RESOURCE CAPACITY ALLOCATION OF INPATIENT CLINICS AMC Academic Medical Centre Amsterdam. F.J. Mak BSc S

Transcription

1 RESOURCE CAPACITY ALLOCATION OF INPATIENT CLINICS AMC Academic Medical Centre Amsterdam September 2011 May 2012 F.J. Mak BSc S Supervisors E.W. Hans PhD MSc University of of Twente A. Braaksma MSc Academic Medical Centre Amsterdam N. Kortbeek MSc University of of Twente University of Twente School of Management and Governance Department of Industrial Engineering and Business Information Systems

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3 Management summary An ageing population, more advanced treatments and a high standard of care led the past decade to an enormous increase in demand for care and costs. Health care managers face the challenging task to organize their processes more effectively and efficiently [17]. Within the Academic Medical Centre of Amsterdam (AMC) the sense of urgency to change is gradually accepted. Different types of research projects are started in order to improve the overall performance and to provide insight in the relations of complex hospital processes. High fluctuations in the demand for care and beds in the clinical wards of the surgical division of the AMC have led to the development of two models. The model of Smeenk et al. [29] makes it possible to predict the number of beds that are occupied each hour of the day given the Master Surgical Schedule (MSS). The model of Burger et al. [9] uses the output of the model of Smeenk to determine the optimal number of dedicated nurses per ward and the number of nurses per flex pool. A flex pool consists of nurses that still need to be assigned to a ward at the start of a shift given the dedicated nurses already assigned and the number of patients present. The models of Smeenk et al. and Burger et al. focus on the clinical wards while the MSS is created in the OR department. In this research we develop an integral method that encompasses resource capacity planning decisions in the OR department and the clinical nursing wards. We have formulated the following research objective: To develop a method which determines the best combination of patient case mix, OR capacity, care unit and nurse staffing decisions in such way that total cost margins are maximised while satisfying production agreements and resource, capacity, and quality constraints. We express our research objective as a mathematical optimisation problem in which we minimise the resource usage in the OR department and clinical wards, while selecting the most profitable case mix. We define several quality and resource constraints. To evaluate the total costs of the objective function we have defined several cost parameters. The solution method we present encompasses a decomposition approach in which we use several models and optimisation tools based on state of the art literature. Our solution approach consist of the following six steps: 1. Set the desired patient case mix and the length of the MSS. 2. Solve an Integer Linear Program (ILP) to create a master surgical schedule and assign elective and acute patient types to wards, while minimising the number of ORs, wards, and the expected number of nurses and beds required. 3. Evaluate the access time service level of the created block schedule with the model of Kortbeek et al. [19]. 4. Determine the number of beds required per ward while satisfying target rejection and misplacement rates with the model of Smeenk et al. [29]. 5. Iteratively use the model of Burger et al. (Step 6) to determine the best flex pool-ward combination. i

4 6. Determine the optimal number of dedicated nurses per ward and the total number of nurses in a flex pool given various target service levels with the model of Burger et al. [9]. To test our approach we performed experiments with real data obtained from the surgical division within the AMC. Our experiments show that our solution approach reduces variation in demand for beds and thereby levels the workload. When we consider a cyclic MSS of four weeks we can reduce the number of beds by 5.2% compared to our model representation of the current situation. From our results we conclude that nurses can be utilised more efficiently by considering less wards with more beds per ward. When we consider three wards with at most 50 beds we require 11.1% less FTE nurses compared to our model representation of the current situation. When we consider a flex pool of nurses between two wards we can achieve an additional reduction of 1.7% in FTE compared to our model representation of the current situation. The benefits of a flex pool mainly depend on how the MSS is organised, the flex pool-ward assignment and the chosen values of the service levels. Our solution approach encompasses a large variety of resource capacity planning decisions that are related to each other. Due to the large number of planning decisions and the complexity between them it is very ambitious to find one optimal solution. The MSS that results from solving our ILP does reduce the expected number of beds and thereby reduces variation in demand for care. Possible improvements lie in the development of an MSS that further improves alignment in demand for beds with the required number of nurses and a tool to automatically select the optimal case mix. The patient-to-ward assignment can be improved by taking the surgery, and, admission and discharge distributions into account. To conclude, the approach we present provides hospital managers with a tool to evaluate and optimise the resource requirements in the OR department and the clinical wards given a patient case mix and the length of the MSS. This tool can be used to (re)design, evaluate and improve current hospital processes and is, due to its generic nature, applicable in a wide variety of hospitals. ii

5 Preface I am proud to present this graduation report, which contains my research carried out at the Academic Medical Centre (AMC) Amsterdam. This report is the last piece of a puzzle, completing my Master s degree in Industrial Engineering and Management. Almost nine months ago, when I first came to the AMC I had high expectations. After a cumbersome first three months, in which I had difficulties defining the scope and accepting an uncertain outcome, I finally found my way with as end result this graduation report. I would like to thank several people that supported me during this project. First, I thank Erwin Hans for providing the opportunity to perform my assignment in the AMC and his role as first supervisor. I enjoyed your enthusiasm and your constructive feedback during the various meetings we had. I thank Nikky Kortbeek and Aleida Braaksma of the AMC for their extensive supervision. Both have encouraged my academic thinking and helped to improve the quality of this research. I enjoyed the weekly sessions and appreciated the discussions we had. I especially thank Aleida for her detailed feedback regarding my report, which definitely improved after each revision. Next, I thank Piet Bakker and Delphine Constant for the possibility to execute my research in the AMC and their contribution during the monthly meetings. I thank all co-workers at KPI for the pleasant time. I enjoyed the cosy atmosphere and the famous "tweede donderdag van de maand" drinks. Finally, I thank my parents for their continuous support throughout my student career. I am glad that you always encouraged me to make my own choices. Last, but certainly not least, I thank my girlfriend, Jojanneke, for supporting me throughout this project. You were always there for me and helped me stay motivated. Amsterdam, May 2012 Frank Mak iii

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7 Contents Management summary i Preface iii 1 Introduction Research context: AMC Problem statement Research objective Research demarcation Research questions Context analysis Division B: surgical specialties Patient flow OR department Inpatient care units Conclusions Literature Techniques for resource capacity planning Methods OR department Methods clinical wards Decomposition approaches Conclusions Solution approach Optimisation problem

8 4.2 Decomposition approach Software implementation Verification & validation Conclusions Computational results Data gathering Demarcation experiments Experimental set-up Experimental design Results Conclusions Conclusions & recommendations Conclusions Discussion Recommendations Further research Bibliography 62 A Mathematical optimisation problem and ILP 67 A.1 Notation A.2 Notation B Data analysis 71 C Financial parameters 73 D Class diagram Delphi 75 E Detailed results 77

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11 Chapter 1 Introduction An ageing population, more advanced treatments and a high standard of care led the past decade to an enormous increase in demand for care and costs. Health care managers face the challenging task to organize their processes more effectively and efficiently [17]. Within the Academic Medical Centre of Amsterdam (AMC) the sense of urgency to change is gradually accepted. Different types of research projects are started in order to improve the overall performance and to provide insight in the relations of complex hospital processes. High fluctuations in the demand for care and beds in the clinical wards of the AMC have led to the development of two models by Smeenk et al. [29] and Burger et al. [9] to evaluate the effect of resource capacity planning decisions on the nursing wards. In this report we continue this research by developing a method that optimises resource capacity planning of the OR department and the inpatient clinical wards of Academic Medical Centre Amsterdam. We structured this chapter as follows. We introduce the Academic Medical Centre Amsterdam and the department of Quality and Process Innovation in Section 1.1. Section 1.2 states the problem. Section 1.3 provides the objective of this research, which we further clarify by means of a theoretical framework. We demarcate our research in Section 1.4. We conclude this chapter with our research questions in Section Research context: AMC This research is carried out in the Academic Medical Centre of Amsterdam (AMC) within the department of Quality and Process Innovation (KPI, Dutch for: Kwaliteit en Proces Innovatie). AMC is one of the eight academic teaching hospitals in the Netherlands and is specialised in providing top clinical care. The AMC is assigned one of the eleven trauma centres and thus has a coordinating role in allocating acute patients. The department KPI falls under direct control of the Board of the Hospital. This department was founded in 2008 to support other departments and nursing wards in the hospital by monitoring and improving their processes. One of the objectives of KPI is to develop generic quantitative models that can be generally applied within the AMC. These quantitative models encourage transparency and provide opportunities for internal benchmarking. Furthermore, this approach will result in standardisation of processes which improves overall efficiency while maintaining quality of care [2]. 1

12 1.2. PROBLEM STATEMENT CHAPTER 1. INTRODUCTION 1.2 Problem statement In this section we define the problem. First, we give the motivation for this research and the problem description in Section Next, we conduct a stakeholders analysis in Section Research motivation and problem description The nursing wards in the Surgical Division experience high fluctuations in demand for beds and care. This demand is highly influenced by the Master Surgical Schedule (MSS) and the Length of Stay (LOS) of patients. An MSS is a schedule that defines the number and type of available ORs, the opening hours and the surgeons or specialist groups to whom the OR time is assigned [15]. According to literature, sixty to seventy percent of all hospital admissions are caused by surgical interventions [15]. In surgical nursing wards this percentage is thought to be even higher. The relationship between the MSS and bed capacity usage at wards is not transparent for most hospital managers, which makes it difficult to match the appropriate amount of staff to the actual demand for care. Understaffing of nurses yields quality loss and leads to increased mortality [26] while overstaffing leads to extra costs for the hospital. Furthermore, in the near future a shortage of nurses in the Netherlands is to be expected [1]. Due to rising expenditures hospital managers are continuously pressured to improve the hospital s operational efficiency. Hospitals need to survive in a competitive environment in which their income increasingly depends on the composition and volume of the case mix. Some patient types yield more revenue than others and are therefore more beneficial to treat. Figure D.1 shows a simplification of the inpatient care chain from a patient oriented view. We distinguish between elective patients, which are planned in advance and come from the outpatient clinics, and non elective or acute patients that arrive due to an emergency. In this research we focus on the Operating Room department (OR) and the clinical wards, marked in the box of Figure D.1. Figure 1.1: Patient flow through the simplified inpatient care chain (departments marked in box are the focus of this research). In order to support ward bed and staffing decisions by health care managers, a decision support model has been developed by Smeenk [29] and Burger [9]. Smeenk based his model on the research of VanBerkel [37]. VanBerkel developed an exact approach that relates the patient daily workload at a ward to the Master Surgical Schedule (MSS). Smeenk extended this research by developing an hourly bed census model, which estimates the number of occupied beds in a ward on an hourly basis given the MSS and the arrival of acute patients. The advantage of this hourly approach is that it provides sufficient accuracy to support nurse staffing decisions. Burger used the output of the hourly bed census model to determine the amount of nursing staff at a ward needed to treat the estimated number of patients. Furthermore, he researched the potential of using a flex pool with nurses. Nurses in a flex pool are assigned to a ward at the beginning of a shift depended on the number of dedicated nurses in a ward and the current workload. Burger showed that the use of flexible staff combined with dedicated staff leads to lower nurse staffing costs, while satisfying predefined service levels. 2

13 1.2. PROBLEM STATEMENT CHAPTER 1. INTRODUCTION The models developed by Smeenk [29] and Burger [9] provide health care managers with a tool to evaluate resource capacity planning decisions on the clinical wards."resource capacity planning addresses the dimensioning, planning, scheduling, monitoring, and control of renewable resources" as stated in Hans et al. [16]. By limiting the scope of a project to a single department suboptimal conclusions may be drawn, particularly when the influences of other departments are ignored [37]. The next step is to extend the research of Smeenk and Burger by developing a tool that supports health care managers in their resource capacity decisions that have an effect on the OR department and the clinical wards Stakeholder Analysis We conduct a stakeholder analysis in order to identify the objectives of the various actors in the care chain (Figure D.1). We discuss the main involved stakeholders (see also Figure 1.2): Figure 1.2: Overview of the various stakeholders in the inpatient care chain of the AMC. Board of the hospital: The board is responsible for the long term strategic goals of the hospital. The strategic horizon encompasses decisions concerning one to five years ahead. The overall objective of the board is to achieve high quality care, efficient use of resources, satisfied employees and a financially healthy organisation. Marketing and control: Marketing and control determines the production volumes for each specialty group in cooperation with the different specialties. In addition they negotiate with health care insurers on the volume, quality and price of care. They negotiate with hospi- Insurers: Insurers represents the interest of their policy holders. tals on the quality, volume and price of care. Patients: Patients demand high quality care at an affordable price. Furthermore, patients are willing to travel further to receive the best possible care. Access time (time from referral until the day of appointment) and waiting time (on the day of appointment) are increasingly important when patients select a hospital. Specialists: The specialist in the outpatient clinic is responsible for the first contact with a patient and performs the surgery. Specialists deliver good quality of care and demand stable working hours. Because the AMC is an academic hospital the specialists are contracted in-house, compared to non academic hospitals where specialist are hired from medical partnerships. OR management: The OR management is responsible for the strategic decisions that affect the OR. On a tactical level they allocate capacity to the various specialties. Besides this, they are 3

