, 2008; Demeulemeester et al. 2013). Figure 6 shows the share of elective and non-elective surgical cases in the literature from 2005 to Note that the sum of the elective and non-elective cases is higher than 100% due to papers that deal with both elective and non-elective patients. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% elective cases non-elective cases Figure 6: The share of elective versus non-elective cases in the literature (Demeulemeester et al. 2013). 17

35 Inpatients and outpatients Another classification of patients are inpatients and outpatients. Inpatients are considered to be patients that stay overnight in the hospital, whereas outpatient only come for a surgery and leave the hospital on the same day. They are taken care of in the ambulatory surgical centers before and after surgery. Outpatients surgeries are typically shorter and less complex to perform (Pham & Klinkert, 2008, Demeulemeester et al. 2013). Add-elective cases and add-on cases Add-elective cases are elective patients scheduled to fill up the remaining operating room time. Addon cases are add-elective patients, emergencies and urgencies (Guerriero & Guido, 2011) Duration of procedures Each operation is characterized by a certain expected duration that is used for the planning and scheduling of the surgical cases (Jebali et al., 2006). The duration of both the perioperative and postoperative stage needs to be estimated. Perioperative stage In order to be able to plan and schedule surgeries, one needs to estimate the length of the perioperative stage, i.e. the stage in which the surgical act is performed. Therefore, the duration of a certain surgery needs to be estimated. The duration of a surgery is dependent on a number of factors, such as the type of surgery, the pathology of the patient and the surgeon s expertise and technical skills. The predicted estimation of the length of surgery is one of the most important input parameters when scheduling surgeries, in order to ensure an effective implementation of the operating room schedule (Guinet & Chaabane, 2003; Jebali et al., 2006; Riise et al., 2012). In order to provide a reliable and accurate planning, an information or calculation system of the durations is necessary. Different approaches have been researched to estimate the duration of a surgery (Jebali et al., 2006). First, log normal distributions can be used to estimate operating times (Zhou & Dexter, 1998). Second, information system can calculate an average based on historical data. However, in most of the cases the estimation is done by the surgeon himself based on experience and trial & error. Philips (1975) states that surgeons request on average 4% more time than needed for their surgical cases. A survey on operating room scheduling in Flanders in 2007 indicates the way in which surgery durations are estimated. Based on this survey, 30% of the estimation of surgery durations are based on software by analyzing historical data. Another 30% of the estimations is done by the surgeon who 18

36 will perform the surgery. In 11% of the cases the surgeries are estimated by the head nurse and in 24% of the cases both by the head nurse and the surgeon. (Cardoen et al., 2007). The estimation of surgery durations Head nurse 11% Other 5% Historical data 30% Head nurse and surgeon 24% Surgeon 30% Figure 7: The estimation of surgery durations (Cardoen et al., 2007). Post-operative stage The duration of the postoperative stage, i.e. the stage in which the patient is transferred to the PACU or ICU, also has to be estimated accurately in order to build an adequate operating room schedule (Guinet & Chaabane, 2003). Few studies address the duration of the postoperative phase. This phase depends on the type of surgery that has been performed, the duration of the surgery, the patient and the type of anesthesia that has been applied (Marcon, Kharraja, Smolski, Luquet, & Viale, 2003a). Log-normal distribution may be used to approximate the duration of the post-anesthesia phase (Dexter & Tinker, 1995) Departments Surgeons belong to a certain discipline. This is important because this discipline determines how much operating room time there is assigned to a certain surgeon at the tactical planning level (Demeulemeester et al., 2013) Uncertainty in operating room scheduling Uncertainty is a major pitfall in operating room scheduling, which defines the need for other scheduling methods compared to machine scheduling systems. Uncertainty results from different aspects in the planning and scheduling process (Guerriero and Guido, 2011). Most operating room schedules are assumed to be deterministic, which means they do not incorporate uncertainty and variability. When uncertainty is incorporated, a model is assumed to be stochastic (Lamiri, Xie, Dolgui, & Grimaud, 2008b). Operating room planning and scheduling is essentially stochastic. 19

37 Different types of uncertainty can be defined in the operating room planning and scheduling process: arrival uncertainty, duration uncertainty and uncertainty caused by incorporating non-elective patients in the scheduling process (Riise, Mannino, & Burke, 2016). Arrival uncertainty Arrival uncertainty results from the variable arrival times of patients or surgeons due to unforeseen circumstances. Also the arrival of emergencies is not predictable and therefore causes a lot of variation (Persson & Persson, 2009). Duration uncertainty Duration uncertainty is caused by the difficulties in estimating the duration of an operation. An accurate prediction of the operating time required for a surgery is necessary for an effective implementation of the operating room schedule (Jebali et al., 2006; Riise et al., 2012; van Oostrum et al., 2010). When complications arise, an operation may take more time than scheduled. This may lead to a later start of the next surgery and consequently for the rest of the surgeries. Research states that most of the delays are caused by surgeons and that their delays take a longer amount of time compared to others (Marjamaa et al., 2008). In order to avoid these delays in the schedule and the negative impact of uncertainty in general, it should be measured and incorporated in the operating room schedule (Cardoen et al., 2010). Authors that consider stochastic duration in the operating room planning and scheduling problem are the following: Denton, Viapiano & Vogl, 2007; Hans, Wullink, van Houdenhoven, & Kazemier, 2008; Marcon, Kharraja, & Simonnet, 2003b; Zhou and Dexter Uncertainty caused by non-elective patients Most of the research on operating room scheduling focuses on elective patients and ignores the difficulties caused by emergencies (Guerriero & Guido, 2011). Besides this, it is often not specified for which type of patients an operating room schedule is developed (Lamiri et al. 2008b). This type of planning approach is called the offline operational operating room planning. Resources and operating room personnel that are known beforehand are scheduled in order to avoid critical resource conflicts. This planning is typically made for a week (Van Oostrum et al., 2010; Wullink et al., 2007). When non-elective patients, i.e. emergencies and urgencies, are considered, the degree of uncertainty in the planning and scheduling of surgical cases increases (van Oostrum et al., 2010). The assignment of the emergencies needs to be monitored and controlled during the day, which results 20

38 in the online operational planning. When emergencies cannot be scheduled within the existing offline planning, adjustments will be needed and this can result in cancellations or rescheduling of surgeries (Cardoen et al., 2010; Van Oostrum et al, 2010; Wullink et al., 2007). It is very difficult to predict when non-elective patients will arrive in the hospital. The surgeries for these patients cannot be planned in advance and they may interrupt the schedule of the elective surgeries (Latorre-Núñez et al., 2016). Two different approaches to deal with non-elective patients have been identified in the literature: opening a special operating room for emergencies or reserving some additional capacity in the existing operating rooms (Wullink et al., 2007). This is further discussed Resource requirements During surgery, scarce resources are often required in order to be able to perform the surgery. Resource units are assigned to surgeries and should be available throughout the performance of the surgery. First, we discuss the physical resources, such as operating rooms and equipment, that are required during surgery. Next, each surgery requires a competent surgeon and other medical staff such as nurses and anesthesiologists in order to be able to perform the surgery. Third, we discuss the post-anesthesia care unit or recovery room, in which a bed needs to be available for the patient after surgery. If the patient requires special care, he is transferred to the Intensive Care Unit (ICU). (Latorre-Núñez et al., 2016). At the end of this paragraph an overview of the occurrence of the resources is given Operating rooms Operating rooms can be classified on the base of two different aspects. The first classification is based on whether it concerns emergency or non-emergency operating rooms (Latorre-Núñez et al., 2016; Wullink et al., 2007). The second classification depends on the equipment that is available in the operating rooms (Jebali et al., 2006). Emergency and non-emergency operating rooms Two different types of operating rooms can be identified, emergency and non-emergency operating rooms. Emergency operating rooms are special operating rooms in which only emergencies are treated. The non-emergency operating rooms are used for all surgeries of elective patients (van Oostrum et al. 2010). 21

39 How to deal with non-elective surgeries? Two different approaches to deal with non-elective surgeries have been identified in the literature: opening a special operating room for emergencies or reserving some additional capacity in the existing operating rooms (Wullink et al., 2007; Van Essen, Hans, & Hurink, 2012). In the first approach, i.e. opening a special operating room for emergencies, all unplanned surgeries will be performed in this emergency operating room. In this way, no extra capacity needs to be reserved in other operating rooms and the planning of elective surgeries will not be interrupted. However, opening an extra operating room is very costly and thus requires to be used efficiently (Wullink et al., 2007). In the second approach, i.e. reserving some additional capacity in all operating rooms for emergency surgeries, the capacity for emergencies is spread over all operating rooms and emergency and nonemergency rooms do not exist (Van Essen et al., 2012). This may lead to disruptions in the planning of the surgeries, when the reserved capacity is not sufficient to respond to all emergencies. When an emergency arrives that has to be operated as soon as possible, other planned operations of elective patients may be postponed to a later moment in time. In the case of urgencies, i.e. non-elective patients that do not need treatment as soon as possible, less problems appear (Guerriero & Guido, 2011). The second approach turns out to be the most efficient, especially in relatively small hospitals. They will not be able to obtain a sufficient utilization rate in the emergency operating room (Wullink et al. 2007; Latorre-Núñez et al., 2016). Research has proven that reserving operating room capacity in all elective operating rooms maximizes the responsiveness for emergencies and the utilization rate of all operating rooms. Waiting times for non-elective patients diminish and overtime reduces (Wullink et al., 2007). However, it is essential that some buffer time is incorporated in the schedule of the operating rooms to be able to deal with the required surgeries of emergencies and urgencies Equipment Equipment that needs to be present in the operating room for surgery may be a bottleneck resource. Stationary equipment may be present in some operating rooms, but other equipment may need to be moved and sterilized between operating rooms (Jebali et al., 2006; Riise & Burke, 2011). Such interchangeable surgical material defines the need for considering availabilities per time period (Roland, Di Martinelly, Riane, & Pochet, 2010). It often happens that operating rooms are allotted to the different disciplines, since special equipment is needed for some surgeries. It may also happen that emergencies are operated in non- 22