14 1.3. RESEARCH OBJECTIVE CHAPTER 1. INTRODUCTION responsible for coordinating the daily operations inside the OR Department. OR planner: The OR planner is part of the OR department and responsible for the allocation of ORs, supporting staff and equipment to the specialties. Each specialty has various preferences and demands. The OR planner strives to meet these preferences and allocate capacity in a fair and transparent way. OR personnel: OR personnel consists of an anaesthetist, assistant anaesthetists and surgery assistants. The OR personnel assists the specialist during surgery. They want to have regular working hours, smooth transitions between surgeries and a balanced workload. Division management: A division manages a cluster of specialties in the hospital. The division management consists of a board and supporting staff. The board is responsible for the long term vision of a division. The supporting staff performs administrative tasks and monitors the financial status for each specialty. Ward management: The management of a ward is responsible for the daily operations on a ward. They decide how to allocate the staff to the various shifts and how many operational beds are available. The ward management wants to provide good quality care, use their resources as efficiently as possible and keep their staff satisfied. Specialty planner: The specialty planners of the nursing wards schedules patients into fixed surgery blocks, and aims to maximise patient throughput and to minimise the number of cancelled surgeries. Nurses: Nurses have direct contact with patients and largely influence the satisfaction of the patients. Nurses want to have steady working hours and a levelled workload. High variations in demand for care make it difficult for nurses to perform their tasks adequately. The stakeholder analysis yields various objectives regarding the inpatient care chain: Maximise quality of care: Each stakeholder in the inpatient care chain demands a high quality of care. Budget restrictions and variable workloads restrict the solution space of this objective. Staying financially healthy: The hospital management needs to make sure that the hospital stays financially healthy. Minimise access time: Patients do not want to wait a long time before they can undergo surgery. By using resources more efficiently, access time can be reduced, which is beneficial for the patients and the hospital s reputation. Minimise underutilisation of resources: The different management layers inside the hospital all want to use their resources efficiently and want to avoid underutilisation. Level the workload: A levelled workload leads to satisfied employees that can provide a more constant quality of care. The above stakeholder analysis makes clear that there are various, conflicting objectives in the inpatient care chain. Due to these conflicting objectives optimisation of the inpatient care chain is very complex. 1.3 Research objective We conclude from the previous section that the objective of the hospital management is to create a levelled workload. This will yield a higher quality of care against lower costs. Section 1.2 shows 4

15 1.3. RESEARCH OBJECTIVE CHAPTER 1. INTRODUCTION that it is interesting to extend the research of Smeenk [29] and Burger [9] to a model that optimises resource capacity planning decisions in the OR department and the clinical wards. Furthermore, a cost based approach is important for selecting the right mix of patient types. Combining this all leads to the following research objective: To develop a method which determines the best combination of patient case mix, OR capacity, care unit and nurse staffing decisions in such way that total cost margins are maximised while satisfying production agreements and resource, capacity, and quality constraints. Figure 1.3: Framework for health care planning and control, Hans et al. [16]. Our research objective addresses several resource capacity planning decisions on different levels of control. Figure 1.3 shows a managerial framework that encompasses all managerial health care areas involved and all hierarchical levels of control [16]. In this research we focus on resource capacity planning decisions on the strategic and tactical level. Strategic decisions are made based on forecasts, while operational decisions are based on known demand. The tactical level encompasses all decisions that are made when demand is partly known. To further clarify the resource capacity decisions we use a taxonomy proposed by Hulshof et al. [17]. They classify the various resource capacity planning decisions for six types of health care services. The two health care services of interest are the surgical care service and the inpatient care services. We select from this taxonomy nine planning decisions on two hierarchical levels that we relate to the decisions in our research objective. Table 1.1 shows these decisions. We briefly explain each planning decision. Patient case mix: The patient case mix decision concerns the selection and volume of patient types to treat. Capacity dimensioning: The capacity dimensioning decision focuses on estimating the number of resources necessary inside the OR department and the clinical wards. Care unit partitioning: Care unit partitioning consists of deciding which wards to open and how to allocate the patient types across them. Capacity allocation: Capacity allocation considers decisions regarding how shared resources should be allocated to the various actors. For example, OR capacity needs to be allocated to specialists that want to perform surgery. Staff shift scheduling: Staff shift scheduling focuses on determining the number of nurses that need to be assigned to each shift. We distinguish between dedicated nurses and flexible nurses. Dedicated nurses can only be assigned to one ward while flexible nurses can be assigned to multiple wards. 5

16 1.4. RESEARCH DEMARCATION CHAPTER 1. INTRODUCTION Patient case mix OR Capacity Care unit Nurse staffing Strategic Case Mix Capacity dimensioning Capacity dimensioning Capacity dimensioning Which patient types How many ORs? How many nursing beds in total? How much nursing staff? and how many patients? How much surgery staff? How much equipment? (five to one year) (five to one year) What are the surgery hours per (five to one year) day? How much equipment? What types of surgical wards? Care unit partitioning (five to one year) How many wards? Which patient group is assigned to which ward? (five to one year) Patient case mix OR Capacity Care unit Nurse staffing Tactical Capacity allocation Capacity allocation Staff shift scheduling How to assign patients to How many beds per ward? How many dedicated nurses groups? How much OR time per patient How much equipment per ward? per ward per shift? group? (Year to three months) How many flexible nurses Which OR time is assigned to per flex pool? which time and date? (Eight to six weeks) (Year to three months) Table 1.1: Overview of the planning decisions (bold), the decision variables (normal) and the planning horizon (italic) inside the inpatient care chain. 1.4 Research demarcation In this section we define the scope of this research. For the OR capacity dimensioning decision we do not consider the various type of surgical wards (holding room, recovery room, etc.) inside the OR department. For the OR capacity allocation decision we assume that the distribution of patients inside a surgery block is known and that surgery blocks have a fixed length. We do not focus on scheduling of patients in a surgery block. We assume that acute patients are scheduled in dedicated emergency ORs and only need to recover at a clinical ward. Furthermore, we do not consider the outpatient clinic. 1.5 Research questions We formulate research questions to reach our research objective in a structured way. The numbers of the research questions refer to the corresponding chapter in which we answer them. Chapter 2. How is the inpatient care chain organised and how does it perform? First, we describe what we consider the inpatient care chain. Next, we give information on the organisation of the OR department and the nursing wards. For both, we describe the patient processes, how capacity is allocated and the current performance. Chapter 3. What methods are known in literature for making resource capacity planning decisions in health care organisations? Our literature review consists of three parts. The first part focuses on techniques to make resource capacity planning decisions. The second part is divided into a part that focuses on planning decisions in the OR department and a part that focuses on the clinical wards. The last part contains methods that focus on planning decisions that encompass multiple departments. Chapter 4. How can we develop a model that optimises the resource capacity planning decisions? First, we define our optimisation problem based on the information obtained in Chapter 2. Next, with the information obtained in Chapter 3 we develop a solution approach to solve the optimisation problem. The last step we perform is to translate this solution approach into the realisation of an 6

17 1.5. RESEARCH QUESTIONS CHAPTER 1. INTRODUCTION actual model. Chapter 5. How can we apply our modelling approach to the AMC? In this chapter we perform experiments on data obtained from the Division B of the AMC. First, we apply the model on the current situation. Next, we demonstrate the performance of our solution approach by various experiments. Chapter 6. What are the managerial implications? We conclude this thesis by describing the managerial implications. We summarise our results into conclusions and give recommendations. Because our research has limitations due to the complexity of the various planning decisions we reflect on our approach and provide directions for further research. 7

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19 Chapter 2 Context analysis In this research we focus on the clinical wards within the surgical division (Division B) of the AMC. First, Section 2.1 provides key figures of this division. Because hospitals are highly complex systems and are poorly understood most of the time, these systems are best described by the flow of their patients [36]. Section 2.2 describes the patient flow of a patient through the inpatient care chain. Section 2.3 continues with process, control and performance information of the OR department. Section 2.4 contains process, control and performance information of the nursing wards in Division B. We end this chapter with conclusions in Section Division B: surgical specialties The case study we conduct focuses on Division B, surgical specialties, of the AMC. Table 2.1 displays key figures of this division for the year Currently, this division is in the process of reorganisation in which the number of wards is reduced to five from seven. Number of specialties 9 Total number of patients per year 8501 Total number of wards 7 Average LOS in days 5.1 Total number of nurses in FTE 161 Total number of beds 176 Table 2.1: General characteristics of Division B: surgical specialties for the year 2010 (Source: Braaksma and Kortbeek [8]). 2.2 Patient flow During the research of Smeenk and Burger [30] an extensive process description of the patient flow, OR department and two nursing wards within this division has been made. The other wards within this division have similar processes. We summarise parts of this description and provide additional information. Figure 2.1 shows the different departments that are involved during the stay of the patient. We make the distinction between elective patients (who are planned) and non elective or 9

20 2.3. OR DEPARTMENT CHAPTER 2. CONTEXT ANALYSIS Figure 2.1: Patient flow through inpatient clinics (Source: Smeenk and Burger [30]). acute patients (who are urgent). First, we describe the elective patient process. Next, we explain the non elective patient process. Elective patients enter a hospital through the outpatient clinic after which they are placed on a waiting list when they require surgery. Close to the date of surgery, elective patients are prehospitalised on a ward. This is done to make sure the patient s conditions are controlled. Before surgery a patient is transported to the holding room of the OR. After surgery the patient is transported to the intensive care unit or to a ward. The admission of non elective patients to the hospital is unplanned. Figure 2.1 distinguishes two types of non electives: semi-urgent and urgent. Semi-urgent patients are admitted from another ward or another hospital and their arrival is known a few hours in advance. Urgent patients are immediately hospitalised and come from outpatient clinics, the emergency department or their homes. When a non elective patient has been admitted, the process is similar to that of elective patients. 2.3 OR department Process description The OR department consists of twenty-five operating rooms (OR) and one emergency OR. Twenty theatres are used for all patients admitted at clinical wards and five ORs are part of the daycare centre. Our research focuses on the former theatres. The personnel in an operating theatre consists of a specialist, a surgery assistant, an anaesthetist and an anaesthetist assistant. We describe each in more detail: Surgeon: The surgeon is the specialist in the OR and performs the surgical procedure. Surgery assistant: The surgery assistant assists the specialist during the surgery. Before surgery the assistant performs preliminary tasks and during surgery he hands over tools. Anaesthetist: The anaesthetist is responsible for the condition of the patient. Before surgery 10

21 2.3. OR DEPARTMENT CHAPTER 2. CONTEXT ANALYSIS Figure 2.2: Description of the OR process from a patient oriented view (Source: OK-handleiding [14]). the anaesthetist checks the condition of the patient and decides if the patient is ready for surgery. During the surgery the anaesthetist continuously monitors the patient s health and has the power to abort the surgical procedure. Anaesthetist assistant: limited decision power. The assistant of the anaesthetist supports the anaesthetist and has Figure 2.2 displays the different steps in the OR process. First, a call is made from the OR centre to the designated ward that a patient can be transported to the holding room. A nurse from the ward brings the patient to the holding room where the patients wait until the operating theatre is available. When the operating theater is ready, the patient is transported into this theater and the time-out procedure is started. The time-out procedure consists of verification of the patient, the surgical procedure, the location of the surgical procedure on the patient and it is checked if all surgical tools needed during surgery are present. Once the time-out procedure is completed the anaesthetist anaesthetises the patient and released him for surgery. When the specialist is finished the patient is transported back to the IC or the recovery room of the OR department Resource capacity planning and control In this section we discuss the strategic patient case mix decision and the planning stages concerning the dimensioning and allocation of OR capacity Patient case mix The patient case mix decision concerns the selection and volumes of patient types to treat. This decision ideally depends on the revenue of a patient type, the waiting lists, the facilities and the contract agreements made with insurers. In the current situation this decision is largely based on choices made in the pasts. The contract agreements normally take place a few months before the start of a new calendar year. In the AMC the department of Marketing and Control (MC) negotiates with insures on behalf of the specialties OR capacity decisions We define OR capacity by the following resources: total number of ORs, total amount of staff, opening hours and specialised equipment like x-ray machines. Capacity dimensioning consists of deciding how much OR capacity is needed to treat all patients. Capacity allocation consists of dividing the available capacity over the various specialties. Table 2.2 describes the stages in which OR capacity is allocated to the specialties in the AMC. Capacity allocation On a tactical level the OR centre receives a request for OR capacity from each division for the upcoming year. This request is based on the annual budget that a division has available to spend on OR capacity. The board of the OR centre balances all requests and assigns each division a total number of surgery hours. These surgery hours are translated to a fixed number of Operating Room Days (ORDs) per year per specialty. In this phase the total annual OR capacity is assigned to the specialties. 11

22 2.3. OR DEPARTMENT CHAPTER 2. CONTEXT ANALYSIS Actor Action Time Period Planning phase OR Centre Total yearly OR days per Three months before a Tactical specialty new year OR Centre OR days assigned to specialtiegical Three months before sur- Tactical session Specialty planner OR days assigned to subspecialties Six weeks before surgical Tactical session Surgeon Patient planned into Thursday week before surgical Offline operational blocks of subspecialties session OR Centre Creation of definite surgical Thursday week before sur- Offline operational schedule gical session OR Centre Daily scheduling of surgeriesion On day of the surgical ses- Online operational Table 2.2: Overview of the AMC planning stages of OR capacity (Source: Smeenk and Burger, OK-handleiding [30, 14]). Next, the ORDs assigned to each specialty are transformed by the OR centre into a Master Surgical Schedule (MSS) that states the number of surgery blocks a specialty can use for each day of a year. This MSS has a rolling horizon of twelve months and does not change in the last three months before execution. The speciality planner of a division receives the final schedule three months before execution and subdivides the ORDs into full, morning and afternoon ORDs. Each specialty planner of the division uses her own method to plan surgery blocks. For example, the specialty planner of General Surgery uses a basic assignment method [29] to divide the available ORDs to the subspecialties. Other specialty planners assign blocks to surgeons and some directly plan patients on a first come first serve basis. During interviews with the specialty planners we received various remarks about the high variation in number of ORDs that a specialty receives from the OR centre each month. This high variation makes it difficult to predict the number of ORDs available and to create a standardised planning method. Operational planning The operational level consists of an offline and online operational planning. On an offline operational level the patients are planned into sub specialty blocks by the specialist that performs the surgery. Once the surgeries are planned, the OR schedule is updated and finished by the OR centre. The last surgery in the operational schedule can be marked Pro Memorie (PM) when there is high variance in scheduled surgery times or when there is a schedule that is too tight. A surgery marked PM has a high probability to be cancelled when other surgeries are delayed. The OR centre is responsible for the online operational planning in which non elective patients are scheduled. Non elective patients at the OR department are classified by four categories: acute, urgent, semiurgent and semi-elective. The first two categories preferably undergo surgery in the emergency OR. When the emergency OR is occupied the OR centre can decide to break into an elective schedule. The semi-urgent and semi-elective surgeries are preferably performed in ORs assigned to that specific specialty Performance In this section we evaluate the performance of the OR department. We use available information from the management information system Cognos. Because a detailed in-depth research in the OR department is not possible within the available time frame we focus on describing the performance indicators present of this information system. First, we introduce five definitions which are needed 12

23 2.3. OR DEPARTMENT CHAPTER 2. CONTEXT ANALYSIS to understand the performance indicators. Next, we describe the performance indicators and show the values for the year We have the following definitions: Budgeted hours: Total surgery hours per year assigned to each specialty by the OR centre. These budgeted hours are allocated but not scheduled yet. Realised budgeted hours: Total hours actually scheduled in the Master Surgical Schedule by the OR centre. This value may differ from the budgeted hours. Planned hours: The total amount of hours per year that a specialty planner or specialist actually plans with elective patients. Realised hours: The total amount of hours per year that a specialty uses the OR during normal working hours. This value does not include overtime, but does includes the treatment of acute patients during regular surgery hours. Overtime hours: The total amount of surgery hours per year that a specialty used in overtime. We give a numerical example of how the definitions can be interpreted. For example, a specialty requests 110 surgery hours but receives 100 budgeted hours from the OR centre. From this 100 budgeted hours, 90 hours are actually assigned by the OR centre to ORDs. The other 10 hours are not assigned due for example a shortage of nursing assistants. A specialty plans 70 hours of the 90 realised budgeted hours, while it in reality uses 80 hours to perform all planned and non elective surgeries. Off these 80 hours 5 hours are used during overtime. We consider the following performance indicators: Budget realisation: This is the percentage of budgeted hours that is actually scheduled by the OR centre to the various specialties. We determine this indicator by dividing the realised budget hours by the budgeted hours. Planning utilisation: This performance indicator states the percentage of hours actual planned compared to the budget realisation. We calculate this indicator by dividing the planned hours by the realised budgeted hours. Realised utilisation: This performance indicator states how efficient the OR is used during assigned hours. We calculate this indicator by dividing the realised hours by realised budget hours. Overtime utilisation: This performance indicator states how much of the OR time assigned is spend on overtime. We determine this indicator by dividing the overtime hours by realised budget hours added with the over time hours. Cancellation rate: Total number of surgeries cancelled by the hospital divided by the total number of scheduled surgeries. Table 2.3 shows the current performance for each indicator per specialty. Section provides more information about the used abbreviations of the specialties of this division. Remarkable is the high number of cancellations for the specialty Cardiothoracic surgery. We see that the planning utilisation is relative low compared to the realised utilisation for all specialty types. A reason for this could be due that in the planning hours the arrival of acute patients is accounted for or be an indication that the tactical planning should be improved. 13

24 2.4. INPATIENT CARE UNITS CHAPTER 2. CONTEXT ANALYSIS Indicator CAP CHI CHP MZK ORT URO Budgeted hours Budget realisation 91.4% 89.6% 92.2% 92.2% 75.5% 92.0% Planned utilisation 72.7% 80.3% 79.3% 79.5% 77.8% 69.1% Realised utilisation 83.0% 87.0% 83.7% 82.5% 86.9% 86.2% Overtime utilisation 10.8% 4.4% 3.7% 3.6% 4.7% 3.6% Cancellation rate 12.8% 5.0% 5.7% 3.9% 7.8% 6.9% Table 2.3: Overview of the performance of the OR department for each specialty for the year 2010 (Source: Cognos 2011). 2.4 Inpatient care units In this section we discuss the nursing wards within division B. Section discusses the actors involved and the patient process. Section describes how the capacity of a nursing ward is allocated and the assignment of nurses to shifts. Section states the current performance of the nursing wards in terms of misplacements and the realised bed census Process The surgical specialties division consists of seven wards to which nine different patient types are assigned. Table 2.4 provides an overview of these types and their designated wards for the year Cluster Specialty Patientgroup Ward CAP CAP Cardiothorcal surgery G3Z CHI CHI General surgery G6N CHI CHI General surgery G6Z CHI CHIss Short stay surgery G5N CHP CHP Plastic surgery G5Z MZK MZK Oral pathology and maxilla surgery G6Z MZK ORT Orthopedic surgery G7Z TRA TRA Traumatology G7N URO URO Urology G5N CHI VAA Vascular surgery G5Z Table 2.4: General characteristics of patient groups of the Division B: surgical specialties for the year 2010 (Source: Braaksma and Kortbeek [8]). A ward can be characterised by the number of beds, the patient types and the total amount of available personnel. The wards all have twenty four operational beds except for ward G5 Noord, which has thirty operational beds [8]. Each patient group requires a different type of medical care and has an unique length of stay (LOS). The staff at a ward can be categorised into nursing, medical, and administrative staff, of which nursing is the largest group. The working hours at a ward are divided into three shifts: day, evening, and night. For a more extensive description of staff, working hours and tasks we refer to the process description by Smeenk and Burger [30]. From a patient-oriented view we can distinguish three steps, admission, stay, and discharge. A patient is admitted at a ward either because he is scheduled for surgery, another procedure, or he 14

25 2.4. INPATIENT CARE UNITS CHAPTER 2. CONTEXT ANALYSIS needs urgent medical care. The LOS of a patient is the time between admission and discharge. During the stay of a patient he undergoes surgery and recovers. When the doctor declares the patient healthy he is discharged from the ward and can go home Resource capacity planning and control We define the capacity of a ward by the following three resources: the number of available beds, specialised equipment and assigned staff. How this capacity is utilised depends on the allocation of beds and the scheduling of nurses. Section discusses control decisions related to the care units. Section states the decisions related to nurse scheduling Care unit decisions In this section, we discuss the resource capacity planning decisions in the care units. First, we provide information on the allocation of patients to care units. Next, we briefly describe how the capacity is determined. We end this section by describing the operational admission planning and control. Care unit partitioning Care unit partitioning consists of two interrelated decisions. How many wards do we need to open and how should we assign the different patient types to them? This decision is made by the division management. Traditionally, each specialty is allocated to an own ward [31]. In the AMC, specialties with a low number of patients per type are combined and specialties with a high number of patients per type have their own individual ward. Capacity dimensioning Capacity dimensioning of a ward consists of determination of the number of operational beds and equipment necessary to treat all patients. This decision is made by division management in cooperation with ward management. Admission control Admission control of patients consists of two operational planning phases. Wards are informed about elective surgeries one or two weeks in advance that are scheduled for surgery. The head nurse then decides if a patient can be admitted to their ward and at what time. If the ward is expected to be occupied by other admissions the patient is assigned to an other ward. On the admission day of the patient the head nurse evaluates the current bed occupation and the other planned admissions. When there is no room to admit the patient the head nurse can reallocate a planned admission to another ward. Acceptance of non elective patients depends on the current bed occupation and the planned admissions for the upcoming days Nurse scheduling decisions In this section we discuss decisions related to the scheduling of nurses. First, we state the capacity dimensioning decision. Next we describe how the number of nurses per shift are determined in the current situation. Capacity dimensioning Capacity dimensioning consists of determining the total amount of staff needed. Because contractual agreements with employees are made for a long term this must be done accurately. The contract negotiations are done by the management of each ward. Staff shift scheduling Staff shift scheduling consists of determining the number of nurses needed per shift. Ward management is responsible for this decision. The offline operational nursing roster is created ten weeks in advance by the planner of the ward, when the demand resulting from the master surgical schedule is still unknown. In at least two wards, the management chooses to assign the same total number of nurses to a shift for each day of the week. This total number is based on the maximum number of operational beds and the patient-to-nurse ratio during each shift. The patient-to-nurse ratio describes how many patients a nurse can care for and depends on the shift. 15

26 2.4. INPATIENT CARE UNITS CHAPTER 2. CONTEXT ANALYSIS On an online operational level nurses can become ill and the number of operational beds at a ward is reduced. Another option is to hire additional workers to replace these ill nurses Performance We describe the performance of a nursing ward by the misplacement rate and the rejection rate, and the realised bed census. We discuss the misplacement rejection rates first. Next, we evaluate the realised bed census. A misplacement occurs when a patient is placed in a non dedicated ward. A rejection occurs when a patient is refused by the hospital. Table 2.4 shows the dedicated or preferred ward for each patient type of Division B for the year We determine the misplacement rates from 2010 as follows. We count the number of admissions of each patient type in their non dedicated ward and divide this by the total number of admission for this specific patient type. We clustered the surgical short stay and general surgery patient types because these could not be separated in the data. Table 2.5 shows the misplacement rates for each specialty. The specialty CAP has a misplacement rate of zero because these patients need specific equipment, which is only available in their designated ward. We can not measure the rejection rate because no information is available about the number of rejections per patient type. Specialty Nr admissions designated ward Total admission Misplacement ratio CAP % CHI % CHP % MZK % ORT % TRA % URO % VAA % Table 2.5: Admission and misplacements for each specialty for the year 2010 (Source: Locati 2011). The bed census states the number of operational beds used during a day of the week. Figure 2.3 shows a box plot of the bed occupancy for each day of the week for the wards under discussion. The green dotted line gives the maximum number of beds on each of the wards. The upper percentile of the bed census for some wards in the box plot can go above this line because wards have the possibility to add additional beds in case of extreme demand. Figure 2.3 shows that the average number of occupied beds lies far below the maximum number of operational beds that is used to staff nurses. When we look at the distribution of demand we denote that the values hive a high variation on each day of the week. The average bed census of ward G7NO is particularly low compared to the maximum and this ward is favourite to misplace patients to. 16

27 2.4. INPATIENT CARE UNITS CHAPTER 2. CONTEXT ANALYSIS (a) G3ZU (b) G5NO (c) G5ZU (d) G6NO (e) G6ZU (f) G7NO (g) G7ZU Figure 2.3: Average bed census for each day of the week for wards of Division B, in which the green dotted line states the maximum number of beds, 8500 admissions, year 2010 (Source: Cognos 2011). 17

28 2.5. CONCLUSIONS CHAPTER 2. CONTEXT ANALYSIS 2.5 Conclusions In this chapter we analysed the OR department and the clinical wards of Division B. We identified the following causes that may yield inefficient resource usage: A fluctuating block schedule The current Operating Room Days (ORDs) assigned to the specialties differs monthly. Variation in the number of surgery blocks scheduled results in variation of demand for care at the inpatient care units. No standardisation in scheduling of patients for surgery Each specialty planner has its individual method of scheduling patients into surgery blocks. Due to lack of standardisation between the specialties it is difficult to predict the impact of a scheduled surgery block on the bed occupation in a ward. Nurses are scheduled based on static demand The scheduling of nurses is based on the maximum number of beds per ward and the patientto-nurse ratio per shift for each day of the week. The actual demand is dynamic and differs each day and each hour. In at least two of the wards, the required number of nurses is based on the maximum demand, which results in overstaffing. Mismatch of planning horizons In order to schedule nurses based on a dynamic demand we need two consecutive steps. First, we need to know the demand resulting from the MSS and the arrival of acute patients. Next, we need to align both planning horizons. Currently, the scheduling of nurses is performed ten weeks in advance. At this moment in time, the demand caused by the MSS is partly unknown, because the scheduling of patients in a surgery block is performed two to three weeks in advance. This mismatch in planning horizons makes it difficult to adequately staff the right number of nurses to a shift. Relatively small ward sizes The ward sizes under consideration are relatively small. Due to these small ward sizes the demand for care is highly influenced by how the surgery blocks, and the patient inside a surgery block, are scheduled. All of the causes above indicate that the processes in the inpatient care chain can be improved and that resources can be used more efficiently. To reduce variation in the demand for beds, several patient types could be combined into the same ward to obtain benefits of the risk pool effect. Other opportunities lie in the integral development of a cyclic MSS that minimises the total resources needed at the clinical nursing wards and better aligns the planning horizon of the scheduling of nurses with the MSS. 18

29 Chapter 3 Literature This chapter describes literature related to resource capacity planning in health care. For a taxonomic classification of planning decisions in health care and a state of the art review we refer to Hulshof et al. [17]. We structured this chapter as follows. First, we provide an overview of the techniques in literature for making resource capacity planning decisions in Section 3.1. Next, we discuss resource capacity planning methods for the OR department in Section 3.2. Section 3.3 states the planning methods for nursing wards. Section 3.4 describes various decomposition approaches for optimisation of multiple planning decisions. Section 3.5 states the conclusions of this chapter. 3.1 Techniques for resource capacity planning Within the Operational Research/ Management Science literature several models are presented to support resource capacity planning decisions. These models can be broadly characterised as analytical or simulation based [35]. Most of the time analytical optimisation methods require only one experimental run to produce optimal or near optimal solutions [18] while simulation optimisation focuses on finding the best input variables from all possible combinations without evaluating each possibility [12]. The strength of simulation models lies in the fact that they are well equipped to capture the broad scope of complex systems [36] while analytical methods have a limited capacity to characterise these systems. A possible weakness of simulation based optimisation is that these models are inexact and require a great deal of time to develop [36]. For a literate review of articles in which analytical and simulation techniques are used in the operating theatres we refer to Cardoen et al. [11]. For a more elaborated overview of simulation techniques in hospital settings we refer to Jun et al. [18]. To overcome the disadvantages of both simulation and analytical models several researchers have developed methods that combine the strength of both techniques. We provide one example: Cochran et al. [13] propose a method for stochastic bed balancing inside an obstetrics hospital. First, an analytical queueing model is developed to asses the flow between units. Next, discrete-event simulation is used to maximise the flow through the balanced system and to study several what-if scenarios. 19