40 emergency rooms when special equipment is needed for a specific surgery or when the emergency operating capacity is not sufficient (Zhang, Murali, Dessouky, & Belson, 2009). In Flanders, 70% of the hospitals have non-identical operating rooms of which 69% differ in size and 50% differ in equipment. These constraints should be taken into account when scheduling operating rooms (Cardoen et al., 2007) Staff Available and skilled surgical staff is a prerequisite in order to be able to perform surgical cases. Three types of medical staff can be identified in the operating room, i.e. surgeons, nurses and anesthesiologists. Competent surgeons are required to perform the surgery. The determination of the number of surgeons employed in the operating theatre is determined at the strategic planning level (Guerriero & Guido, 2011). Surgeon s preferences need to be taken into account here in order to ensure the schedule will be accepted by the surgeons (Ozkarahan, 2000). Nurses are needed for assistance during surgery and for supervision in the recovery room. Anesthesiologists are needed to perform anesthesia on each patient that requires surgery (Latorre-Núñez et al., 2016) Recovery room beds The operating room planning and scheduling process affects the other divisions of the hospital as well. Due to the direct link between the operating rooms and the recovery room, a shortage in the number of recovery room beds available can appear when unforeseen circumstances occur (Riise & Burke, 2011). Therefore, the capacity of the recovery room needs to be taken into account when scheduling surgeries (Demeulemeester et al., 2013). Each patient requires a bed in the postanesthesia care unit (PACU) for awakening after surgery. When the PACU is full, patients will be forced to recover in the operating room or the start of other surgeries will be blocked until the PACU has some beds available (Iser, Denton, & King, 2008; Augusto, Xie, & Perdomo, 2010). When the operating room is seen as an isolated part of the hospital, the efficiency of the other divisions may worsen when optimizing the operating room as a single entity (Demeulemeester et al., 2013). The recovery room bed availability is taken into account in the elective case planning and scheduling problem by several authors (Riise & Burke, 2011). For example, Jebali et al. (2006) consider recovery room bed limitations. When there are no recovery rooms beds available after surgery has been performed, the next surgeries will be delayed. Around 50% of the literature on operating room planning and scheduling deals with the operating room planning and scheduling problem considering the availability of PACU (Demeulemeester et al., 2013). 23

41 ICU Similar to the PACU, patients can be transferred to the Intensive Care Unit (ICU) after surgery when they require extra medical care. The congestion of the ICU can also be a bottleneck and may affect the functioning of the operating room (Demeulemeester et al., 2013). This has been researched by Kolker (2009) and Litvak, van Rijsbergen, Boucherie and van Houdenhoven (2008). They both aim at improving the service level for patients by reserving some capacity for emergency arrivals Suite of regular wards When the patient is transferred to the suite of regular wards after he is released from the operating theatre, a bed is required. This bed capacity needs to be taken into account in the assignment and sequencing of surgical cases (Adan & Vissers, 2002; Santibanez et al., 2007; Ma et al., 2009; Testi & Tanfani, 2009). Therefore, the number of patients that are operated and require a bed in the suite of regular wards on a certain day needs to be smaller or equal to the number of beds available on that day (Testi & Tanfani, 2009). Adan & Vissers (2002) also take into account the length of stay of the patients when deciding upon the mix of patients admitted to the hospital on a certain day Occurrence of the different resources The critical resources considered depend on the stage of the surgical case assignment and sequencing problem. In the assignment phase, the level of detail of the critical resources is usually lower, thereby considering fewer resources compared to the sequencing phase. This is because the information uncertainty in the assignment phase is higher compared to the sequencing phase, which is more on the short term (Riise & Burke, 2011). Resources often considered in the assignment phase are operating rooms and surgeons. In the sequencing phase specific material requirements etc. add detail to the scheduling of operations (Riise & Burke, 2011). Table 6 distinguishes between different types of resources i.e., staff, recovery room beds, Intensive Care Unit, stay beds and equipment or surgical materials, considered by the different papers. Operating room availability is inherently considered in all the contributions. Staff, i.e. surgeons, nurses or anesthesiologists, are the resources taken into consideration by more than 50% of the literature. Recovery room beds and equipment or surgical material are considered by respectively 16 and 17 out of 42 contributions. The incorporation of the ICU as a resource in the planning and scheduling problem is less popular. The stay bed constraint is considered by none of the contributions to the surgical case assignment and sequencing problem. 24

42 Type of resource Number of contributions in the literature [42] Staff 23 Equipment/surgical material 17 Recovery room beds 16 No resources considered 13 ICU 3 Stay beds 0 Table 6: Overview of the types of resources considered in the literature on the surgical case assignment and sequencing problem Objectives used in the surgical case assignment and sequencing problem Operating room planning and scheduling is complex since the preferences of all the stakeholders, i.e. surgeons, nurses, patients, etc. need to be taken into account. It is not possible to satisfy the objectives of all the stakeholders simultaneously. The objectives at the strategic and tactic planning horizon are more focused on the budget and profitability of the hospital. At the operational level, also surgeons and patient preferences are taken into account (Guerriero & Guido, 2011). In the following paragraph the different performance criteria used in previous research on the surgical case assignment and sequencing problem are described Operating room utilization The goal of the surgical case assignment and sequencing problem is to ensure the medical resources that are the main source of revenues for the hospital are utilized in an efficient way. Therefore, operating room utilization is one of the most used objective in the surgical case assignment literature (Vijayakuram et al., 2013). Utilization can be subdivided into two categories, underutilization or undertime and overutilization or overtime. There is a trade-off between over-and underutilization as overutilization implies overtime costs and underutilization or idle time leads to undertime costs (Cardoen et al., 2010). When the objective is to optimize the operating room utilization, this mostly refers to the minimization of undertime in the operating rooms. However, the operating room utilization optimization can also refer to the minimization of both under- and overtime (Jebali et al., 2006), as has been done by Cardoen et al. (2010) and Demeulemeester et al. (2013). They use utilization as a common concept and split this up in over-and underutilization and relate overutilization to overtime and underutilization to undertime. In this master thesis we use the concept operating room utilization to refer to undertime or underutilization. Overtime will be discussed in the next paragraph. 25

43 A second remark is related to the difference between undertime and underutilization. The distinction between undertime and underutilization is vague and oftentimes both concepts are used in different ways. However, there is a clear difference between these concepts. Undertime considers the timing aspect whereas underutilization refers to the workload of a resource, for example the operating room, and should be measured against a predefined target utilization level percentage of for example 100% (Cardoen et al., 2010). Utilization is calculated by dividing the total duration of the planned surgeries by the total available operating time (Guerriero & Guido, 2011). Underutilization occurs when the assigned operating room workload is lower than the available operating time and thus a utilization percentage of less than 100% is achieved (Guerriero & Guido, 2011). Thus, an underutilized operating theatre means that the operating rooms are utilized under their target utilization level. Consequently, it is possible to have an underutilized operating theatre with overtime in some operating rooms (Cardoen et al., 2010). The utilization of the operating theatre should be maximized, since underutilization results in extra avoidable costs for the hospital. Fixed costs of surgical centers are enormous compared to variable costs, therefore unused capacity should be avoided (Cardoen et al., 2010). However, one should not aim at obtaining a utilization rate of 100% since this would lead to very high uncertainty. Time buffers are not available when for example a surgeon arrives late. These aspects make the utilization rate an important strategic decision that should be made with caution. In practice we assume that a utilization rate in the range 85% - 95% is convenient for the planning of elective surgeries. In this way emergencies can be scheduled appropriately (Jebali et al., 2006; Tyler, Pasquariello, & Chen, 2003) Overtime Overtime refers to the time surgeries are performed beyond the regular opening hours of the operating room. Overutilization occurs when the assigned operating room workload is higher than available operating time and thus a utilization percentage of more than 100% is achieved (Guerriero & Guido, 2011). Overtime is the most widely used objective in the literature of the surgical case assignment and sequencing problem. Overtime should be avoided since it leads to high unavoidable personnel costs and staff dissatisfaction. A fully utilized operating room without time buffers will consequently lead to high overtime costs (Cardoen et al., 2010) Minimizing makespan Makespan minimization is often used in manufacturing job shop problems. It can be defined as the time difference between the start of the first job and the finish of the last job that has to be performed. In operating room scheduling, the makespan is defined as the time between the first 26

44 patient entering the operating room and the last patient leaving the operating room (Dekhici & Belkadi, 2010; Riise et al., 2016). It should be noted that the makespan objective is mainly used in the sequencing phase, when determining the order of surgical cases (Demeulemeester et al., 2013). When used in this context, surgical cases, i.e. jobs, need to be scheduled while minimizing the overall makespan (Riise et al., 2016). Makespan minimization often leads to a dense schedule. Therefore, small deviations in the schedule or emergency arrivals may cause serious problems, for which a rearrangement of the schedule is required (Augusto et al., 2010). The robustness of the schedule should be taken into account when minimizing the makespan (Demeulemeester et al., 2013) Patient waiting time Decreasing patient waiting times is an often used objective in operating room scheduling. Gupta and Denton (2008) define two types of waiting time, i.e. direct and indirect waiting time. Direct waiting time can be defined as the difference between the hospitalization date and the surgery date of a patient (Guerriero & Guido, 2011). This objective is often modelled using the hospitalization day cost that is incurred for each day spent in the hospital (Guinet & Chaabane, 2003; Jebali et al., 2006). Increasing direct waiting times highly affects the customer satisfaction. Patient waiting time is among one of the main complaints in health care leading to many research focusing on reducing the waiting time for patients (Cardoen et al., 2010). The second type of waiting time, i.e. indirect waiting time, can be defined as the time between the request for surgery and the actual date scheduled for surgery (Gupta & Denton, 2008; Riise & Burke, 2011). High indirect waiting time does not only affect the patient satisfaction, it may also cause serious problems in terms of safety concerns when a patient does not receive surgery within a certain time frame (Gupta & Denton, 2008). Besides waiting time for patients, also surgeons waiting time or operating room idle time is often used as an objective, since both the surgeon and operating room are very expensive resources for the hospital (Cardoen et al., 2010) Patient priority Patient priorities can be used to determine the planning and scheduling of surgical cases. Patient priorities are often defined in terms of priority rules. They are used to ensure that high priority cases such as urgencies are scheduled early in the day. These priority rules can be general, such as for example the first-come first-served rule, the longest processing time rule or the shortest processing time rule. Priority rules can also be hospital or surgeon specific, for example scheduling children first, 27

45 scheduling latex allergic patients first, grouping surgeries of the same type, giving priority to short or long surgical cases, scheduling patient with a high travel distance later in the day, etc. In this way, surgeons try to obtain individually optimal block time assignments and sequences (Ozkarahan et al., 2000; Cardoen et al., 2007; Latorre-Núñez et al., 2016) Financial The cost objective is more general since all of the above objectives can be defined in terms of costs. Consequently, minimizing the costs of operating rooms and thus avoiding unused capacity is a widely studied objective in operating room scheduling. Pursuing a minimization of costs gives hospitals the opportunity to improve the objectives mentioned before. In this way, costs may provide a reasonable penalty for the objective function (Jebali et al., 2006; Fei, Chu, & Meskens, 2006; Lamiri et al., 2008a, 2008b, 2008c, 2009; Roland et al., 2006, 2010; Landa, Aringhieri, Soriano, Tanfani, & Testi, 2015) Occurrence of the different objectives After investigating the majority of the contributions in the literature of the surgical case assignment and sequencing problem, some conclusions about the occurrence of the different objectives can be drawn. The number of occurrences of each objective is summarized in Table 7. Table 8 gives an overview of the major objectives and their definitions as they are used in this master thesis. Objective Number of contributions in the literature [42] Overtime 26 Operating room utilization 20 Patient priority 17 Waiting time 15 Makespan 11 Costs 10 Number of PACU nurses 2 Preferences staff 2 Number of beds used 2 Total completion time 2 Table 7: Overview of the different objectives in the literature on the surgical case assignment and sequencing problem. 28

46 Objective Utilization Underutilization Overutilization Makespan Waiting time Definition The total duration of the planned surgeries divided by the total available operating time. Assigned operating room workload is lower than available operating time. Assigned operating room workload is higher than available operating time. Time between the first patient entering the operating room and the last patient leaving the operating room. Time between the hospitalization date of a patient and the surgery date. Table 8: Major objectives and their definitions. Overtime and operating room utilization are the most used objectives. As already has been explained in paragraph , overtime and operating room utilization sometimes refer to the same concept (Cardoen et al., 2010; Demeulemeester et al., 2013). Therefore, it is possible that a paper is categorized within the objective of operating room utilization when overtime is also minimized. Operating room utilization and overtime can be optimized in different ways. Some papers maximize the utilization by maximizing the total duration of the scheduled surgical cases (e.g. Doulabi et al., 2014; Marques, Captivo, & Pato, 2012, 2014), which is satisfactory for both the hospital and the surgeons. Surgeons want to perform as many operations as possible in their available time (Marques et al., 2014; Vijayakumar et al., 2013). The management of the hospital wants to maximize its revenues, which is achieved by an optimal allocation of the available capacity. Others aim at minimizing both the under-and overtime or only the overtime, which can be done by using a cost function (e.g. Guido and Conforti, 2016; Jebali et al, 2006; Roland et al., 2006, 2010). The hospital and the nurses want to avoid overtime as much as possible, since it leads to high personnel costs, nurse dissatisfaction and work fatigue (Roland et al., 2010). However, surgeons often want to use some overtime in order to be able to perform more surgical cases. Patient priorities are the third most used objective, reflecting the patient preferences and mainly impacting the ordering of surgical cases. Waiting time is the fourth most used objective, being used in 15 papers. All patients have right on minimal waiting times. An adequate scheduling should aim at minimizing this waiting time taking into account the patients characteristics (Cardoen et al., 2009a). Makespan is used as an objective in eleven contributions. Both surgeons and the hospital benefit from makespan minimization, since more time is available to schedule surgeries in this way. 29

47 Costs as an objective is used in connection with the objectives of Table 8, for example the minimization of overtime costs. Other objectives used are total completion time, number of beds used, number of PACU nurses and the preferences of the staff, but they occur less frequent. These finding are in line with a survey on the performance criteria of operating theatre planning in Flanders by Cardoen et al. (2007). The survey indicates that 64% of the respondents find the utilization of the operating rooms the most important objective. About 43% indicate that the avoidance of overtime is an important objective. Also throughput and the preferences off the staff are considered to be important (Cardoen et al., 2007) Overview of the different approaches used to address the surgical case assignment and sequencing problem The problem characteristics of the previous paragraph provided an overview of the task characteristics, resource requirements and performance criteria of the operational phase in general. As already has explained in the beginning of this paragraph, a distinction can be made between the two stages of the operational phase, namely the assignment and sequencing stage. Consequently, the characteristics of these stages differ. Therefore, an overview of the contributions to the surgical case and sequencing is provided here together with their characteristics, distinguishing between four different configurations of the problem. First, the assignment and sequencing stage are discussed separately. Next, the contributions that deal with the operational phase using two sub-steps are discussed. Finally, the contributions that consider the problem as an integrated model are discussed. The emphasis of this overview lies on this last configuration of the problem, since the model proposed in Chapter 5 will mainly be based on these contributions. The characteristics discussed are the following: the scheduling system, i.e. block or open scheduling system, the mathematical model used, the objectives pursued, the resources considered, the length of the planning horizon and the type of stochasticity that is eventually discussed. After each configuration of the problem a short overview is provided in order to be able to compare the different configurations Assignment phase (advance scheduling) A lot of research focuses solely on the assignment phase. In this phase, also called the advance scheduling phase, the assignment of specific surgical cases to a certain time block is determined. The sequencing of the operations is determined in the next phase. After each surgeon has been assigned a certain number of time blocks resulting in the master surgery schedule, the surgical cases of the surgeon need to be assigned to their time blocks. When assigning surgical cases to time blocks, 30

48 several considerations need to be taken into account which make the assignment a critical step in the operating room planning and scheduling process (Jebali et al., 2006). Typically, surgeons and operating rooms are considered as critical resources in this phase (Risse & Burke, 2011). The first important contribution to the surgical case assignment phase has been done by Ozkarahan (2000). He focuses on the assignment phase and uses a goal programming approach which is aimed at minimizing idle time and overtime in order to maximize satisfaction of surgeons, patients and staff. His multi-criteria integer goal programming model intents to satisfy all these conflicting goals simultaneously. The model takes into account recovery room bed, equipment and staff availabilities and uses a one-day planning horizon. Ogulata and Erol (2003) also use a multi-criteria binary integer programming model in order to maximize the utilization of the operating rooms, balance the distribution of operation days and minimize the patient waiting times. They pursue the same goals as Ozkarahan (2000). However, they only take into account staff restrictions and ignore other possible resource constraints. They break the problem down into three distinctive phases in order to facilitate the solving process and apply a weekly planning horizon. Chaabane, Meskens, Guinet and Laurent (2008) view the hospital as a multi-service production system, constrained by a limited number of resources. They target an open scheduling system, which is characterized by an empty schedule that is filled up with surgical cases based on their arrival time. The objective is to improve patient satisfaction, by minimizing waiting times and to minimize the total costs, by focusing on the minimization of overtime. The only resource constraints incorporated are staff restrictions. They model the problem as a capacity constraint assignment problem. Another important contribution to the assignment literature has been done by Fei et al. (2008, 2009). Fei, Chu, Meskens, & Artiba (2008) solve the surgical case assignment problem in a single objective optimization model in order to minimize the total operating cost. They base their work on a previous model proposed by Fei, Chu, Artiba, & Meskens (2004). They start from a binary integer programming model and use a block scheduling strategy, which they transform into a set partitioning problem. Fei et al. (2009) developed a binary integer programming model and take into account due dates, operating room opening hours and staff availability. As opposed to Fei et al. (2008), they use an open scheduling strategy and multifunctional operating rooms. The objective is to maximize the utilization and minimize the overtime cost of operating room. By minimizing both over-and under-utilization 31

49 the objectives are similar to those used by Ozkarahan (2000), Guinet and Chaabane (2003) and Jebali et al. (2006). It should be noted that none of the approaches presented above incorporate any type of stochasticity, i.e. surgery durations, arrival times or non-elective surgical cases (emergencies), since they lead to a huge amount of uncertainty and increase the complexity of the problem. Hans et al. (2008) try to overcome this drawback. They model the assignment problem as a stochastic knapsack problem. The objective is to maximize the operating room utilization and minimize the overtime, which implies minimizing the risk on cancelled patients. They incorporate duration uncertainty and apply a block scheduling approach (Marques et al., 2012). Lamiri et al. (2008b & c) also incorporate uncertainty by taking into account emergency surgeries in their planning of surgical cases. They aim at minimizing overtime, waiting time and hospitalization costs. Conclusion Table 9 gives an overview of the characteristics of the papers that address the surgical case assignment problem. Scheduling system block scheduling open scheduling Mathematical model 7 integer programming 3 multiobjective simulation model stochastic optimization Criteria Resources Length of planning horizon Stochasticity 9 overtime 9 staff 5 daily 1 none 7 3 OR utilization 6 none 4 weekly 9 emergencies 2 2 costs 3 PACU beds 1 duration 2 / waiting time patient priority 6 equipment 1 total operating time 3 ICU / makespan / stay beds / / Table 9: Overview of the characteristics of the assignment phase. From this table some conclusions can be drawn. Most of the papers use a block scheduling approach. Second, integer programming turns out to be the most popular modelling approach, sometimes in combination with multi-objective optimization. The most popular objective is overtime minimization, operating room utilization and patient waiting time are also used frequently. The patient waiting time considered in this phase is in most cases the indirect waiting time, i.e. the time between the request for surgery and the actual date scheduled for surgery (Gupta & Denton, 2008; Riise & Burke, 32

50 2011), as has been described in paragraph Four out of ten papers do not consider any resources. The most popular resource considered is the availability of the staff. Since the assignment phase involves less detail compared to the sequencing phase, most assignment models consider a weekly planning horizon. Stochasticity is not considered in most of papers Sequencing phase (allocation scheduling) Other research only focuses on the sequencing phase, also called the scheduling phase or intervention scheduling (Riise & Burke, 2011). After all the surgical cases have been assigned to a certain time block, the sequence of the operations within a certain time block needs to be determined. Each operation is scheduled individually by assigning a certain expected start time to it, taking into account the resource availabilities (Lee & Yih, 2014). In this way the human and material resources needed are synchronized (Guinet & Chaabane, 2003). The first contribution to the literature of the sequencing of surgical cases has been done by Sier, Tobin and McGurk (1997). They developed a mixed integer nonlinear model that aims at scheduling surgical cases over multiple operating rooms on a daily planning horizon. The multi-objective function aims at minimizing the waiting time for children, minimizing the waiting time for prioritized patients, scheduling surgeries with long duration first and minimizing the penalty for equipment conflicts, block overruns and collisions. They take into account equipment and staff availabilities. Hsu, de Matta, & Lee (2003) treat the problem in a different way. They tackle it as a variant of the two-stage no-wait flow shop scheduling problem taking into account the PACU capacity. In the first sub problem they minimize the makespan (i.e. the completion time of the last patient), in the second sub problem they minimize the number of nurses required in the PACU. They use a block scheduling approach and take into account the availability of staff and surgical materials (Guerriero & Guido, 2011). Pham and Klinkert (2008), Arnaout and Sevag (2008), Augusto et al. (2010), Dekhici and Belkadi (2010) and Lee and Yih (2014) all use similar approaches based on the classic machine scheduling problem configuration aimed at minimizing the makespan. Pham and Klinkert (2008) represent the model as a mixed integer linear programming model that includes emergencies and the capacity of both the PACU and ICU. Patient priorities are taken into account. Arnaout and Sevag (2008) take into account the uncertainty of surgery durations, but do not consider any resource availabilities. They use an open scheduling approach and work on a weekly planning horizon. In contrast, Augusto et al. (2010) and Dekhici and Belkadi (2010) use a block scheduling approach and consider recovery room bed availability as a bottleneck resource (Latorre-Núñez et al., 2016). Lee and Yih (2014) also aim at 33