30 3.2. METHODS OR DEPARTMENT CHAPTER 3. LITERATURE 3.2 Resource capacity planning methods for the OR department This section describes resource capacity planning methods that focus on planning decisions of the OR department. We structured this section as follows. First, we discuss recent literature reviews. Next, we describe articles that focus on the patient case mix decision. This section ends with literature about the Master Surgical Schedule (MSS). Literature reviews A large number of articles is written about operation room planning and scheduling. In order to obtain an understanding of the research conducted we consulted three recent literature reviews that encompass operating room planning and scheduling. All these reviews are published in the past two years. All of these authors choose a different method to structure their article. Cardoen et al. [11] organise the literature by the use of six descriptive fields: patient characteristics, performance measures, decision delineation, research methodology, uncertainty and applicability of the research. Cardoen et al. conclude with directions for further research. They emphasise to conduct more research on scheduling of non elective patient types, incorporation of uncertainty and stochasticity and a better integration of the operating room planning with downstream facilities and resources. However, they realise that this last recommendation widens the scope of the problem setting, yielding increased difficulty, to obtain reasonably fast results, that are likely to be general [11]. Guerriero and Guido [15] classify the reviewed articles in terms of strategic, tactical and operational decision levels. Guerriero and Guido conclude with the following five objectives that operation research techniques aim at inside the operating room theatres: increased patient throughput, increased satisfaction of patients, surgeons and staff, increased utilisation of OR resources, reduction of cancellations and reduction of time loss due to the late starts and changeovers. The third review studied is from May et al. [23] which categorises the reviewed articles by the planning horizon and the domain of the problem studied. May et al. distinguish six different planning horizons ranging from very long term (12-60 months) to contemparous (on the same day). Furthermore they distinguish six different domain areas: capacity planning, process re-engineering, surgical services portfolio, procedure estimation, schedule construction and schedule execution monitoring and control. May et al. concludes that the economic and project management aspects of the surgical scheduling process might be the most promising lines of research in the foreseeable future [23]. They mention that many interesting models have been proposed but that none appear to have had widespread impact on the actual practice of surgery scheduling. Patient case mix The patient case mix decision concerns which patient types to treat and how many of them. Mulholland et al. [24] propose a Linear Program (LP) to optimise financial performance for the department of surgery. The objective of their model is to maximise financial outcomes for the hospital and physicians while deciding on the procedure mix. The procedure mix for each specialty could be increased or decreased by 15% for each patient type. The outcomes of the LP show that by adjustment of the procedure mix professional payments could be increased by 3.6% and hospital total margin by 16.1%. Ma et al. [21] propose an Integer Linear Program (ILP) and a branch and price algorithm to solve the strategic case mix problem optimally. They assume that hospitals are profit maximisers that will select an optimal casemix given a minimum and maximum number of patients per patient type and various resource constraints. They consider three resources: surgeons, operation rooms and beds. The patient specific parameters they use are the reward of a treatment and a deterministic surgery duration and length of stay. Their results show the applicability of their model but it has not been tested with real data. Master surgical schedule A Master Surgical Schedule (MSS) is a schedule that defines the number and type of available ORs, the opening hours and the surgeons or specialist groups to whom the OR time is assigned [15]. A cyclic MSS is a schedule that is repeated after a certain time period [5]. Cyclic master surgical scheduling is a promising approach for hospitals to optimise resource utilisation and patient flows [34]. We describe two articles that state the creation of an MSS. We discuss more advanced methods in Section 3.4. Testi et al. [32] propose a three phase approach for the scheduling of operating rooms. The first stage consists of determining the number 20

31 3.3. METHODS CLINICAL WARDS CHAPTER 3. LITERATURE of sessions per type to schedule based on the demand and the available operating room time. In the second stage an ILP is solved that assigns the session time from the first stage to available OR days. The objective of this ILP is to maximise surgeon s preferences. The last stage consists of a simulation model in which various heuristics are used to assign patient types to surgical blocks. Van Oostrum et al. [33] propose a two phase master surgical scheduling approach at the tactical level. First, they create a set of operating room days (ORDs) in which patient types are scheduled by means of column generation with the objective to reach a target utilisation. Next, an ILP is formulated to assign ORDs to actual days of the MSS with the objective to level the bed occupation. 3.3 Resource capacity planning methods for the clinical wards In this section we discuss planning methods that are related to the clinical wards. We have categorised the articles into three categories. First, we discuss articles on the care unit partitioning decision. Next, we describe reviews that focuses on the capacity allocation decision. This section concludes with articles that contains nurse rostering decisions. Care unit partitioning Care unit partitioning is rarely described in the literature. To the best of our knowledge one article by Villa et al. [38] describes the redesign of the hospital wards. Villa et al. studied three different hospitals in Italy that redesigned their patient flow logistics around patient care needs. They considered four areas of interest: the organisation of the wards, the hospital s physical lay-out, the capacity of the planning system and the organisational roles supporting the patient flow management. In traditional hospitals, patients are assigned to a ward according to the relevant clinical specialty. In the Italian hospitals studied the organisation of the wards was changed from a specialty focus to a length of stay focus. They distinguish five types of wards ranging from week surgery wards that are closed during the weekend to post-acute wards that only accept patients with a length of stay longer than three weeks. Specialists argue that due to these multidisciplinary wards they could share facts and experiences with other colleagues which was beneficial for their work. Villa et al. conclude that the three hospitals under investigation, after the changes, decreased their patient s average length of stay and increased utilisation. The authors do not especially state whether these organisational changes caused this effect or this is caused by other interventions. Another remark can be placed by the argumentation on how these ward configurations are determined because no quantitative formulation is given. Capacity allocation Literature about capacity allocation inside care units mainly focuses on the number of beds necessary. In some countries (e.g. France) health authorities issue a ratio about the number of beds a hospital needs. Nguyen et al. [27] propose a simple method to determine the number of beds necessary inside a hospital. Their method consists of a score based on three parameters: the number of misplacements caused by a lack of space, the number of days with no possibility to admit unscheduled admissions and the number of days with at least U unoccupied beds, where U is a predefined threshold. This method is applied to three clinical wards and the outcomes show that the method performs well compared to the given ratios from health authorities. A disadvantage of this method is that due to its simplicity it does not consider the stochastic nature of patient arrivals and length of stays. A more advanced model is proposed by Cochran et al. [13]. They use a queueing model to study blocking behaviour. Blocking means that patients cannot advance through the system because beds or units are still occupied. The authors use an exponential length of stay distribution for their patients. Their outcomes show that 38% more flowthrough of patients through the departments can be achieved with only 15% more beds. VanBerkel et al. [37] propose an exact approach to determine the workload inside the clinicial wards based on the MSS. VanBerkel et al. consider surgery blocks of patient types in which a variable number of patients undergoes surgery. Furthermore, they consider a stochastic length of stay distribution derived from historical data. The output of the model is a stochastic distribution of the demand for beds inside the clinical wards, which is called the bed census. Based on this demand it is possible to determine the appropriate amount of beds needed to meet a fixed percentile of demand. Smeenk [29] extended 21

32 3.4. DECOMPOSITION APPROACHES CHAPTER 3. LITERATURE the research of VanBerkel et al. into the development of a bed census model on an hourly basis that also takes acute patients into account. Because the bed census is hourly it is suitable to support nurse staffing decisions and it can be used to study the effect of various discharge and admission policies. Nurse rostering A comprehensive literature study about nurse rostering has been conducted by Burke et al. [10]. First, they discuss other literature reviews and describe the role that nurse rostering plays within the longer term hospital personnel planning. Next, they discuss articles that describe solution approaches from an operations research technique point of view. Burke et al. consider mathematical programming, goal programming, multi criteria analysis, artificial intelligence methods, heuristics and meta heuristics. They conclude that a lot of methods are developed but that very few of the developed methods are suitable for directly solving real world problems [10]. A more recent research conducted by Burger [9] focuses on the determination of the optimal number of nurses based on the expected workload inside the clinical wards. The research of Burger is closely related to the research of Smeenk [29]. Based on the output of Smeenk, a stochastic bed census, Burger proposes a method to determine the optimal number of dedicated nurses per ward and flexible nurses per flex pool. Dedicated nurses are assigned to one ward and flexible nurses inside a flex pool are assigned to a ward at the beginning of a shift, dependent on which ward has the highest demand and the number of dedicated nurses already assigned. The optimisation model of Burger distinguishes two service levels. The minimum service level states the minimum fraction of patients that is covered by the nurses assigned to this shift. The overall service level denotes the fraction of the time there are sufficient nurses present during this shift. Burger computes three bounds to determine the number of dedicated nurses per ward and the number of nurses inside a flex pool. The first bound considers only the use of dedicated nurses. The second and third bound consider cases in which flexible nurses can be used. These three bounds together with a decision mechanism select the optimal number of dedicated and flexible nurses to schedule per shift. 3.4 Decomposition approaches for optimisation of multiple planning decisions We want to determine the best combination of multiple planning decisions. In this section we discuss relevant literature that encompasses more than one planning decision. First, we discuss a literature review that discusses articles that encompass multiple departments. Next, we describe literature that focuses on the development of a Master Surgical Schedule in combination with bed levelling and other downstream resources. Literature review Vanberkel et al. [36] have conducted a survey that analyses quantitative health care models that encompass multiple departments. The selected articles are grouped into five different main care services to which surrounding departments are modelled: Emergency medical care, surgical care services, inpatient bed wards, ambulatory care and diagnostic services and pharmacy. For each of the reviewed articles the quantitative approach is given. We select the most relevant articles and discuss them in the next two sections. MSS & bed levelling A lot of current research focuses on the development of an MSS while levelling the downstream requirements at the clinical wards in terms of beds. In the research of Vanberkel et al. [37] a simulated annealing heuristic is used to swap given surgery blocks to level the number of beds at a wards. Bosch [7] uses a two phase decomposition. First, efficient operating room days are developed based on the method of Van Oostrum et al. [33]. Next simulated annealing is used to level the number of beds. Belïen and the Demeulemeester [5] propose several different approaches to level the bed occupancy resulting from the MSS. They consider demand constraints like each surgeon obtains a specific number of operating room blocks, and capacity constraints that 22

33 3.5. CONCLUSIONS CHAPTER 3. LITERATURE limit the number available blocks on each day. Furthermore, they consider stochastic multinomial distributions for the length of stay and number of operated patients inside a surgery block. Their solution approach consists of the development of an MILP and several heuristics. They conclude that a simulated annealing based heuristic performs best, but has a very long computational time. Another solution that performs well is a meta-heuristic approach in which the true objective is evaluated. This meta heuristic approach consists of solving an MIP several times, and incremently adding additional constraints. MSS, beds & other resources Belïen et al. [6] present a decision support system for cyclic master surgery scheduling with multiple objectives. They consider three main objectives while developing the MSS: the MSS needs to be simple and repetitive, the demand for beds need to be levelled and an OR is best allocated exclusively to one group of surgeons. They use MIP techniques involving the solution of multi-objective and linear quadratic optimisation problems. The outcome of this research consists not of a complete solution but gave hospital managers the possibility to study several what-if solutions. Santibanez et al. [28] study trade-offs between OR availability, bed capacity, surgeons booking privileges and waiting lists. They propose a MIP model to schedule surgical blocks for each specialty into ORs while considering OR time availability and post-surgical resource constraints. Santibanez et al. conclude that it is possible to reduce resource requirements needed to care for patients after surgery while maintaining the throughputs of patients. Their method uses a deterministic probability for the patients length of stay. Adan et al. [3] incorporates a stochastic length of stay based on historical data in the development of a two stage planning procedure for the selection of elective and acute patients while allocating at best the available resources. They consider four resources: operating time, intensive care beds, nursing hours at the ICU and medium care beds. The first stage consists of solving an MILP with CPLEX in which the deviation from target resource utilisation is minimised and capacity for emergency patients is reserved. The second stage consists of the development of operational strategies to deal with the actual flow of elective and acute patients. In the last stage a simulation study is performed in which simulation results show a trade off between hospital efficiency and patient service satisfaction. 3.5 Conclusions To make resource capacity planning decisions various methods have been developed. We can choose between simulation optimisation, analytical exact methods or a combination. Cyclic master surgical scheduling is a promising approach to predict and stabilise processes which improves utilisation of resources. Creation of a master surgical schedule (MSS) is mostly done by analytical methods such as solving variants of an LP, column generation and the use of heuristics. The research of VanBerkel et al. and Smeenk provides exact values to determine the downstream workload given the MSS. The research of Burger proposes a method to determine the optimal number of nurses needed. Limited research has been conducted on the patient ward assignment. This research contributes to the literature by studying the care unit partitioning decision and the integration of various analytical models, to optimise the resource usage of the inpatient care chain. 23