51 minimizing the makespan. They incorporate emergency surgeries, duration uncertainty and the bed availability in the PACU. Marcon and Dexter (2006) and Cardoen et al. (2009a, 2009b) use various sequencing rules to schedule surgical cases. Marcon and Dexter (2006) use rules such as longest case first, shortest case first, etc. The objectives pursued are related to the performance of the operating rooms and PACU such as operating room overtime, PACU completion time, PACU delays and PACU staffing (Guerriero & Guido, 2011). They indicated the longest case first rule has a negative impact on the performance of both the operating room and PACU. They take into account the recovery room bed availability and the uncertainty of surgery durations. Cardoen et al. (2009a) aim at scheduling children and high priority patients first, patients coming from a considerable distance later in the day and leveling the bed occupancy at the recovery room. The problem is solved by using a deterministic mixed integer linear programming model where they assign start times to each surgical case within each operating room within each time block for a given day. They apply a block scheduling approach and take into account the limited availability of recovery rooms beds and medical equipment on a one-day planning horizon (Guerriero and Guido, 2011). Lamiri, Augusto and Xie (2008a) schedule patients in a hospital operating theatre considering three types of resources: porters, operating rooms and recovery room beds. They aim at minimizing the total patient s completion times in order to minimize the total cost of the operating theatre. Meskens, Duvivier and Hanset (2013) take into account the preferences of the personnel (surgeons, nurses and anesthesiologists) and include it as a constraint (Latorre-Núñez et al. 2016). They incorporate recovery room beds and equipment restrictions. The objective is to minimize the makespan, minimize overtime hours and maximize the satisfaction of the surgeons by taking into surgeon s preferences. This multi-objective scheduling problem is translated into a constraint programming model. 34

52 Conclusion Table 10 gives an overview of the characteristics of the papers that address the surgical case sequencing problem. Scheduling system block scheduling open scheduling Mathematical model 13 integer programming 2 multiobjective simulation model stochastic optimization Criteria Resources Length of planning horizon Stochasticity 12 overtime 8 staff 6 daily 10 none 8 4 OR utilization 3 none / weekly 6 emergencies / 4 costs 2 PACU beds 8 duration 5 4 waiting time patient priority 3 equipment 6 total operating time 6 ICU 1 makespan 9 stay beds / 1 Table 10: Overview of characteristics of the sequencing phase. From this table some conclusions can be drawn. Similar to the assignment phase, most of the papers use a block scheduling approach and apply integer programming. The objectives pursued in the sequencing stage are different from those of the assignment stage. Minimization of the makespan is the most popular objective. Besides this, also overtime and patient priority are often used. In contrast to the assignment stage, operating room utilization is not used frequently. Also the length of the planning horizon used differs from the assignment stage. Here, 10 out of 14 contributions used a daily planning horizon, mainly due to the complex nature of this problem. With respect to the consideration of resources, we can state more resources are considered in this stage. Only 3 out of 14 contributions do not consider any resources. As opposed to the assignment stage, the incorporation of the capacity of the PACU is popular, with eight papers considering the number of PACU beds, compared to only one paper in the assignment stage. Besides this, also staff and equipment are popular resources considered. The incorporation of uncertainty remains low, however, there is a small increase compared to the assignment stage mainly with respect to the incorporation of duration uncertainty The assignment and sequencing problem addressed by two sub-problems The assignment and sequencing of surgical cases are highly interrelated. Therefore, a significant number of authors have tried to tackle the problem using a two-step method. They consider both the 35

53 assignment and sequencing problem as two related sub-problems (Ozkarahan 1995; Guinet and Chaabane 2003; Jebali et al. 2006; Fei et al. 2006; Fei et al. 2010; Landa et al. 2015). The majority of the works mentioned hereafter had an important contribution to the surgical case planning and scheduling literature (Fei et al. 2006; Fei et al. 2010; Guinet & Chaabane 2003; Jebali et al. 2006; Landa et al. 2015; Ozkarahan 1995). Ozkarahan (1995) starts from a job shop problem to solve the operating room planning and scheduling problem for elective surgical cases. He transforms it into an integer linear programming model that minimizes the makespan. Guinet and Chaabane (2003) address both the assignment and sequencing step in two distinctive sub-problems. They tackle the problem as multi-service production system, constrained by a limited availability of resources, such as recovery room bed, equipment and staff availabilities. However, they do not incorporate time-of-day availabilities and priorities (Vijayakuram et al., 2013). They developed a NP hard linear integer programming model to address the assignment of surgical cases. The objective of the first phase is to improve patient satisfaction and resource efficiency by minimizing the hospitalization costs and overtime costs over a two weeks planning horizon (Marques et al. 2012). The sequencing problem is modelled using a two-stage hybrid flow shop problem in which stage one represent the operating room and stage two the recovery room. Here, the objective is to minimize the makespan on a daily planning horizon. Another important contribution to the assignment and sequencing literature has been done by Fei et al. (2006, 2010). Fei et al. (2006) address both the allocation and the sequencing phase by dividing it into two sub-problems modelled as a binary set partitioning master problem and a two-stage flow shop problem, considering the recovery room availability. They minimize the total undertime and overtime operating cost. They apply a weekly planning horizon for the assignment problem and a daily planning horizon for the sequencing problem. They assume a block scheduling strategy in which the groups of surgeons are assigned to time blocks. The major drawback of this approach is that they do not take into account the surgeons availability, emergencies and other resource constraints. The same approach is mainly used by Jebali et al. (2006), but they overcome the drawbacks of Fei et al. (2006) by incorporating surgeons availability and resource usage constraints. They aim at improving the patient satisfaction and resource efficiency, on a daily planning horizon. They do not only minimize overtime and undertime operating costs, but also incorporate the hospitalization costs in order to minimize patient waiting times. This is presented as a cost function in a mixed integer programming model for the two steps by using a two-stage hybrid flow shop problem, which is similar to the model presented by Guinet & Chaabane (2003). 36

54 Fei, Chu, & Meskens (2010) aim at maximizing the operating room utilization and minimizing overtime cost and idle time between operations on a weekly planning horizon. Similar to the model in Fei et al. (2006), they use a set-partitioning integer programming model. Each surgical case is assigned a date for surgery. They try to implement the open scheduling strategy in order to improve the performance of the model compared to Fei et al. (2006). The method turned out to be successful since less idle time, less overtime and a higher utilization are obtained. Surgeons availability is taken into account in this model which increases its accountability compared to the model used in Fei et al. (2006). None of the above approaches take into account stochastic aspects in their problem formulation, this drawback is solved by Landa et al. (2015). Landa et al. (2015) both plan and schedule surgical cases on a weekly planning horizon taking into account the inherent hierarchy between the two decision levels. Their objective is to minimize the patient waiting time costs and to maximize the operating room utilization by creating a cost function. They formulated a binary integer programming model to solve the problem. They take into account stochastic surgery durations in their problem formulation. Conclusion Table 11 gives an overview of the characteristics of the papers that address the surgical case assignment and sequencing problem as two sub-problems. Scheduling system block scheduling open scheduling Mathematical model 3 integer programming 3 multiobjective simulation model stochastic optimization Criteria Resources Length of planning horizon Stochasticity 6 overtime 4 staff 3 daily 3 none 5 4 OR utilization 4 none / weekly 4 emergencies / / costs 3 PACU beds 3 duration 1 / waiting time patient priority 3 equipment 2 total operating time / ICU 1 makespan 1 stay beds 1 / Table 11: Overview of characteristics of the surgical case assignment and sequencing problem considered as two subproblems. 37

55 From this table some conclusions can be drawn. The division between the block and open scheduling approach is more balanced in this configuration of the problem. Integer programming remains the most popular modelling approach, in combination with a multi-objective function. Overtime and operating room utilization are the most frequently used objectives. Three papers present the objective function as a cost function. Only one out of the six papers does not consider any resources. The incorporation of PACU beds and staff availabilities is the most popular. Due to the consideration of both the planning and scheduling of surgical cases is this configuration, both weekly and daily planning horizon are used. Stochasticity has only been addressed by Landa et al. (2015) The assignment and sequencing problem using an integrated approach The decomposition of assignment and sequencing problem as tackled above has several disadvantages. Cardoen et al. (2009a) state the quality of the surgery scheduling in the sequencing step highly depends on the assignment that has been obtained in the advance scheduling step. This is because the sequencing is constrained by the assignment step, in which the sequencing objectives were not taken into account (Riise & Burke, 2011). Jebali et al. (2006) indicated that a rescheduling step where modifications can be made on the previous assignment phase, gives better results compared to using the input of the assignment phase as a given. The consideration of the surgical case assignment and sequencing problem in an integrated way has received more attention only recently. Research on this topic is more limited since tackling the problem in an integrated way increases the complexity of synchronization and thereby the computational effort. However, some papers have considered the assignment and sequencing phase in one single step considering simultaneously the advance and allocation scheduling on a short term planning horizon of one week or one day (Doulabi et al. 2014; Riise & Burke, 2011; Roland et al., 2006, 2010). Most contributions use an integer programming approach using a binary decision variable denoting a surgical case will start in a certain operating room, on a certain day and on a certain start time (Doulabi et al. 2014; Marques et al. 2012, 2014; Riise and Burke, 2011; Roland et al., 2006, 2010; Vijayakumar et al., 2013). Roland et al. (2006) had an important first contribution to the integration of the assignment and sequencing phase. They use an integrated approach by creating a binary integer model that tackles both the assignment and the sequencing phase. They aim at minimizing the opening costs of the operating rooms and the overtime. Their approach is based on a recourse constrained project scheduling model by taking into account renewable and non-renewable resources, however, the PACU is not taken into account. Roland et al. (2010) extend this problem by considering human resource availabilities. 38

56 Velásquez and Melo (2006) maximize the preferences of the surgeons to perform surgeries. Resource availabilities such as operating rooms, staff and equipment are taken into account and sets of possible combinations of resources are defined. In this way, a variety of resources and preferences can be used. They schedule elective surgeries on a planning horizon of a few days up to one week in advance using a decision variable that determines if a certain surgery is scheduled at a start time using a certain resource combination. They highlight the inherent difficulty of solving the operating theatre scheduling problem because of the conflicting objectives of management, patients and staff imposed. Persson and Persson (2009) tackle the problem using an open scheduling system. They combine both optimization and simulation on a four-week planning horizon. They focus on the minimization of patient costs and surgery costs taking into account the medical priority of patients, the queuing time and the available resources. Similar to Roland et al. (2006, 2010), the objective function is a cost function penalizing overtime. However, in Persson and Persson (2009), also post-operative care resources are considered, besides equipment and staff. They do not take into account any stochasticity aspects. They start their approach by selecting patients from waiting lists generated by the hospital (Latorre-Núñez et al., 2016). Their research was aimed at verifying the implications of a new law imposed by the government. Riise and Burke (2011) solve the assignment and sequencing problem taking into account the availability of operating rooms and surgeons. They recognize the importance of dealing with the assignment and sequencing phase in an integrated way, avoiding an isolated approach. They do not incorporate patient priorities or other resource availabilities such as equipment or surgical materials (Vijayakumar et al. 2013). They developed a binary integer programming model using a decision variable which is equal to one when a certain surgery starts in a certain period on a certain day in a certain room. The objective function minimizes the patient waiting time, surgeon overtime, waiting time for children in the morning and the cost of not scheduling a surgery. The multi-objective function uses a normalized weighted sum approach in order to ensure all terms are displayed in the same order of magnitude. Riise and Mannino (2012) model the surgery scheduling problem as a general project scheduling model using an extension of the resource constrained project scheduling problem. They formulate the problem as a mixed integer linear programming model, aimed at providing a general framework for the surgery scheduling problem that is subject to a lot of variation among hospitals. They assign start times to all activities, while reserving capacity for renewable resources. Stochasticity is not considered. The objective function minimizes the makespan and contains a component that 39