34

35 Chapter 4 Solution approach In this chapter we describe our solution approach to reach our research objective. Section 4.1 formulates the research objective as a mathematical optimisation problem. In this section we elaborate on the relationships between the various planning decisions and constraints we take into account. Section 4.2 describes our decomposition approach to solve this optimisation problem. We describe the technical implementation of our model in Section 4.3. Section 4.4 contains the verification and validation. Section 4.5 concludes with our conclusions. 4.1 Optimisation problem In the upcoming sections we systematically formulate our research objective as a mathematical optimisation problem. Section contains the assumption we make. Section describes the relationship between the resource capacity planning decisions and the constraints we take into account. We formulate the objective function and describe cost parameters in Section We conclude this section with an overview of the performance indicators in Section Assumptions To reduce modelling complexity we make the following assumptions: Acute patients arrive directly at the ward according to a non-homogeneous Poisson distribution. This means that acute patients in our model do not use the OR and directly arrive at a ward. A surgery block only contains patients of the same type. This implies that each patient from the same surgery block is assigned to the same ward Decisions Figure 4.1 shows the relationships between the various planning decisions as described in Section 1.3. Some decisions are marked green and others are marked orange. In Section to Section we explore the mathematical relations between the planning decisions. Based on these relations we show that when we make the decisions marked in orange we automatically make the decisions marked in green. Section discusses decisions regarding OR capacity allocation. 25

36 4.1. OPTIMISATION PROBLEM CHAPTER 4. SOLUTION APPROACH Section describes the decisions inside the care units. We conclude this section with nurse staffing decisions in Section To accommodate the reader throughout these sections we have summarised the mathematical notation of the parameters and decision variables in Appendix A. Figure 4.1: Overview of the relationships between the various resource capacity planning decisions OR capacity allocation Figure 4.2 shows the relationships between the OR capacity allocation planning decisions and the decision variables. We first introduce the mathematical description of the decision variables and the constraints. Figure 4.2: Overview of the OR capacity allocation planning decisions and their decision variables Decision variables The OR capacity allocation decision is concerned with the assignment of specific patient type surgery blocks to days and the length of the Master Surgical Schedule (MSS). We define b j,s as the number of surgery blocks assigned to patient type j scheduled on day s of the MSS. The length of the MSS is defined as S. Furthermore, we introduce b M j,s and ba j,s as the number of morning blocks and afternoon blocks. We allow full surgery blocks and half surgery blocks to enhance flexibility of OR capacity allocation. Furthermore, patients that are scheduled for surgery in the afternoon have a different admission distribution compared to patients that are scheduled in the morning. We define the number of patients within a surgery block by the distributions C j (k), Cj M (k), CA j (k), which state the probability that in a surgery block assigned to patient type j, k surgeries are carried out. From the surgery block assignment we derive the relationship between the OR capacity dimensioning and the case mix decision. The capacity dimensioning decision can be quantified by non renewable resources that require a high investment. Non renewable resources are for example the number of ORs to open or the number of X-ray machines needed. We define Ω r as the number of non renewable resources. The index r states the specific resource under consideration. The parameters RE j,r, REj,r M and REA j,r specify how much of resource r a surgery block of patient type j uses. We denote x j as the volume of each patient type j to treat. Then Ω r and x j are given by the following 26

37 4.1. OPTIMISATION PROBLEM CHAPTER 4. SOLUTION APPROACH three equations: {RE j,r b j,s + REj,r M b M j,s} Ω r s, r (4.1) j {RE j,r b j,s + REj,r A b A j,s} Ω r s, r (4.2) j x j 365 S s { } b j,s E[C j ] + b M j,s E[Cj M ] + b A j,s E[Cj A ] j (4.3) In (4.3) we use the less or equal sign because we assume that at most the expected number of patients can be treated. Solution space resource constraints The OR capacity allocation decision is bounded by several restrictions. The number of non renewable resources is limited by a physical maximum which we denote as Ω max r. An actual example of this limitation is the current shortage in surgery assistants. The volume of each patient type to treat is based on production agreements between the hospital and insurers. To take this agreement into account we assume that of each patient type j at least a minimum, Xj min, and at most Xj max, can be treated. We define the following equations to satisfy the capacity restrictions and the production agreements: Ω r Ω max r r (4.4) X min j x j X max j j (4.5) Solution space performance constraint The surgery block scheduling assignment influences the OR access time of patients. We define the OR access time as the time between the specialist diagnoses that a patient needs surgery until the day of surgery. We define the access time service level as the fraction of patients θ(a j ) with an access time that is not greater than A j. We count A j in weeks. For example, an access time service level of θ(2) = 0.95 states that 95% of the patients that require surgery undergo this surgery within two weeks. To determine the access time service level, we need to know the average number of appointment requests µ j,d, for patient type j generated on day d of the week and the distribution of the number of patients within a surgery block and the scheduling of these surgery blocks. The relationship between day d of the week and day s of the surgical schedule is shown by (4.7). We require the following constraint to make sure that our block scheduling decision obeys the target service level θ norm (A j ): f access j (A j, µ j, C j, C M j, C A j, b j, b M j, b A j ) θ norm (A j ) j (4.6) d = s mod 7 (4.7) Care unit decisions Figure 4.3 shows the capacity planning decisions related to care units. The workload at a care unit depends on the Master Surgical Schedule and the assignment of patients to wards. First, we derive the relationships between the decision variables. Next, we discuss the constraints. 27

38 4.1. OPTIMISATION PROBLEM CHAPTER 4. SOLUTION APPROACH Figure 4.3: Overview of the care unit planning decisions and their decision variables Decision variables We need to assign the various patient types j to wards w. We define the binary variables ass j,w (1 if patient type j is assigned to ward w) as the dedicated ward and mis j,w (1 if patient type j is assigned to ward w) as the misplacement wards. The dedicated ward is the preferred ward for a patient type and a misplacement ward is a ward to which the patient may be misplaced when the dedicated ward reaches its capacity. Based on the patient-ward assignment we define a binary variable u w (1 when ward w is opened) that states which wards are opened. Then Ψ gives the total number of wards opened. These relations are shown by the following equations. bigm u w j ass j,w w (4.8) Ψ = w u w (4.9) The resources under consideration in a ward are the total number of beds needed per ward w. We define m w as the number of beds needed on ward w. To determine this number of beds we need to know the patient-ward assignment ass j,w, mis j,w and the block scheduling decision b j,s. We define φ as the misplacement policy, which states decision rules for misplacement of patients. Furthermore we denote E j (n) as the probability that a patient of type j arrives on day n, in which n { 1, 0}. This means that a patient arrives on the day of surgery or the day before surgery, to take into account the pre hospitalisation process described in Chapter 2. The time slot arrival distribution is denoted as Wn(t). j This distribution denotes the probability that a patient type j is admitted on day n during time slot t. We assume a non-homogeneous Poisson arrival distribution for acute patients with rate λ j,d,t, where j denotes the patient type, d the arrival day of the week and t the time slot in which this patient arrives. The discharge distribution of patients is denoted P j (n) for the length of stay distribution and Mn(t) j for the time slot discharge distribution. Here n and t state the probability that a patient is discharged on day n or at the beginning of time slot t. We assume that patients of one type have the same discharge distribution. The last inputs we need are the target rejection and misplacement rates, χ and η. The target rejection rate states the maximum fraction of patients that are allowed to be rejected by the hospital. The target misplacement rate states the maximum fraction of patients that may be misplaced to a non dedicated ward. We introduce these target rejection rates to make sure that our resources are utilised efficiently while ensuring quality care for patients. We have defined all the inputs necessary to formulate a function to determine the optimal number of beds in ward w: m w = f beds w (b, b M, b A, ass w, mis w, φ, C, C M, C A, E, W, λ, P, M, χ, η) w (4.10) Solution space resource constraints The decision variables of the care unit are limited by several constraints. The number of wards to open is bounded by the maximum number of wards physically possible within the building, which we define as Ψ max. Furthermore, each ward has a limited capacity in terms of the maximum number of beds possible, which we denote by Mw max. The maximum number of wards and maximum capacity 28

39 4.1. OPTIMISATION PROBLEM CHAPTER 4. SOLUTION APPROACH possible are satisfied by the following equations: Ψ Ψ max (4.11) m w M max w w (4.12) Another important constraint that limits the patient-ward assignment possibilities is the prevention of conflicts. Some patient types are not allowed to be assigned to the same ward due to medical reasons. We introduce the binary parameter NP i,j that has a value of 1 if patient types i and j are not allowed to be allocated together. The following constraint ensures that this restriction is satisfied. ass i,w + ass j,w 2 NP i,j i, j, w (4.13) We allow that a patient type can be assigned to one dedicated ward only. This implies that the maximum ward sizes should be large enough to make sure that a patient type completely can be assigned to a single ward. This restriction is satisfied by (4.14). We introduce AS that states the maximum number of patient types that may be assigned to the same ward. We ensure this with (4.15). ass j,w = 1 j (4.14) w ass j,w AS w (4.15) j Nurse staffing decisions Figure 4.4 shows the relations between the nurse staffing decisions. The number of nurses necessary depends on the wards that share a flex pool and the bed census. A flex pool consists of multi skilled nurses that can work at different wards. These multi skilled nurses are assigned to a ward at the beginning of a shift dependent of the workload and the number of dedicated nurses assigned. The advantage of having a flex pool is that wards are better able to adapt to fluctuating demand for care. Figure 4.4: Overview of the nurse staffing decision variables The bed census Z w s,t(x) is defined as a probability distribution over the number of patients x inside a ward w during a time slot t on day s of the MSS. We can define a similar function as (4.10) to determine the bed census. Additional input for this equation is the number of operational beds m w. 29

40 4.1. OPTIMISATION PROBLEM CHAPTER 4. SOLUTION APPROACH We do not need the target rejection and misplacement probabilities. The function to determine the bed census is given below: Z w s,t(x) = f Census (b, b M, b A, ass w, mis w, φ, C, C M, C A, E, W, λ t, P, M(t), m w ) w, s, t (4.16) We first discuss the decision variables related to nurse staffing. resource constraints we consider. Next, we introduce the various Decision variables We need to determine which wards share the same flex pool. We denote flex f,w as the binary variable that denotes whether ward w uses flex pool f. Once the flex pool assignment is made and the bed census Zs,t(x) w for each ward is known we determine the optimal number of nurses in terms of dedicated, n ded, and flexible, n flex, staff. To determine the number of nurses necessary we introduce target service levels that prevent that the workload for nurses gets too high and the patients safety cannot be guaranteed any more. We introduce the patient nurse ratio rs,τ w that specifies how many patients a nurse can care for during each shift τ on day s assigned on ward w. A patient nurse ratio of two specifies that for every two patients at least one nurse is necessary. We distinguish two types of desired service levels. The target overall service level α denotes the fraction of the time that the number of patients on a ward does not exceed the number of nurses staffed times the patient nurse ratio. An overall service level of 95% states that one in twenty time slots the nurse coverage may be insufficient. The minimum service level β states the minimum fraction of patients per time slot that should be covered by the staffed number of nurses times the patient-nurse ratio. Futhermore we introduce γ which states the fraction of nurses on a ward that should be dedicated. The bed census in combination with the flexpool-ward assignment and the target nurse service levels are used as inputs for the following function to determine the number of nurses in a flex pool and the number of dedicated nurses per ward. (n ded w,s,t, n flex f,s,t ) = f Nurses (Z w s,t, flex f,w, α, β, γ, r w s ) w, f, s, t, (4.17) The last step is to determine the total amount of FTE in terms of dedicated and flexible nurses. We do this by taking the sum of all dedicated nurses per ward and all flexible nurses per flex pool. n ded = w n flex = f n ded w,s,t (4.18) s s t n flex f,s,t (4.19) t Solution space resource constraints We define several quality constraints that limit the solution space. A nurse needs specialised training to treat a patient type j. Therefore it is not realistic that each nurse in a flex pool is qualified to treat all different patient types. We denote the binary parameter NF f,j that is 1 if a nurse from flex pool f is not skilled to treat patient type j. Then (4.20) makes sure that only skilled nurses are assigned to patients they can care for. ass j,w + flex f,w 2 NF f,j f, j, w (4.20) We assume that a ward can only be assigned to one flex pool. Assigning more flex pools will not only give a larger solution space but also complicates the situation from an operational perspective. How are hospital managers on an operational level going to decide the number of nurses needed per 30

41 4.1. OPTIMISATION PROBLEM CHAPTER 4. SOLUTION APPROACH ward when they have the possibility to choose from multiple flex pools? To satisfy this we define: flex f,w = 1 w (4.21) f Optimisation objective Section gives the mathematical link between the decision variables and constraints of the planning decisions. For each resource and patient case mix we define cost parameters to evaluate the total costs. Based on these cost parameters we formally define the objective of this research: { min j F patient j x j + r Fr nonrenewable Ω r +F F T Ed nded +F F T Ef n flex + {Fw ward u w +F bed m } w } w In this objective function various cost parameters are present. We explain each parameter briefly: Revenue per patient type (F patient j ): Revenue or cost of a patient type j. When the revenues of a patient type j are higher than the costs this value is negative. We only consider patient specific costs that are not accounted for by the other cost parameters. The resource specific costs per day are included in the revenue of a patient type. Non renewable resource costs per year (Fr nonrenewable ): The costs of investing in non renewable resources. For example the number of ORs and X-ray machines. This includes depreciation costs. Cost of one dedicated nurse in FTE per year (F F T Ed ): Cost of one FTE of a dedicated nurse per year. Cost of one flexible nurse in FTE per year (F F T Ef ): Cost of one FTE of a nurse assigned to a flex pool per year. Because nurses in a flex pool have extra skills and education this cost parameter may differ from F F T Ed. Fixed costs to open a ward per year (Fw ward ): Fixed costs to open ward w per year. These fixed costs consist of overhead, depreciations and supporting staff other than nurses. Fixed costs of a bed per year (F bed ): The fixed costs to place a bed in a ward. Based on the stated cost parameters we can evaluate the objective function for various resource configurations. In Section 4.2 we explain our decomposition approach to make the resource capacity planning decisions and to minimise the objective function Performance indicators To measure the performance of the block scheduling decision on the resource utilisation we define the non renewable OR resource utilisation given by (4.22). OR resutil r = Ω r S { REj,r b j,s {REM j,r bm j,s + REA j,r r (4.22) ba j,s }} s j To measure the performance of resources in the clinical wards we first determine the average bed census of each time slot. The average bed census Z w s,t is given by (4.23). m w Z s,t w = {x Zs,t(x)} w s, t, w (4.23) x=1 31

42 4.2. DECOMPOSITION APPROACH CHAPTER 4. SOLUTION APPROACH Once we know the average bed census we can define the operational bed utilisation by taking the sum of the average bed census for each time slot and day of the MSS: Bed util w = s t Z w s,t S T m w w (4.24) The performance indicators for the number of nurses are given by the realised α and β service levels for which we use the mathematical notation as given by Burger [9]. 4.2 Decomposition approach This section describes our solution approach to solve the optimisation problem defined in Section 4.1. We use a decomposition approach because the solution space is extremely large and the relationships between variables are complex and non linear. Figure 4.5 shows the different steps of our decomposition approach. In the following sections we will discuss each step in more detail. Throughout this section we assume that the reader has taken knowledge of the variables and parameters that are introduced in Section 4.1. For a complete overview of the variables and parameters used in this section we refer to Appendix A. Figure 4.5: Decomposition approach to solve the optimisation problem Set number of blocks and length of MSS We set the number of patients per type, x j, to treat and the length of the MSS, S, as input parameters. We derive the ratio between full, morning and afternoon blocks from historical data. From x j and S we then can determine the number of full blocks, Bj norm, morning blocks, B norm,m j, and afternoon blocks, B norm,a j, that we need to schedule Block scheduling in the OR and patient-ward assignment In this step we create a block schedule and assign patient types to wards. From Section we know the number of blocks to schedule and the length of the MSS. The block scheduling decision can be characterised as a bin packing problem, in which a list of items is assigned to a minimum number of bins. Surgery blocks can be seen as the items and the number of ORs to open as the minimum number of bins. The patient ward-assignment is defined in the literature as a bin packing problem with conflicts. In this case we want to assign each patient type (items) to a ward (bins) without exceeding maximum capacity and violation of the constraint that some patient types cannot be allocated to the same ward. The patient-ward assignment and the block scheduling decision are interrelated. Both decisions together are main contributors to the resource usage inside a ward. Therefore it is not clear which decision should be taken first in a decomposition. Within the literature bin packing problems are 32

43 4.2. DECOMPOSITION APPROACH CHAPTER 4. SOLUTION APPROACH defined as Integer Linear Programs (ILP) and can be solved with commercial software like CPLEX. By nature bin packing problems are NP-hard, which means they are hard to solve when instances get too large. When instances get too large we can use an taboo search or graph theory algorithm to solve the bin packing problem with conflicts, see for example [22] and [25]. We formulate an ILP that combines the block scheduling in the OR and the patient ward assignment. The objective of the block scheduling decision and the patient ward assignment consists of minimisation of: Number of non renewable resources Ω r. Number of Wards Ψ to open. Expected number of beds m µ w on a ward. Expected number of nurses n µ w,s,τ needed on ward w on day s during shift τ When we minimise the number of beds and the number of nurses needed per ward, we automatically level the workload inside the wards. Minimisation of these resources is translated into the following objective function: min { r κ nonrenewable r Ω r + w {κ wards w u w + κ beds m µ w + κ nurses s τ } n µ w,s,τ } (4.25) In this objective function the various κ values determine the weight factors of the resources. We can choose for example to use the cost parameters defined in Section to determine the values for κ. Let us define the decision variables b j,s,w, b M j,s,w, ba j,s,w as the number of surgery blocks of patient type j assigned to ward w and scheduled on day s. The prefixes M and A make the distinction between morning and afternoon blocks. We combine the patient-ward assignment and the block scheduling decision into one decision variable because these are interrelated and multiplication of variables causes a non linear problem. The downside of this approach is that the number of variables rapidly increases which makes it harder to solve this ILP to optimality. To make sure that we assign all blocks that we set in Section of our decomposition approach the following three equations should be satisfied: b j,s,w = Bj Norm j (4.26) s s w w b M j,s,w = B Norm,M j j (4.27) b A j,s,w = B Norm,A j j (4.28) s w (4.29) and (4.30) ensure that the number of non renewable OR resources are set when a block is scheduled on a given day. {RE j,r b j,s,w + REj,r M b M j,s,w} Ω r r, s (4.29) j w {RE j,r b j,s,w + REj,r A b A j,s,w} Ω r r, s (4.30) j w 33

44 4.2. DECOMPOSITION APPROACH CHAPTER 4. SOLUTION APPROACH When a surgery block of patient type j is scheduled on day s and assigned to ward w, (4.31) ensures that the variable ass j,w becomes 1 for this patient-ward combination. The parameter AS limits the maximum number of patient types that can be assigned to the same ward. (4.32) ensures that a patient type is only assigned to one ward. (4.33) ensures that the constraint related to the maximum number of non renewable OR resources present is not violated. (4.34) and (4.35) satisfy that a ward is opened when a patient type is assigned to it and (4.36) ensures that patient types with conflicts are not allocated to the same ward. {b j,s,w + b A j,s,w + b M j,s,w} bigm ass j,w j, w (4.31) s ass j,w = 1 j (4.32) w Ω r Ω max r r (4.33) AS u w j ass j,w w (4.34) Ψ = w u w (4.35) ass j,w + ass i,w 2 NP i,j j, i, w (4.36) j L max j n= 1 G j (n) {E[C j ] b j,s n,w + E[C M j j L max j n=0 ] b M j,s n,w + E[C A j ] b A j,s n,w}+ G j (n) {λ j,s n ass j,w } m µ w s, w (4.37) We define (4.37) to calculate the expected number of beds on a ward, m µ w. This constraint is based on the ILP formulation given by Adan et al. [3]. The first part of (4.37) consists of the expected number of elective patients and the latter part states the number of acute patients. G j (n) states the probability that a patient is still present inside the hospital on day n. Where n { 1,..., L max } counts the number of days before or after surgery, with surgery on day n = 0. The demand that results from a scheduled surgery block is calculated based on the average number of patients inside a block. The number of acute patients of type j that arrive on day s n is given by λ j,s n. To determine the expected maximum number of beds needed we take the sum over all patient types j and the maximum length of stay of a patient type j and calculate the number of beds occupied on each day, s, of the MSS. To take the pre operative stay into account we take the sum from n = 1. Because it occurs that the LOS of some patients is longer than the MSS we use the convention that the subscript s n should be treated as modulo S. This means that day -1 is the same as day S 1. The maximum number of beds physically possible on a ward is given by Mw Max. In (4.37) we calculate the maximum number of expected beds needed. In the real world there is some variation in beds. To make sure that we take this variation into account we define the stochastic capacity scale factor, ɛ, that prevents us from exceeding the maximum capacity. m µ w ɛ M Max w (4.38) 34

45 4.2. DECOMPOSITION APPROACH CHAPTER 4. SOLUTION APPROACH To estimate the required number of nurses per day we define a similar equation as for the estimated number of beds. We denote the estimated number of nurses needed by N w,s,τ. The number of nurses needed depends on the number of patients present and the time of the shift τ. In 4.39 we denote r j,τ as the nurse care ratio that specifies how many patients of type j a nurse can care for. j L max j r j,s,τ n= 1 G j (n) {E[C j ] b j,s n,w + E[C M j j r j,s,τ L max j n=0 ] b M j,s n,w + E[C A j ] b A j,s n,w}+ G j (n) {λ j,s n ass j,w } n µ w,s,τ s, w, τ (4.39) Elective surgery only takes place on working days so we need to make sure that no blocks are assigned to weekend days. We satisfy this requirement with the following constraints. We assume that the length of the MSS is a multiple of seven days and that Monday is day 1. b j,s,w = 0 and b j,s+1,w = 0 s = (h 1) h = 1... ( S ) j, w, s 7 b M j,s,w = 0 and b M j,s+1,w = 0 s = (h 1) h = 1... ( S ) j, w, s 7 (4.40) b A j,s,w = 0 and b A j,s+1,w = 0 s = (h 1) h = 1... ( S ) j, w, s 7 To finalise our ILP we define the following integer constraints: b j,s,w, b M j,s,wb A j,s,w, n µ w,s,τ, m µ w, Ψ, Ω r {0, 1, 2...} (4.41) ass j,w, u w {0, 1} (4.42) Access Time model by Kortbeek et al. In section we propose a method to create a block schedule and assign patients to wards. Our optimisation problem in Section 4.1 states that the outcome of the block scheduling decision should satisfy the desired target OR access time service level. The access time service level is a function of several inputs and given by (4.6). To evaluate the access time service level we use the methodology proposed by Kortbeek et al. [19] for this function. First, we summarise the model of Kortbeek et al. [19]. Next, we discuss how we use this model to determine the realised OR access time service level. Kortbeek et al. propose[19] a methodology to design cyclic appointment schedules for outpatient clinics with scheduled and unscheduled arrivals. Their method is based on an algorithm that links two models. The first model evaluates the access time for scheduled arrivals and the second model evaluates the day process of scheduled and unscheduled arrivals. The access time is evaluated given the distribution of the number of appointment requests per day and the number of available appointments slots per cycle. A cycle is a fixed time period of several days that is continuously repeated. For each cycle the backlog distributions are determined. Backlog is defined as the number of appointment requests that have been made while the actual appointment has not yet taken place. Based on the derivation of moment generating functions the steady state transition probabilities for the backlog at the start of each day in the cycle are determined. Based on these transition 35

46 4.2. DECOMPOSITION APPROACH CHAPTER 4. SOLUTION APPROACH probabilities it is possible to derive the expected access time service level. The methodology proposed by Kortbeek et al. needs the distribution of the number of appointment requests per day and the number of available planning slots per cycle. When we translate this analogy to the access time of the OR we need to know the distribution of the number of surgery requests per day and the capacity of a surgery block. The model of Kortbeek assumes deterministic appointment durations. To model the capacity of a surgery block we use the expected number of patients in a surgery block, denoted E[C j ]. We derive the distribution of the number of surgery requests per day from historical data. When the access time service level is below our predefined target we can adjust the assignment of blocks to days or we need to increase the number of blocks scheduled. Our ILP, described in Section, is designed in such a way that it levels the expected number of beds needed on a ward. This means that the surgery blocks are most likely evenly distributed over the MSS cycle. It is therefore reasonable to expect that changing the block scheduling decision will only yield small improvements in the access time service level. Increasing the number of blocks that are scheduled provides better results Hourly Bed Census model by Smeenk et al. The hourly bed census is based on the Master Surgical Schedule and the patient-ward assignment. To determine the number of beds needed on a ward w we defined (4.10). For this function we can use the model developed by Smeenk et al. [29] which provides the number of beds needed per ward given the target rejection and misplacement rates. In this section we provide a brief summary of the model developed by Smeenk. For more detailed information regarding the model of Smeenk, we refer to the thesis of Smeenk [29]. Figure 4.6 shows the three consecutive steps to obtain the bed census per ward. First, the demand of elective patients is determined. Next, the demand resulting from the arrival of acute patients is calculated. In the last step both demand distributions are combined. We discuss each step in more detail. Figure 4.6: Overview of the different steps to obtain the bed census on a ward Elective steady state: First, the demand for beds at a ward resulting from a single surgery block is calculated. This is done for each surgery block of each patient type. In the next step a single MSS is considered in isolation in which multiple surgery blocks are present. The single cycle demand is created by means of discrete convolutions, because patients from a block scheduled on previous days are still recovering in a ward when new patients arrive. The last step is to consider multiple overlapping MSS cycles by taking discrete convolutions of the single cycle demand. Acute Steady State: The steady state demand for acute patients is determined in a similar way as for elective patients. The first step is to determine the influence of a single acute patient type that arrives on a day d. Next, the influence of multiple patient types in a cycle is considered. The cycle length for acute patients is R days. The steady state cycle is calculated with discrete convolutions of the single patient types. The final step is to consider multiple overlapping cycles by taking discrete convolutions of the single cycle demand. We need discrete convolutions because patients from a previous cycle may still be recovering in the next cycle(s). 36

47 4.2. DECOMPOSITION APPROACH CHAPTER 4. SOLUTION APPROACH Ward Steady State: Both steady state distributions are combined by discrete convolutions to obtain the steady state ward demand distribution. This demand distribution states the probability that x patients are present on day s during time slot t on ward w. The method proposed by Smeenk also has the possibility to misplace patients between multiple wards given predefined control rules. Given the number of beds on a ward the model computes the realised rejection and misplacement rates. To determine the number of beds needed per ward we initially set the number of beds at a predefined value. Then, we evaluate the corresponding rejection and misplacement rates for this number of beds and compare this with our target rates. If the realised rates are above the targets we need to increase the number of beds. When the realised rates are below the targets we decrease the number of beds. We do this iteratively until we have reached the minmial number of beds for which the targets are satisfied. For the exact mathematical formulation of the misplacement policy and the rejection and misplacement rates we refer to the thesis of Smeenk [29] Flex pool decisions In this section we explain our procedure to evaluate all flex pool assignment possibilities among the different wards. We use the model of Burger [9] to determine the number of dedicated nurses per ward and the number of nurses within a flex pool. This model needs as input the wards that share the same flex pool and the bed census derived from the model of Smeenk. Section discusses the model of Burger in detail. Because the number of flex pools is low and the number of wards that share the same flex pool is limited we can use total enumeration to evaluate all flex pool combinations. In this section we describe our four step method that makes use of the model of Burger. We illustrate our method by means of an example. 1. Set number of flex pools that can be used. 2. Create a set of all combinations of 2 to Ψ wards sharing a flex pool, in which Ψ denotes the number of wards opened. Remove combination of wards that are not allowed to share the same flex pool together because the nurses are not trained to treat all their patient types. 3. For each combination in the set determine the number of dedicated nurses needed and the number of nurses within a flex pool. We use the model of Burger to do this. 4. Select the combination that minimises the number of nurses needed. Now we give our example: 1. We set the number of flex pools used at two. Let us consider four wards, of which ward 1 and ward 2 cannot share the same flex pool. 2. We compute all possible combinations in which two, three or four wards share the same flex pool. We remove the combinations in which ward 1 and ward 2 share the same flex pool. 3. For each combination determined in step 2 we calculate the number of dedicated nurses per ward and the number of nurses in a flex pool. 4. We select the combination that minimises the number of nurses needed. We have the option to have one flex pool for three wards or two flex pool of two wards. One flex pool of four wards is not an option because ward 1 and ward 2 can not share the same flex pool Nurse staffing model by Burger et al. The number of dedicated nurses and flexible nurses needed per shift depends on the demand for beds by patients and the flex pool ward assignment. Dedicated nurses are always assigned to the 37