57 maximizes the number of patients scheduled on a short time period. In this way priority is given to certain patients, for example children or diabetics. The general model is applicable to a wide range of surgery scheduling problems and must be adapted towards the needs of a specific hospital. Marques et al. (2012) tackle a case study of a hospital in Lisbon. They jointly consider the assignment and sequencing of elective surgeries within a weekly planning horizon using an integer linear programming model. The model is closely related to that of Guinet and Chaabane (2003). The objective is to increase the efficiency of hospital s surgical theatre by maximizing the utilization of the operating rooms. A block scheduling approach is applied. Constraints prohibit the overlap of surgeries for operating rooms and surgeons. Resource limitations such as beds, nurses, staff, materials and PACU beds are not considered. Four different priority levels for patients are defined, on which the selection from a weekly waiting list is based. Marques et al. (2014) continue the research of Marques et al. (2012) incorporating an extra objective, i.e. maximizing the number of surgical cases scheduled. Vijayakumar et al. (2013) address the problem as a multi-period, multi-resource, priority-based case scheduling problem and developed a mixed integer linear problem, with a decision variable equal to one when a surgery is scheduled in an operating room, on a certain day, on a certain start time. They aim at minimizing the costs of over-utilization. However, they state that usually the costs of resources are fixed and overtime is not possible. Therefore, the objective becomes to maximize the number of surgeries scheduled, for a fixed surgical time available and a fixed set of resources taking into account patient priority. This objective is similar to that of Marques et al. (2014). The model takes into account resource availabilities (i.e. surgical materials and nurses), patient priorities and variations in surgery times but does not incorporate the PACU. The block scheduling approach is applied. Doulabi et al. (2014) also focus on the integration of advance and allocation scheduling. They developed a binary integer programming model assigning and sequencing surgical cases on a weekly planning horizon, using a constraint programming based column generation approach. They also present the model as a compact formulation using binary decision variables. An open scheduling approach is used and duration stochasticity is not considered. Surgeons availability and recovery room beds are not seen as bottleneck resources and all operating rooms are identical. The objective is to maximize the operating room utilization. They distinguish between mandatory and optional surgeries. In their model, they use a coloring constraint that states that a surgeon can only be available in one operating room at a time on a given day. The model representation is based on Fei et al. (2009), apart from a surgeon overlapping constraint. 40

58 The most recent contribution to the literature on the integration of the assignment and sequencing phase has been done by Latorre-Núñez et al. (2016). They view the scheduling problem as a twostage hybrid flow shop problem, in which the two stages represent the operating room and PACU beds, respectively. Each job (i.e. surgery) is assigned to both stages (i.e. one operating room and one bed of the PACU). The problem is formulated as an MILP and aims at minimizing the makespan. The model schedules surgical cases with consideration of all resources, i.e. nurses, anesthesiologists, other professionals, physical resources (instruments, etc.), recovery room beds and emergency surgeries, for the first time in the literature. Priorities are assigned to certain surgical cases. Emergency surgeries are incorporated using a constraint that imposes a certain maximum waiting time allowed (van Essen et al. 2012). Conclusion Table 12 gives an overview of the characteristics of the papers that use an integrated approach. Scheduling system block scheduling open scheduling Mathematical model 10 integer programming 2 multiobjective simulation model stochastic optimization Criteria Resources Length of planning horizon Stochasticity 8 overtime 5 staff 9 daily 6 none 10 6 OR utilization 7 none / weekly 11 emergencies 2 / costs / PACU beds 5 duration 1 / waiting time patient priority 4 equipment 8 total operating time 7 ICU 2 makespan 3 stay beds / / Table 12: Overview of characteristics integrated surgical case assignment and sequencing problem. From this table some conclusions can be drawn. Most of the papers use a block scheduling approach. Second, integer programming turns out to be the most popular modelling approach, often taking into account multiple objectives. The multi-objective function approach is more popular in this configuration of the problem compared to the previous configurations that have been discussed. Besides this, the multi-objective function approach is gaining increasing attention during the last years (Demeulemeester et al., 2013; Riise and Mannino, 2016). Looking at the objectives pursued by the papers considering the operating room assignment and sequencing problem in an integrated way, we see that two objectives turned out to be the most 41

59 popular. Seven out of twelve contributions premise the maximization of the operating room utilization (Doulabi et al. 2014; Marques et al. 2012, 2014; Vijayakumar et al. 2013) of which two assume fixed resources and aim at maximizing the number of surgeries to schedule (Marques et al. 2014; Vijayakumar et al., 2013). Another seven out of twelve contributions take into account priority rules to schedule surgical cases, which have not been considered by the literature tackling the problem as two sub-problems. Four contributions tackle the minimization of waiting time as an objective, often in combination with other objectives. Five out of twelve contributions premise the minimization of overtime as objective (Roland et al. 2006, 2010; Persson & Persson, 2009; Riise & Burke, 2011). Another three out of twelve contributions aim at minimizing the makespan (i.e. the closing time of the last operating room) (Riise and Mannino, 2012; Latorre-Núñez et al. 2016). The resources taken into consideration in most of the papers are surgical equipment and staff. The recovery room and ICU are taken into account less frequently. In most of the contributions, stochasticity is not considered. Two third of the literature plans the surgical cases on a weekly planning horizon. An overview of the characteristics of all contributions considered in this literature review can be found in Appendix A. The authors that are mentioned in the list in Appendix A are indicated with an asterisk in the reference list. 42

60 4 CASE CHAPTER 4 STUDY This master thesis is performed in dialogue with the Algemeen Ziekenhuis Herentals. Therefore, the situation with regard to the planning and scheduling of surgical cases in the Algemeen Ziekenhuis Herentals is discussed in this chapter. The model presented in Chapter 5 will be adapted towards the situation at the AZ Herentals as described hereafter. The data of their operating theatre will be used as input for the proposed model in Chapter 5. First, a general introduction to the hospital is given. Second, the resources involved in the planning and scheduling of operations are discussed. Third, the different decision levels with respect to the surgical case planning problem are discussed. 4.1 GENERAL INTRODUCTION TO THE ALGEMEEN ZIEKENHUIS HERENTALS The hospital discussed in this case study is the Algemeen Ziekenhuis Herentals (AZH), a regional hospital situated in the province of Antwerp. The AZ Herentals is a modern and dynamic hospital, it can be perceived as one of the most high-quality hospitals in the province of Antwerp. Their policy is focused on quality, patient safety, expertise, cost-effectiveness, reducing information asymmetries for patients and collaboration with external and internal partners. They occupy 243 beds, 600 employees and 100 doctors. They offer all basic services concerning surgery, general internal medicine, geriatrics, maternity, pediatrics and rehabilitation. Besides this, they also have an emergency department, an intensive care unit (ICU), an operating theatre and a renewed day hospital. They are well-known for their elaborated orthopedics department which consists of a sports hospital and a post rehabilitation center. In 2008 they started renewing the infrastructure of the hospital and expanding the current capacity. These renovations and expansions will be completed in In 2011, they signed an alliance with the AZ Turnhout, named HETU in order to offer more specialized treatment. This expansion and organizational changes all are situated within the ten-year master plan to become one of the leading hospital in the province of Antwerp (URL: < Over time, the AZH has known an evolution with regard to the organization of its services. In the past, there existed a care manager, a responsible person for the operating theatre etc. for all disciplines together. Recently, they are evaluating more towards a care-oriented approach in which each discipline has a care manager, a person responsible for the consultations, etc. In this way the 43

61 disciplines are becoming more and more independent entities (interview with Doctor Stefaan Verfaillie, Orthopedic surgeon, 29 th April 2016). This evolution is in line with what Degadt and Van Herck (2003) and Lega and DePietro (2005) identified. They found out that functional-like organizations are transforming towards more carefocused organizations because of the need for coordination in the different hospital disciplines. 4.2 CURRENT SITUATION AT THE AZH In this paragraph the current situation at the operating theatre of the AZ Herentals is explained. First, the actors involved in the functioning of the operating theatre are discussed. Second, the available infrastructure and its characteristics are examined. Finally, the different types of surgical cases existing at the AZH are discussed The different actors in the operating theatre of the AZH Different actors, i.e. nursing staff, anesthesiologists, surgeon and patients are involved in the operating theatre. These actors are required in order to ensure a well-functioning operating theatre The nursing staff The nursing staff is the biggest stakeholder group in the operating theatre of the AZH. They are also responsible for the highest cost in the operating theatre. Each operating room employs two nurses. These nurses need to prepare the patient and the operating room for surgery, clean up the room after surgery and assist the surgeon during surgery. Besides this, also two to three nurses are employed in the recovery room permanently in order to take care of patients that have been operated Anesthesiologists The AZH employs nine anesthesiologists but they do not all work full-time in the hospital. Anesthesiologists take care of the anesthesia that is applied on patients. This can be either a general anesthesia or a local anesthesia, depending on the surgery characteristics. The type of anesthesia influences the time a patient has to stay in the recovery room and consequently also influences the occupation and workload in the recovery room Surgeons The AZH employs 31 surgeons that perform surgeries. They can be subdivided among the different disciplines: orthopedics (ORT), vascular and thoracic surgery (VAT), urology (URO), general surgery (CHI), oral and maxillofacial surgery (THK), gynaecology (GYN) and ear, nose and throat disorders 44

62 (NKO). Table 13 gives an overview of the different disciplines and the number of surgeons per discipline. Abbreviation Discipline Number of surgeons ORT Orthopedics 9 VAT Vascular and thoracic surgery 2 URO Urology 2 CHI General surgery 3 THK Oral and maxillofacial surgery 4 GYN Gynaecology 7 NKO Ear, nose and throat disorders 3 Table 13: The different disciplines in AZH and the number of surgeons per discipline. It should be noted that the AZH still has other disciplines than the ones in Table 13, but these are not considered here because those disciplines do not perform a considerable amount of surgeries and insufficient data is available for these disciplines Patients As already described in the literature review in paragraph 2.3, patients can arrive in the operating theatre from three different sources, i.e. the suite of regular wards, the emergency department and the ambulatory surgical centers or the day hospital. The same situation applies for the AZH. On average, 10% of the patients come from the emergency department, 50% from the suite of regular wards and 40% from the day hospital Categories of surgical cases Three different categories of surgical cases, i.e. planned surgeries, emergencies and urgent-planned surgeries, can be distinguished in the AZH (based on data made available by Katrijn Vanlommel, 1 st of March 2017). Planned Planned surgeries are surgeries that can be planned long in advance and are not required to be performed as soon as possible. Patients can make an appointment for a planned surgery in two ways. The first way is during consultation with the surgeon in the hospital. A date for surgery is agreed upon. When the patient still needs to consider the possible dates for surgery at home, the surgeon 45

63 registers the surgical case already in the database. Afterwards, the patient calls to the hospital to confirm the date and the secretary plans the surgery in the agenda of the surgeon. Emergencies Emergencies or urgencies need to be performed as soon as possible. In most cases, they arrive in the hospital via the emergency department. Rarely, they arrive coming from the suite of regular wards. Emergencies are scheduled within the existing schedule and marked as urgent. There does not exist a separate emergency operating room since this would be inefficient for the hospital as the amount of emergency surgeries is not enough to fill the capacity of one operating room. It is generally agreed in the hospital that emergencies need to be taken care of in the discipline they belong to. For example, when an emergency arrives with a broken hip, this patient needs to be operated in one of the time blocks of the Orthopedics department. Only for very urgent matters a discipline will give some of its operating time to the other discipline for which the emergency appears (interview with Doctor Stefaan Verfaillie, Orthopedic surgeon, 29 th April 2016). Urgent planned Another category of surgeries besides the planned surgeries and the urgencies are the urgent cases that can be planned for in advance. These cases do not arrive in the hospital via the emergency department or do not need surgery immediately. However, these surgical cases need to be planned as soon as possible in the system. They marked as urgent-planned and have priority over the planned surgeries. 4.3 THE DIFFERENT DECISION LEVELS IN THE AZH The same decision levels with respect to the planning and scheduling of operating rooms can be identified at the AZ Herentals Strategic planning level The strategic planning level determines the available capacity in the long term. We restrict ourselves to two factors that are relevant for this research, i.e. the operating room capacity and the availability of overtime. Operating rooms The AZH currently has eight operating rooms and two rooms for small surgical cases, also called the day hospital. From these eight operating rooms, two opened only two years ago. Currently, they are able to perform 50 to 60 surgeries on average per day. The Orthopedics department is the biggest 46

64 department of the hospital, using more than 50% of the total available operating time and is still expanding quickly. The number of people treated in the day hospital has increased rapidly during the last years. The surgeons try to treat as many patients as possible in the day hospital, if the patients health allows, in order to have more beds available in the regular wards. The hospital aims at increasing this capacity further, by expanding the current day hospital with two new operating rooms within one year. These new rooms will be situated near the ambulatory surgical centers, in order to provide a better accessibility for the surgeons when releasing patients from the hospital and thus working more efficiently. The operating theatre opens at 8:00 AM and closes at 16:30 PM. This leads to a weekly available operating time of 340 hours, excluding the capacity at the day hospital. During the weekend usually two operating rooms are open for emergencies. Not all the operating rooms are multifunctional. For the performance of prosthesis, laminar airflow is required. This laminar airflow is only available in five operating rooms, namely in operating room five, six, seven and eight (Interview with Doctor Stefaan Verfaillie, Orthopedic surgeon, 29 th April 2016). Overtime In the AZH it is currently possible to work overtime, as it is allowed by the Klinisch Werkstation software to plan operations taking longer than the regular opening times of the operating rooms. Surgeons see this as an improvement on the previous system, in which it was not possible to plan for overtime. Even when they needed only five minutes extra to perform a surgery, they were not able to plan this surgery. However, as imposed by the hospital s regulations, working overtime is not allowed as a regular activity. When the planning for the next day is verified, the planning team may require the surgery to be cancelled because of an excessive amount of overtime. It is possible to plan for surgery up to 24 hours before the next day, after this period, the nurses required for the surgery have to give their permission to perform the surgery. They often refuse to perform the surgery since they want to leave the hospital at the regular closing time of the operating room (Interview with Doctor Stefaan Verfaillie, Orthopedic surgeon, 28 th February 2017). 47

65 4.3.2 Tactic planning level The tactic planning level results in the master surgery schedule, where time blocks are assigned to the different departments. At the AZ Herentals, this planning phase is performed by a team of planners at the managerial level. The division of operating room time over the different departments is performed based on the demand for the different departments and the number of surgeons. Also the efficiency is taken into account by calculating the utilization rate. This division does not change frequently and is mainly based on historical data, evolution and experience. After the division among departments, the available time is further divided among the surgeons of a specific department. The current master surgery schedule is presented in Table 14 (based on data made available by Katrijn Vanlommel, 1 st of March 2017). Monday Tuesday Wednesday Thursday Friday OR AM PM AM PM AM PM AM PM AM PM 1 URO URO THK THK URO URO VAT VAT CHI CHI 2 ORT ORT ORT ORT CHI CHI ORT ORT ORT ORT 3 CHI CHI ORT ORT ORT ORT ORT ORT ORT ORT 4 ORT ORT ORT ORT ORT ORT ORT ORT ORT ORT 5 ORT ORT ORT ORT ORT ORT ORT ORT VAT VAT 6 VAT VAT GYN GYN GYN GYN NKO GYN NKO NKO 7 ORT ORT THK/NKO VAT ORT ORT CHI GYN ORT ORT 8 ORT ORT CHI CHI ORT ORT VAT GYN ORT ORT Table 14: The master surgery schedule of the AZH. In the Orthopedics department it may happen a surgeon possesses two operating rooms at the same time. In this way the surgeon is able to perform more operations compared to when he has two operating rooms available in two different time blocks. Whether a surgeon has two rooms at the same time depends on the type of surgeries he has to perform, since the surgeon only has one anesthesiologist at his disposal. It is not possible to work in two operating rooms at the same time when children are involved, since the anesthesiologist is obliged to stay with the child during the surgery. Moreover, when long surgeries are planned it is also not efficient to work in two operating rooms. This method turns out to be efficient in the case short operations with a long preparation phase need to be performed (Interview with Doctor Stefaan Verfaillie, Orthopedic surgeon, 28 th February 2017). Based on the master surgery schedule the total operating time per discipline per week can be calculated. This is shown in Table

66 Abbreviation Discipline Number of assigned time blocks Total operating hours per week ORT Orthopedics ,5 VAT Vascular and thoracic surgery 2,5 10,6 URO Urology 4 17,0 CHI General surgery 8 34,0 THK Oral and maxillofacial surgery 9 38,3 GYN Gynaecology 7 29,8 NKO Ear, nose and throat disorders 3,5 14,9 Table 15: The operating room time per discipline. Figure 8 displays the division of the operating time over the different disciplines in terms of percentage. It is remarkable that 58% of operating time is assigned to the Orthopedics department, which is still expanding every year. The Oral and maxillofacial surgery department, Urology department and Ear, nose and throat disorders department only account for 12 % of the total operating time. Figure 8: Division of the operating time over the different disciplines in terms of percentage Operational planning level As already described, the operational planning phase concerns both the assignment of surgeries to operating rooms and time blocks of the master surgery schedule, as well as the sequencing of individual surgeries within these time blocks. In the following paragraph, the way this process is currently performed in the AZ Herentals is described. 49

67 Planning systems The AZ Herentals has known three different planning systems: 1. Manual system 2. E-care operating room planning system 3. Klinisch Werkstation software They started from a completely manual system and have evaluated towards a software-driven approach. In the manual system, they worked with a book that contained the division of the time blocks of half a day over the surgeons. Each day, the corresponding page was copied out of the book and hang up on the wall. In this way, surgeons had to write the surgeries they wanted to perform in the book. Second, they adopted the E-care operating room planning system, which provided a framework for planning and scheduling surgeries. This system was not well- integrated with the other departments of the hospital and patient information needed to be inserted manually. This is the main reason why they shifted towards the current system, i.e. the Klinisch Werkstation software (Interview with Doctor Stefaan Verfaillie, Orthopedic surgeon, 28 th February 2017). The tool Klinisch Werkstation, is a project of the University Hospital of Leuven and is evolved in the Nexuz health platform, a medical cooperation of hospitals using the Klinisch Werkstation software. Currently it is the most used software in Flanders, with 20 hospitals being part of the Nexuz health network. One of the strengths of the software is that it is integrated within the hospital, linking patient information by using an electronic patient dossier (EPD). (URL: Surgical case assignment at the AZ Herentals The surgical case assignment problem in the AZH is located at the level of the specific department, for example the Orthopedics department. This phase is performed by the doctor himself by using the Klinisch Werkstation software. When a patient makes an appointment for surgery during a consultation, the surgeon needs to select a certain surgery type and link the specific patient information to it. As a consequence, each surgeon has a database consisting of the operations he or she has to perform and the specific patient information linked to each operation. Each surgery has a standard duration, depending on the type of operation. However, a surgeon is able to adapt these standard durations, if he is convinced he needs more or less time to perform a certain surgery. The surgeon will manually assign each surgery to a certain time block, depending on the availability of operating time in each time block and the urgency of the surgery. It is also possible to assign overtime to the time blocks. Every day the planning is evaluated by the person responsible of the operating rooms, the head of the department and the person responsible of the material. 50

68 It is clear that the scheduling of surgical cases by the doctors is done by using trial & error and based on their experience. Doctors try to perform as many surgical cases as possible within a time block, since they are paid per surgical case performed. The hospital tries to achieve a high efficiency and patient satisfaction resulting from the operating theatre by minimizing the waiting time for patients and by dividing unused time blocks among the other surgeons. They also strive towards the minimization of the number of days a patient lays in the hospital, and are currently one of the best performing hospitals in Flanders. It often happens that the bed capacity in at the departments is not sufficient. Therefore, they try to bring as many patients as possible to the day hospital in order to minimize the lead time (Interview with Doctor Stefaan Verfaillie, Orthopedic surgeon, 28 th February 2017) Sequencing of individual cases at the AZ Herentals Once the surgical cases have been assigned to the time blocks, the sequence of the surgical cases needs to be decided upon. Here, the preferences according to the start time of a certain operation in a certain room are expressed. Based on an interview with Dr. Verfaillie it turned out that a certain priority order is followed by the doctors in AZH in order to define the sequence in which they schedule surgical cases. This ordering is generally agreed upon in the AZH. Patients with latex allergy are always treated first. Second, children get priority. Children need to be sober on the day of surgery. Because it is difficult for them to wait for a long time without eating or drinking something, it is tried to schedule them early in the day. Third, planned urgencies, i.e. surgeries that can be planned for but should be performed as soon as possible, are scheduled. Next, they try to schedule patients coming from the day hospital. In this way, the surgeons are able to release these patients during their lunch break. If a surgeon also has to perform surgeries in the afternoon, those patients have to wait for a long time in the ambulatory surgical centers before they can go home. Next, the surgeons try to schedule surgeries with short durations first. Finally, longer surgeries are scheduled. It also happens that a surgeon has to perform two surgeries per day for which special material is needed. If this occurs, the material needs to be sterilized after the first time it has been used. Consequently the surgeon will try to schedule this operation early in the morning. The type of anesthesia is another factor that may affect the scheduling of surgeries. 51