48 4.3. SOFTWARE IMPLEMENTATION CHAPTER 4. SOLUTION APPROACH same ward and flexible nurses are assigned to the ward that has the highest shortage of nurses at the start of a shift. In Section 4.1 we defined (4.17) to calculate the optimal number of nurses needed given two service levels. We can use the model of Burger [9] et al. to determine the optimal number of dedicated and flexible nurses per shift. In this section we briefly summarise the model of Burger. For more detailed information regarding the model of Burger, we refer to the thesis of Burger [9] Burger assumes that shifts are non-overlapping and nurses only work full shifts. Furthermore, a nurse can only be assigned to one ward during a shift. The model of Burger consists of the determination of three solution bounds and uses two target service levels and a minimum fraction of dedicated nurses per ward. The overall service level α states the fraction of the time that the number of patients on a ward does not exceed the number of nurses staffed times the patient nurse ratio, rs,τ w. The minimum service level β states the minimum fraction of patients per time slot that are covered by the staffed number of nurses times the patient-nurse ratio. The first bound solution (Non Flexible) is calculated while considering no flexibility. This means that the first bound gives the optimal number of dedicated nurses needed per shift and ignores the use of flex pools. The second bound solution (Lower Bound flexible) considers the use of flexible nurses but relaxes the constraint that these nurses need to be assigned to one ward for the whole shift. The third bound solution (Upper Bound flexible) is computed based on the maximum number of patients that are possibly present on a ward during a shift. The maximum number of patients present during a shift cannot be derived from the bed census so these are determined in an alternative way given the MSS and the patient arrival and discharge distributions [9]. Burger uses decision rules to select the optimal number of dedicated and flexible nurses given the solutions of the three bounds. Figure 4.7 summarises the steps mentioned above. Figure 4.7: Overview of the different steps to obtain the number of dedicated nurses per ward and the number of flexible nurses inside a flex pool per shift per day 4.3 Software implementation The mathematical relationships between the various variables and the objective function from Section 4.1 are programmed in Delphi. For the functions (4.2.3), (4.2.4), and, (4.2.6) we use the models of Kortbeek, Smeenk, and Burger to evaluate the access time, determine the number of beds and the number of nurses required. These models were already programmed in Delphi so we had to link them together and make minor adjustments. We integrated these into one large evaluation and optimisation tool. The class diagram of our Delphi program is shown in Appendix D. We formulated the ILP in a program called OmpShell and solved it with CPLEX. OmpShell is a program developed by Erwin Hans which makes it easy to input an ILP in the mathematical notation. The output of the ILP can easily be imported into our evaluation/optimisation tool and then be processed further to evaluate the effect on the resource usage in the inpatient care chain. The programmed models of Smeenk and Burger have some limitations due to computational complexity which limits our solution space. We briefly summarise these limitations: 38

49 4.4. VERIFICATION & VALIDATION CHAPTER 4. SOLUTION APPROACH The maximum number of wards that can exchange patients is two. This means that a patient can only be misplaced to one other ward. It is not easy to extend this to more than two wards. The misplacement and rejection rates that the model of Smeenk computes do not provide accurate values. To resolve this, we neglect the possibility of misplacements. To determine the optimal number of beds per ward we take the 90% demand percentile of the bed census. Within the AMC the model of Smeenk is still improved and an update can easily be integrated into our optimisation tool. The maximum number of wards that can share a flex pool is limited to two. It is not easy to extend this to more than two wards. 4.4 Verification & validation In Section we discuss the verification of our model. validation. Section describes the process of Verification Verification is concerned with determining whether our model has been correctly translated into an computer program [20]. To verify whether our model does not contain any bugs or errors we checked the output of the ILP solution to verify whether no constraints are violated. To test if we integrated the models of Kortbeek, Smeenk and Burger correctly, we created single data files that contain the same input as in our integral model. The single output of the models show exact the same results as used in our integrated tool. This indicates that we combined the models correctly Validation Validation is the process of determining whether a model is an accurate representation of the real world, for the particular objectives of this study [20]. Validation of our proposed method is difficult due to two reasons. First, the current situation is highly diverse and not standardised. Each department in our case study has different ways to plan and schedule patients which is impossible to simulate with our model. This lack of standardisation makes it difficult to incorporate this into our model. Second, our modelling approach aims at optimisation which is completely different from the current practice. Therefore it is not possible to use our model to obtain the same results as in the current practice. To validate our model we propose a theoretical simulation study in which we use the outcomes of our solution approach and compare this with real data. 4.5 Conclusions In this chapter we formulated our research objective as a mathematical optimisation problem. To solve this optimisation problem we propose a decomposition approach optimise the resource usage in the inpatient care chain while considering various decisions. To reduce modelling complexity we had to make a few simplifications compared to the current situation. We propose an ILP to make the patient-ward assignment and to create a master surgical schedule. We used various existing models from literature to determine and optimise the downstream resources requirements in terms of beds and nurses required per nursing ward. Compared to the various literature related to resource capacity planning in the OR department and the nursing wards this is one of the first researches that focuses on decisions and optimisation that affects the whole chain. 39

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51 Chapter 5 Computational results In this chapter we demonstrate the performance of the approach developed in Chapter 4 by conducting several experiments. Section 5.1 describes our data gathering approach to obtain the values of relevant distributions and parameters. In Section 5.2 we clarify which parts of our decomposition approach we exclude when we conduct our experiments. Section 5.3 discusses our experimental set-up. Section 5.4 describes our experimental approach to model the current situation and the various experiments we perform. Section 5.5 shows the results of the experiments we conduct. We end this chapter with conclusions in Section Data gathering We obtain data from the various systems present in the AMC to test our solution approach. The surgery, admission and discharge distributions are gathered from two separate systems: Locati and OK-Plus. Locati contains all the admission and discharge data of the patients that reside on a ward. OK-plus contains all data from the surgeries. There is no direct link between an admission and a surgery so we have to perform several steps to create the relevant data sets. Appendix B discusses these steps in more detail and provides the relevant figures for our case study. Due to the poor quality of the data, we had difficulties matching admitted patients to an accompanying scheduled surgery block, since we consider patients who do not have had an surgery as acute. This had as result, that 51% of the total patients considered in this case study are of the acute patient type, which is remarkable because we are considering the surgical department in which most admissions are caused by surgical procedures. Relevant data regarding the distribution of the number of surgery requests for each patient type was not possible to obtain within the available time frame of this research. We had various conversations with several departments in the AMC to obtain the relevant cost parameters present in our objective function (Section 4.1.3). More detailed information regarding these conversations and a derivation of the required cost parameters is provided in Appendix C. 5.2 Demarcation experiments In the experiments we conduct in this chapter we limit ourselves compared to the full potential of our decomposition approach, which is described in Section 4.2. We exclude: Access time service level: As stated in Section 5.1 we do not have available data regarding the distribution of surgery requests per patient type. 41

52 5.3. EXPERIMENTAL SET-UP CHAPTER 5. COMPUTATIONAL RESULTS Rejection and misplacement rates: We do not consider evaluation of the rejection and misplacement rates as stated in Section 4.3. The realised model in Delphi of Smeenk does not provide accurate values for these rates. Costs and revenues per patient type: We do not have available data concerning the costs and revenues per patient type, which we need to evaluate the effect of a chosen patient case mix on the total costs. Therefore we focus on minimisation of the total resource costs for a given patient case mix. Minimisation of expected number of nurses in the ILP: We programmed the constraint (4.39) that corresponds to the expected number of nurses required per shift per day in the program Ompshell, but due to the large number of variables (over 3000) the program would not run. 5.3 Experimental set-up Figure 5.1 shows the two computer programs we developed to perform our experiments. We solve our ILP, described in Section for the creation of a block schedule and the patient-ward assignment, with the program OmpShell that uses CPLEX. The integrated tool consists of the models developed by Kortbeek (Section 4.2.3), Smeenk (Section 4.2.4), and Burger (Section 4.2.6) in Delphi to determine the number of beds and nursing staff needed, and to derive several performance indicators. Figure 5.1: Overview of the two models we use We perform our experiments on a computer with Windows XP professional, with an Intel Core 2 Duo 3.00 GHz processor and 4 GB of available RAM. We use CPLEX version 12.1 and Delphi version XE. For all experiments we conduct we use the default parameters stated in Table 5.1. We explicitly mention it when we deviate from these parameters. Table 5.2 contains the managerial output we obtain for each experiment. We measure the total number of nurses in our integrated Delphi model in Full Time Employee (FTE). Parameter Value Parameter Value S Length of MSS 28 ɛ Bed capacity scale factor 0.7 T Number of time slots 24 κ nonrenewable OR Weight of an OR 500,000 T Number of shift types 3 κ beds Weight of a bed 9,000 χ Target rejection rate - κ wards Weight of a ward 1,250,000 η Target misplacement rate - F F T Ed Cost of a dedicated nurse 50,000 α Target overall service level 0.95 F F T Ef Cost of a nurse in flex pool 50,000 β Target minimum service level 0.80 Max runtime ILP 30 min γ Minimum fraction of dedicated nurses per ward 0.66 Table 5.1: Default parameter values used in our experiments 42

53 5.4. EXPERIMENTAL DESIGN CHAPTER 5. COMPUTATIONAL RESULTS ILP solution method General Number of ORs to open Number of wards to open Runtime Integrality gap Integrated Delphi model General Expected number of patients treated per year OR utilisation Total number of beds Total number of nurses Total number of flex pools used Total number of dedicated nurses Total number of nurses per flex pool Runtime model Smeenk Runtime model Burger Ward specific Expected number of beds Patient-ward assignment Ward specific Min number of beds based on 90%th demand percentile Max number of beds based on 90%th demand percentile Average bed utilisation Total FTE dedicated nurses per year Flex pool-ward assignment Realised average overall and minimum service levels per year Table 5.2: Managerial output of our experiments 5.4 Experimental design We have structured our experimental design as follows. In Section we determine the managerial output described in Table 5.2 given the resource capacity planning decisions made in the current situation. In Section we systematically show the effects of the resource usage in the inpatient care chain for the various optimisation steps of our decomposition approach. In Section we consider two types of scenarios that affect the LOS distribution and the number of acute patient types on which we have limited influence. We continue in Section with a sensitivity analysis of the weighing factors used in the ILP and of the various parameters used to determine the required number of nurses. Section concludes with alternative interventions, which consist of adapting OR planning decisions and a different admission and discharge process Division B: Current situation In our experiments we focus on the current situation of Division B (see Chapter 2 for detailed information). In 2012 the board of division B has decided to change the allocation of patient types to wards, compared to the situation of 2011 described in Table 2.4. To take the actual situation of 2012 into account we assume the ward lay-out and patient-to-nurse ratios as described in Table 5.3. Patient-to-nurse ratio Ward Patient type(s) Beds Nurses (FTE) Morning shift Evening shift Night shift G5 Noord URO, VAA G6 Noord CHI, CHIss G6 Zuid CHI, CHIss, MZK G7 Noord TRA G7 Zuid ORT, CHP Table 5.3: Overview of patient types assigned to the various wards in Division B 2012 and the patient-to-nurse ratio for each ward (Source: [8]). 43