69 Priority Surgeries 1 Latex allergy 2 Children 3 Planned urgencies 4 Day hospital surgeries 5 Short surgeries 6 Long surgeries Table 16: Priorities of patients. Based on this priority ordering, the surgeons will schedule their surgical cases. It should be noted that surgeons may also have personnel preferences regarding the scheduling of their surgical cases, these are not considered here. 52

70 5 MATHEMATICAL CHAPTER 5 MODEL As mentioned earlier, this master dissertation concerns the operational phase of the operating room planning problem and focuses on the integration of the assignment and sequencing problem. In this chapter the mathematical model to solve this problem in an integrated way is presented. Most approaches in the literature solve this problem separately, considering the assignment and sequencing phase in two distinct models (Fei et al. 2006; Fei et al. 2010; Guinet and Chaabane 2003; Jebali et al. 2006; Landa et al. 2015). The aim in this study is to overcome the negative consequences of splitting this problem into two sub-problems. Besides this, the model presented is adapted towards the current situation in the AZ Herentals and the assumptions they use are incorporated in the model. Based on the characteristics and assumptions discussed in Chapter 3 and 4, a mixed integer programming model is presented in order to assign and sequence the surgeries of the AZ Herentals. The model is mainly based on the contributions of Marques et al. (2012), Riise and Burke (2011), Roland et al. (2006, 2010) and Vijayakumar et al. (2013) 5.1 DATA Every surgeon j belonging to the set of surgeons J will perform a certain number surgical cases i belonging to the set of surgical cases I. The operations i will be performed in a certain operating room r, belonging the set of operating rooms R. The surgical case i belonging to the set of operations I will take place on a certain day d consisting of the set of days D. The operation i will start on a start time s, which consists of the set of start times s. The time between two consecutive start times is referred to as a time period. Every start time s lies in in a certain time block t and is an element of the set of start times belonging to time block t. Every time block t belongs to the set of time blocks T. The collection Ω j represents the set of operations i belonging to surgeon j. 5.2 DECISION VARIABLE x i,r,t,s = 1, if operation i is assigned to operating room r on time block t at start time s. = 0, otherwise. 53

71 5.3 VARIABLES I i J j set of operations index for a unique operation i set of surgeons index for a unique surgeon j D set of days. D = {Mon, Tues, Wedn, Thurs, Fri} = {1, 2, 3, 4, 5} = D = 5. d T index for a day. set of time blocks. One time block corresponds to either the morning or the afternoon. Each day consists of two time blocks. There are ten time blocks in a week. T = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and T = 10. t S index for a time block. A time block in the morning starts at 8:00 AM and ends at 12:15 AM. A time block in the afternoon starts at 12:15 PM and ends at 4:30 PM. start time. The start time s defines the start time of an operation. One start time represents a quarter of an hour. There are S time periods within a time block. S = {1,,17} and S = 17. s index for a time period (start time). s = 1,, S. Dur i end t the expected duration of the operation i in number of time periods (number of quarters of an hour). the end time of time block t (4:30 PM). After this end time no operations can be performed. D open r,t regular opening duration of time block t in room r, in number of time periods (number quarters of an hour). Bed d,dp the average number of beds available per day and per nursing department. LOS i the expected length of stay of the patient that undergoes surgery i. η i,j = 1, if surgeon j is assigned to perform surgery i. = 0, otherwise. 54

72 Ω j set of surgical cases assigned to surgeon j (Ω j = i η i,j j) λ j,r,t = 1, if surgeon j is allocated to room r on time block t. This variable results from the master surgery schedule. = 0, otherwise. A r E = 1, if laminar airflow is available in operating room r. This airflow is required for the performance of prosthesis and is not available in every operating room. = 0, otherwise. r i E = 1, if special equipment is required for surgery i (i.e. if it concerns prosthesis). = 0, otherwise. w i,d,dp = 1, if surgery i is planned on day d and belongs to department dp. = 0, otherwise. z i,d = 1, if surgery i is planned on day d. = 0, otherwise. y i,t = 1, if surgery i is planned in time block t. = 0, otherwise. 5.4 MODEL Objective function Maximize i r t s Dur i x i,r,t,s Constraints Subject to i x i,r,t,s 1 r R t T s S (1) r, t, s x i,r,t,s 1 i I (2) r, t open i s(x i,r,t,s Dur i ) D r,t (3) i, r, t, s (η i,j λ j,r,t ) j x i,r,t,s (4) s τ=s Dur i +1 r, t, s, τ 0 i x i,r,t,τ 1 (5) 55

73 j, t, s, τ 0 Ω j s r x i,r,t,τ 1 (6) i τ=s Dur i +1 i, r, t, s [s + Dur i > end t ] x i,r,t,s = 0 (7) i, r, t, s r i E x i,r,t,s A r E (8) d, dp, τ 0 i τ=d LOSi 1 w i,d,dp Bed d,dp (9) i, d, dp w i,d,dp = z i,d Dep i,dp (10) i, d z i,d = y i,2d + y i,2d 1 (11) i, t y i,t = r s x i,r,t,s (12) i, r, t, s x i,r,t,s [0, 1] (13) 5.5 DESCRIPTION OF THE OBJECTIVE FUNCTION The objective function aims at maximizing the sum of the scheduled surgery time in the operating rooms over the planning horizon of one week. In this way, the operating room utilization is maximized and consequently the operating room undertime is minimized. The same objective is pursued in the formulations proposed by Marques et al. (2012), Vijayakumar et al. (2013) and Doulabi et al. (2014). 5.6 DESCRIPTION OF THE CONSTRAINTS (1) Each operation i can start at most once in an operating room r, on a certain time block t and on a certain start time s (Doulabi et al., 2014; Fei et al., 2009; Riise & Burke, 2011; Roland et al., 2006; Testi & Tanfani, 2009; Vijayakmumar et al., 2013). Not all the surgeries from the waiting list can be planned within the weekly planning horizon (Doulabi et al., 2014). (2) At most one operation can start at start time s in a certain time block t and in a certain operating room r. (3) The sum of the expected durations of the surgical cases assigned to a certain operating room r and a certain time block t must be smaller or equal than the regular opening duration of operating room r on time block t. This constraint ensures the total operating time does not 56

74 exceed the maximum opening hours of the operating room (Fei et al., 2009; Jebali et al., 2006; Ma et al., 2009; Roland et al., 2006; Riise & Burke, 2011; Testi & Tanfani, 2009). (4) The operation i can be performed in room r on time block t when surgeon j is assigned to perform operation i and when surgeon j is allocated to room r on time block t (Jebali et al., 2006). This constraint links the master surgery schedule from the tactic phase to the model presented here. (5) Constraint that prevents the assignment of overlapping operations to the same operating room. In each room at each start time s of each time block t at most one operation can be performed. Since the decision variable x i,r,t,s is only equal to one at start time s, the expression τ = s Dur i + 1 guarantees that no overlapping operations are assigned to a certain time period of a certain room when the duration of the surgery is more than one, by summing over the multiple time periods. This can be explained by an example in Table 17. Assume that s = 4 and two operations with Dur 1 = 2 and Dur 2 = 3 need to be planned in a certain operating room r and in a certain time block t. The under script of the equation is calculated by s Dur i + 1, starting from s = 4. This is repeated for all the operations i in I. For example, when x 1,r,t,3 is equal to one, x 2,r,t,4 cannot be equal to one since operation i = 1 takes two time periods and will not be finished at start time four. Consequently, operation i = 2 cannot start at start time four (Marques et al. 2012; Roland et al., 2010; Riise & Burke 2011; Vijayakumar et al. 2013). i Dur i s s Dur i + 1 x i,r,t,s (index for (duration of (start time) operation) surgery i) = 3 x 1,r,t, = 4 x 1,r,t, x 2,r,t,2 5 3 x 2,r,t,3 6 4 x 2,r,t,4 Table 17: Example: Operations to be planned for s = 4. s x i,r,t,τ 1 i τ=s Dur i +1 4 x i,r,t,τ 1 i τ=4 Dur i +1 x 1,r,t,3 + x 1,r,t,4 + x 2,r,t,2 + x 2,r,t,3 + x 2,r,t,4 1 r, t Table 18: Example: calculation of overlap constraint for s = 4. 57

75 (6) Constraint that prevents the assignment of overlapping operations to a certain surgeon. Each surgeon j can perform at most one operation at a given start time s on a given time block t. Since the decision variable x i,r,t,s is only equal to one at start time s, the expression τ = s Dur i + 1 guarantees that no overlapping operations are assigned to a certain time period for a certain surgeon when the duration of the operation is more than one. This constraint is important when a surgeon has access to two operating rooms at the same time block. The same example as for constraint (5) applies (Riise & Burke, 2011; Roland et al., 2010; Vijayakumar et al. 2013). This constraint increases the complexity of the problem as it imposes that start times need to be selected for each surgery such that the constraint is satisfied (Risse & Burke, 2011). This constraint is also referred to as coloring constraint (Doulabi et al. 2014). (7) Constraint that it is not possible to plan operations ending after the regular opening time of a certain time block. s + Dur i is equal to the end time of a certain planned surgery. When s + Dur i > end t, the decision variable x i,r,t,s will be enforced to be zero. (8) Constraint that takes into account the availability of equipment in an operating room. For certain surgeries special equipment is required to perform a surgery. In the AZ Herentals laminar airflow is required for prosthesis (Latorre-Núñez et al., 2016; Roland et al. 2006, 2010; Vijayakumar et al., 2013). (9) Constraint that accounts for the number of beds available in the regular wards. The constraint takes into account the average length of stay (LOS) of each patient and calculates the number of beds required on each day in each department. This amount needs to be smaller or equal than the number of beds available per day and per department. In this formulation, the patients that have been operated earlier but still need a bed on a respective day are also taken into account. For each department dp and for each day d, the helping variable w i,d,dp is summed starting from the earliest day surgery i can be performed while needing a bed on day d, until the latest day surgery i can be performed while needing a bed on day d. (Adan & Vissers, 2002; Testi & Tanfani, 2009). (10) Helping variable w i,d,dp is equal to one if surgery i is planned on day d and if surgery i belongs to department dp. 58

76 (11) Helping variable z i,d is equal to one if surgery i is planned in a time block belonging to day d. (12) Helping variable y i,t is equal to one if surgery i is planned in time block t. (13) The decision variable x i,r,t,s is binary. 5.7 ASSUMPTIONS o A planning horizon of one week is used, in which only the surgical demand during the weekdays is taken into account. No elective surgical cases are planned during the weekend. o A block scheduling approach is assumed, implying that the total available operating time is subdivided into time blocks assigned to a certain operating room and surgeon. This master surgery schedule is used as input for the mathematical model and is based on the current master surgery schedule of the AZ Herentals. o The opening hours of the operating rooms are the following: 08:00 A.M. 16:30 P.M. In the AZ Herentals there is no break between the two blocks, the operating theatre is open continuously. In the model, we subdivide the opening hours into two time blocks, those who are used in the master surgery schedule. o The AZ Herentals consists of eight operating rooms and two rooms for small surgical cases of the day hospital, which are not considered here. o Not all operating rooms are multifunctional. Not every kind of operation can be performed in each operating room. Prosthesis can only be performed in an operating room with laminar airflow. Operating rooms five, six, seven and eight possess laminar airflow. o Working overtime is not allowed in the operating rooms. o The setup time between two surgical cases is 10 minutes on average. The setup consists of the following activities: changing the patient to a normal bed, cleaning the operating room and applying anesthesia on the next patient. This setup time is not taken into account in the model but is included implicitly in the expected surgery durations. 59

77 o Recovery room beds are not considered as a bottleneck resource. o The beds in the Intensive Care Unit are not considered as a bottleneck resource. o Emergencies are not considered, the surgeries considered are elective surgeries and urgencies that can be planned in advance. Emergencies cannot be planned for in advance. o The expected durations of surgical cases are coming from a standardized table based on historical information and are assumed to be deterministic. The duration is defined as the time between the patient entering and the patient exiting the operating room. o Renewable resources such as nurses and anesthesiologists are not seen as a bottleneck resource and are always available. o Non-renewable resources such as surgical materials and pharmaceuticals are not considered as a bottleneck resource. Laminar airflow for prosthesis is considered as a bottleneck resource, as mentioned above. o Each surgeon has one anesthesiologist at his disposal. o It is possible that a surgeon possesses two operating rooms during the same time block. o All patient that are scheduled for surgery on a certain day are available and ready for surgery on that day. o No pre-emption: a surgical case cannot be interrupted. o A discrete representation of time is followed, in which the smallest time unit is a quarter of an hour. 60

78 6 ANALYSIS CHAPTER 6 In this chapter the model as presented in Chapter 5 will be solved to optimality. First, the data treatment that was required in order to be able to solve the model is discussed. Next, a short overview of the current scheduling situation at the AZ Herentals will be given. Finally, the proposed model will be solved to optimality using different configurations and stakeholder perspectives. 6.1 DATA TREATMENT In this paragraph the data as input for the proposed model is discussed. Two data files have been delivered by the AZ Herentals. The first file consists of data on the surgical cases that have been performed in the AZ Herentals in January 2017, which is discussed in paragraph The second file consists of the standard durations that are used by the surgeons to plan the surgical cases, which will be discussed in paragraph Data on surgeries performed In the AZ Herentals the data concerning the performed surgical cases are registered in the data storage center of the Klinisch Werkstation software. These data have been inserted by the surgeons and the person responsible of the operating theatre. These data consist of a patient identification number, information concerning the type of surgery performed, the date of surgery, the hospitalization date, the operating room, time metrics and information on the personnel involved, i.e. the surgeon, the nurses and anesthesiologists. The data used in this case study has been stored and cleaned from incomplete and surplus information in Microsoft Excel Emergency surgeries have been deleted from the file. The data file has also been used in order to derive the master surgery schedule. An example of the data file is presented is Appendix B.1 (based on information received from Katrijn Vanlommel, coordinator Klinisch Werkstation software, 1 st of March 2017). Different time measures for each surgery performed have been stored in the database of the AZ Herentals. Figure 9 depicts the various time measures. Two main time periods can be defined, i.e. the time in the operating theatre and the time in the operating room. In the AZ Herentals, the time in the operating theatre consists of the time for preparation of surgery and the time in the operating room. The time in the operating room consists of the time for induction and the actual surgery time. 61

79 Figure 9: Time measures in the AZ Herentals. The explanation of the time measures can be found in Table 19. Time in/out operating theatre Time in/out operating room Duration of surgery Time start/end preparation Time start/end induction Time start/end surgery Time start/end of PACU The time the patient enters/leaves the operating theatre. The time the patient enters/leaves the operating room. The total time the patient is in the operating room. The time the preparation for surgery starts/is finished and surgery can be performed. The time the anesthesiologist starts/end the induction for anesthesia. The time the surgeon starts to perform/has finished the surgery. The time at which the patient arrives in the PACU to wake up/ leaves the PACU. Table 19: Explanation of time measures Data on standard durations The data presented in Appendix B.1 consists of the durations of surgical cases ex-post. However, in order to be able to plan for surgeries in advance one should consider the durations ex-ante. The AZ Herentals uses standardized tables that include the durations for each type of surgery for all the medical disciplines. These standard durations have been collected based on historical information and trial and error. In Appendix B.2 an example of such a standardized table for the Orthopedics department is provided. These durations are used to by the surgeons to plan their surgical cases. They will also be used in the mathematical model of this work (based on information from Katrijn Vanlommel, coordinator Klinisch Werkstation software, 10 th of March 2017). 62

80 6.2 DESCRIPTION OF THE CURRENT SITUATION This paragraph discusses the current scheduling practices at the AZ Herentals. In the standard and realized utilization rate is discussed and the differences are examined. Paragraph discusses the average patient waiting times and paragraph concludes by giving an overview of the current scheduling practices at the AZH Standard versus realized utilization rate In this paragraph the efficiency of the current planning and scheduling system is discussed, by examining the difference between the planned durations and the realized durations. First, an overview of the utilization rates of the current planning system is given. The utilization rate is a measure of the efficiency of the planning system. The utilization rate is calculated hereafter as the sum of the durations of the planned operations divided by the total available operating room time Utilization rate of planned durations We calculated the utilization rate of the planned operations in January 2017 for the different departments of the AZ Herentals. These calculations are presented in Figure 10. The utilization rate of the standard or planned durations varies significantly over the different disciplines. The gynaecology discipline (GYN) has a utilization rate of 73%, which is the lowest of all disciplines. This indicates the gynaecology discipline does not use all the available operating time to plan their surgical cases. Therefore, it may be an indication of too many operating time being assigned to the gynaecology discipline compared to the other disciplines. General surgery (CHI) has the highest utilization rate, i.e. 103%. This indicates general surgery uses even more than the available operating time to assign surgical cases. This is only possible when overtime is used in order to schedule for these additional surgeries and may be an indication the general surgery discipline is in need of more available operating room time. The average utilization rate over all the disciplines is 86,50%. 63

81 Figure 10: Utilization rate of standard durations AZ Herentals Utilization rate of realized durations The data available also permits the calculation of the utilization rate of the realized durations, i.e. the effective durations of the performed surgeries. These calculations are displayed in Figure 11. It is clear from the graph that the utilization of the realized surgeries is less variable compared to the utilization rate of the standard durations. Also for the realized durations the general surgery discipline (CHI) has a significantly higher utilization rate compared to the other disciplines. Figure 11: Utilization rate of the realized durations of the AZ Herentals Comparison of the standard versus the realized durations The availability of both the standard and realized durations enables us to compare the standard durations with the realized durations, as shown in Figure 12. All disciplines have a higher standard utilization rate than the realized utilization rate, except for the gynaecology discipline. This means all disciplines except gynaecology need less surgery time in reality compared to the standard durations used to plan the operations, on average. 64

82 Significant differences between the departments arise. For example, the oral and maxillofacial surgery department (THK) has a standard utilization rate of 99% compared to a realized utilization rate of 84%. This indicates the surgeons of the oral and maxillofacial surgery department (THK) do not need all the planned operating room time to perform their surgical cases. When we compare this to the orthopedics discipline (ORT), we denote a standard utilization rate of 85% and a realized utilization rate of 81%. The difference between the standard and realized utilization rate is much smaller here. Those findings may be an indication of the standard durations of the orthopedics department being more accurate compared to those of the oral and maxillofacial surgery department (THK). When it is assumed the standard durations are accurate, this may be an indication of the oral and maxillofacial surgery discipline (THK) working more efficient compared to the orthopedics discipline. Figure 12: Standard and realized utilization rate. Besides the differences between the different departments, there are also differences between the standard and realized durations of the individual surgeons. Therefore, a high difference between the standard and realized durations of a certain department may be caused by one or two single surgeons of that department. As an indication for this, we compared the realized and standard durations of one day for two surgeons of the orthopedics department. This comparison is displayed in Figure 13 and 14. Surgeon one clearly has higher realized durations compared to the standard durations, meaning he needs more operating room time than has been scheduled for. In contrast, surgeon two needs less operating time compared to what has been scheduled for. 65

83 Figure 13: Realized and standard durations of surgeon 1. Figure 14: Realized and standard durations of surgeon 2. We can conclude the standard durations are overestimated if we compare them with the realized durations and significant deviations occur between both the departments and the surgeons. An update of these standard durations and continuously tracking the work efficiency of the surgeons would enable the hospital to plan surgical cases with more precision, leading to less deviations from the planned schedule. It should be noted that the above calculations do not consider possible deviations from the master surgery schedule by the different disciplines. It is possible that disciplines mutually exchange operating room time, when they have too many or too little time. This makes it hard to draw clear conclusions from these calculations. Additional information and research is required in order to 66

84 denote how many operating time each discipline needs to receive. This is a more strategic decision and beyond the scope of this research Average patient waiting times The average patient waiting time in the AZ Herentals is 1 hour and 45 minutes. We observe some differences between the average waiting times of the different disciplines. An overview of the average patient waiting time per discipline is provided in Figure 15. The exact numbers can be found in Appendix C.1. The average waiting times for children and day hospital patients per discipline can be found in Appendix C2 & C3. Figure 15: Average patient waiting time per discipline. The smallest average waiting time, i.e. 1 hour and 7minutes has been obtained by the vascular and thoracic surgery discipline (VAT). The oral and maxillofacial surgery discipline (THK) has with almost 1 hour and 47 minutes the largest average waiting time of the hospital. The orthopedics department (ORT), which accounts for more than 50% of the operating room time, has an average waiting time of 1 hours and 37 minutes. The differences between the average waiting times of the disciplines can partly be attributed to the differences between the average durations for surgery of each department. An overview of the average surgery duration per discipline is given in Table