54 5.4. EXPERIMENTAL DESIGN CHAPTER 5. COMPUTATIONAL RESULTS We need to make a few adjustments to be able to evaluate the situation of 2012 with our model. The AMC does not use a cyclic master surgical schedule which is needed for our model. To be able to compare our solution approach to the results of the current situation we select the block schedule of the first four weeks of September 2010 and assume this is cyclic for the rest of the year. We choose the month September because there are no holidays in these weeks and these are the busiest of the year. We consider a surgical specialty as one unique patient type. Our modelling approach assumes that each patient type has its own surgery block. In the current situation surgical short stay patients (CHIss) undergo surgery in the same surgery blocks as general surgery patients (CHI). To model this, we assume that surgical short stay patients and general surgery patients are the same patient type, which have the same admission, discharge, and LOS distribution. Another assumption in our model is that each patient type needs to be assigned to one ward. In the current situation the patient types surgery (CHI) and surgical short stay (CHIss) are assigned to two wards. To model this, we create two surgical patient types of which one patient type is assigned to ward G6 Noord and the other to ward G6 Zuid. We adapt the block scheduling in such a way that all even blocks in the cycle of the general surgery type scheduled are assigned to ward G6 Noord and all uneven blocks to ward G6 Zuid Optimisation interventions In this section we describe experiments to systematically obtain the effects of the various interventions of our decomposition approach compared to the experiment of the current situation. In Section we create a cyclic MSS which levels the expected number of beds. Section contains experiments regarding the patient-ward assignment. We continue in Section with an experiment to determine the optimal flex pool configuration. Section concludes with a combination of all optimisation steps. For each experiment described in this section we use the default parameters from Table 5.1. For the planning decisions we use the settings stated in Table 5.4. When we deviate from these default settings we will mention this explicitly. Planning decisions Default settings Block scheduling decision 4-Weekly MSS based on September 2010 Patient-ward assignment We use the assignment from Table 5.3 Misplacement policy Flex pool assignment No misplacements are allowed No flex pools are used Table 5.4: Default planning decisions used for optimisation experiments Master Surgical Schedule In this experiment we use our developed ILP method to create an MSS with a cycle length of four weeks. We set the maximum number of blocks to be scheduled equal to the number of blocks scheduled in the first four weeks of September We set the maximum number of beds possible per ward equal to 50 to make sure that our optimal solution is not limited by insufficient numbers of beds. The resulting MSS is used as an input for our integral tool to evaluate the other resource needs. Along with this experiment we also conduct an experiment to determine the run time of our ILP. Our ILP contains many variables that makes it difficult to obtain an optimal solution quickly. To speed up the process we choose to manually stop CPLEX after a fixed time period. To determine the appropriate running time value versus integrality gap we experiment with run times of 30 minutes, 1 hour, 2 hours, 4 hours and 8 hours. We use the outcome of the running time experiment in all other experiments. 44

55 5.4. EXPERIMENTAL DESIGN CHAPTER 5. COMPUTATIONAL RESULTS Patient-ward assignment In this experiment we use our ILP to make the patient-ward assignment and the block scheduling decision. We perform four experiments. The first experiment consists of the option to open at most five wards with a maximum number of 32 beds per ward. Next, we consider the possibility to open at most three wards with a maximum size of 50 beds. For both experiments we use 0.7 as value for the bed capacity scale factor ɛ to match the number of expected beds needed to the actual number of beds. We derived this value based on initial experiments. Table 5.5 summarises the adaptations we make. In the third experiment we consider the decisions in Table 5.4 but assume that all patients are assigned to the same ward. We use the MSS based on four weeks of the MSS. In the last experiment we use the same settings as in the third experiment but we use our ILP to create an MSS. Planning decisions Block scheduling decision Patient-ward assignment Default settings Based on the outcomes of the ILP Based on the outcomes of the ILP Table 5.5: Adaptations to the planning decisions for experiment patient-ward assignment Flex pool To determine the influence of sharing a flex pool of nurses between couples of two wards we use the flex pool optimisation method described in Section We evaluate all combinations of wards that could share a flex pool and select the combination that requires the least amount of nurses. Table 5.6 summarises the experimental settings. Planning decisions Default settings Block scheduling decision 4-Weekly MSS based on September 2010 Patient-ward assignment We use the assignment from Table 5.3 Misplacement policy Flex pool assignment No misplacements are allowed At most two wards can share a flex pool Table 5.6: Default planning decisions used for flex pools experiments All optimisation steps combined In this intervention we use combinations of the interventions described earlier. We use our ILP to make the block scheduling decision and the patient-ward assignment. We assume that we can open at most five wards with a maximum of 32 beds per ward and we determine the optimal combination of wards that share a flex pool. Table 5.7 summarises our experimental set up. Planning decisions Block scheduling decision Patient-ward assignment Misplacement policy Flex pool assignment Default settings Based on outcomes of the ILP Based on outcomes of the ILP No misplacements are allowed At most two wards can share a flex pool Table 5.7: Optimisation decisions for the inpatient care chain 45

56 5.4. EXPERIMENTAL DESIGN CHAPTER 5. COMPUTATIONAL RESULTS Scenarios In this section we consider two types of scenarios that influence the resource usage in the inpatient care chain. We use the term scenarios because we consider situations that we can not directly control. Section discusses changes in the length of stay (LOS) distribution of patients. Section concludes with adaptations in the total number of acute patients. For each scenario discussed in this section we perform three experiments. The first experiments consist of our model of the current situation (decisions described in Table 5.4). Next, we consider an experiment that consist of five wards with at most 32 beds per ward (decisions described in Table 5.8). And the latter experiment consist of five wards with at most 32 beds per ward and the use of flex pools (decisions described in Table 5.7). For all three situations we use the default parameter settings from Table 5.1. Planning decisions Block scheduling decision Patient-ward assignment Misplacement policy Flex pool assignment Default settings Based on outcomes of the ILP Based on outcomes of the ILP No misplacements are allowed No flex pools are used Table 5.8: Experimental settings for the scenario 5 wards with at most 32 beds Reduction or increase in Length Of Stay In this experiment we determine the influence of the LOS on the resource usage in the clinical wards. We consider a reduction in LOS of one day and an increase of one day for all patient types. To obtain the new LOS distributions, we shift the probability that a patient is discharged on day n one day up or down. For example, when we increase the length of stay, the probability that a patient stays two days after surgery equals the probability that a patient stays one day in the current situation Reduction or increase in number of acute patients In this scenario we investigate how the number of acute patients influences the resource usage in the inpatient care chain. We consider an increase and a decrease of 25% of the total number of acute patients. For a 25% increase we multiply all arrival rates λ j,d,t by 1.25, whereas we multiply the arrival rates by 0.75 for a 25% decrease Sensitivity analysis In this section we describe experiments in which we conduct sensitivity analysis on the weight factors in the ILP, the target service levels, and the patient-to-nurse ratios. Section contains two alternative weight factor configurations that we use in our ILP. Section discusses variations in the overall service level, the minimum service level and the fraction of dedicated nurses per ward. Section concludes with alternative patient-to-nurse ratios based on values from literature ILP weighing factors In our ILP solution method we use the cost parameters described in Appendix C as weighing factors. We determine the influence of these weighing factors on the block scheduling decision and 46

57 5.4. EXPERIMENTAL DESIGN CHAPTER 5. COMPUTATIONAL RESULTS the patient-ward assignment by performing two experiments. In the first experiment we assume that the fixed costs of an OR and a ward are equal. In the second experiment we assume that the costs of a bed are equal to the costs of a dedicated nurse. We use the default parameters described in Table 5.1 with the adaptations mentioned in Table 5.9. For both configurations we solve the ILP to create a block schedule and assigning patient types to wards. We consider the option to open five wards with at most 32 beds per ward. Weighing factors Experiment κ nonrenewable OR κ beds κ wards Exp ,000 9, ,000 Exp ,000 50,000 1,250,000 Table 5.9: Overview of two experiments with alternative weighing factors Service level adaptations In this section we change the target service level requirements necessary to determine the optimal number of nurses. We consider three experiments with the values for the target service levels shown in Table For the other parameters we use the settings from Table 5.1. We evaluate each experiment for our model of the current situation (planning decisions in Table 5.4) and the situation in which we determine the optimal flex pool combination (planning decisions in Table 5.6). α Overall service level β Minimum service level γ Minimum fraction ded. nurses Exp Exp Exp Table 5.10: Overview of the variations in service level requirements Alternative patient-to-nurse ratios Within the state of California minimum patient-to-nurse ratios are stated by the health authorities [4] for various patient types. Table 5.11 shows these patient-to-nurse ratios for surgical patients. In this experiment we change the patient-to-nurse ratios into one uniform ratio that is valid for all surgical patient types, this differs compared to the current situation. We evaluate the effects of adapting these values on the total number of nurses needed in the current situation (settings in Table 5.4) and the current situation in which we determine the optimal flex pool combination. For the other parameters we use the default values given in Table 5.1. Morning shift Afternoon shift Night shift Table 5.11: Overview of the patient-to-nurse ratios based on values used in California [4] Additional interventions This section describes additional interventions regarding control decisions of the OR department and the nursing wards. Sections to contain additional interventions that influence 47

58 5.4. EXPERIMENTAL DESIGN CHAPTER 5. COMPUTATIONAL RESULTS the MSS and Section describes an alternative admission and discharge process for elective patients. For all interventions we assume the default settings described in Table 5.1 and Table When we deviate from these settings we will mention this explicitly. Planning decisions Block scheduling decision Default settings Based on outcomes of the ILP Patient-ward assignment We use the assignment from Table 5.3 Misplacement policy Flex pool assignment No misplacements are allowed No flex pools are used Table 5.12: Default planning decisions used for optimisation experiments Length of the MSS In this experiment we vary the length of the MSS, S. We consider MSSes of one week (S = 7) and two weeks (S = 14), and compare these with the results of an MSS with a length of four weeks (Section ). To obtain the number of blocks that need to be scheduled on a weekly or two weekly basis we divide the total number of blocks scheduled in the first four weeks of September by two and four. If the resulting number of blocks is fractional we round it up to the nearest integer Steady cyclic demand The number of surgery blocks scheduled in September, where we base our experiment on, is higher than the average amount through the year. In this experiment we assume that the number of blocks that need to be scheduled in a four week cycle equals the yearly average. That is, we take the total number of blocks scheduled in 2010 and divide this by thirteen. We set the number of blocks to be scheduled equal to this result. If the resulting number of blocks is fractional we round it up to the nearest integer Increase in number of surgery blocks In this experiment we increment the total number of blocks scheduled in September 2010 by 20% and 40%. If the resulting number of blocks is fractional we round it to the nearest integer. The outcomes of these experiments show how an increase in demand affects the resource usage along the inpatient care chain More patients of one type inside a surgery block From our context analysis described in Chapter 2 we conclude that in the AMC, the scheduling of patients for surgery is non standardised and that there is room for improvement. In this scenario we assume that we can perform one extra surgery in each surgery block. We shift the surgery distributions in such way that the probability of two surgeries performed in the new situation is equal to the probability of one surgery performed in the old situation. We do this for each number of surgeries. In this intervention we determine how the resource usage is influenced when the surgical scheduling of patients is improved. 48

59 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS Short surgeries and long surgeries In all our experiments we assume one type of surgery block per patient type. In this experiment we create a block schedule based on two types of surgery blocks per patient type. We differentiate between blocks with long surgeries and blocks with short surgeries. Surgery blocks with long surgeries contain one or two patients and surgery blocks with short surgeries contain three to six patients. We would expect that more detailed surgery information gives us the possibility to improve the scheduling of these surgery blocks and thereby reduce the resource needs in the rest of the inpatient care chain. We alter the surgery distributions of both surgery block types in such a way that overall, the expected number of patients and the probability of x surgeries performed stays the same. In case of fractional numbers of blocks we round the number of blocks with long surgeries down and the number of blocks with short surgeries up to the nearest integers Weekend surgery In this experiment we relax the constraint that prevents weekend surgery. The outcomes of this experiment show how additional weekend work influences the resource usage along the inpatient care chain Admission and discharge policies Our data analysis shows that the current admission and discharge processes are intertwined. Admissions occur in the morning while discharges occur in the afternoon, which results in peak bed utilisation during "lunch". In this experiment we alter the admission and discharge distributions of elective patient types in such way that discharges occur in the morning and admissions in the afternoon. We do not alter the admission and discharge distributions of acute patient types because these can not be influenced. For this experiment we assume the settings from Table Results In this section we present the results of the experiments described in Section 5.4. Section provides the results for the current situation and the various optimisation steps. Section contains the results of the two scenarios and Section provide the results of the experiments regarding the sensitivity analysis. The outcomes of the additional interventions are shown in Section Section concludes with an graphical overview of the improvement potential of the various interventions compared to our model of the current situation. We provide a more detailed overview of the results in Appendix E Current situation and optimisation Table 5.14 shows the results for the current situation (experiment described in Section 5.4.1). In our model representation of the current situation we need 152 beds and FTE dedicated nurses to treat all patients. When we compare these values to the actual values of 138 beds and FTE dedicated nurses (Table 5.3), we notice that our values are substantially higher. This can be explained by three reasons. First, the total number of patients treated in reality is lower than in our modelling of the current situation because fewer patients are scheduled over the whole time horizon. Second, we had to make some modifications to make the current situation suitable for our model (Section 5.4.1). Third, there are some inconsistencies in the data. The results that we present in this section are based on a model of the current situation. Because 49

60 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS Runtime in seconds Model Smeenk Model Burger no flex Model Burger with flex Current situation n/a MSS 5 Wards n/a Patient one ward n/a Flexpool optimisation Table 5.13: Overview of the run times of the model of Smeenk and Burger for various experiments. we do not model the current situation exactly, the results of the various optimisation steps cannot be interpreted as results that are directly applicable to the current situation. The main objective of these results is to show how the resource usage can be improved when various optimisation steps are considered. Table 5.13 shows the run time for the various optimisation steps of our developed Delphi Model. In Appendix E additional performance information regarding the run time and the integrality gap of the ILP solution can be found. Table 5.14, Figure 5.2, and Figure 5.3 show the results for the various optimisation steps. If we take a close look at the experiment ( ) in which we create an MSS while considering the patientward assignment fixed we see that we have an reduction of one OR, six beds, and 2.1 FTE dedicated nurses compared to our model representation of the current situation. Because the number of ORs used in the optimal scenario is one less than in the current situation, we conducted an additional test in which we added a constraint to our ILP that forces the usage of eight ORs. In this scenario we have more flexibility when assigning surgery blocks to days, which should lower the required number of beds. Table 5.14 shows that in this case we need two beds less, and 0.2 FTE dedicated nurses less compared to the experiment optimal ORs ( ). Figure 5.2 displays the 90% demand for beds at time slot 0 for the current situation and Figure 5.3 shows the 90% demand for beds at time slot 0 resulting from our ILP solution ( MSS 8 ORs). We choose to display time slot 0 because at this moment in time no admissions and discharges take place. A limitation of this choice is that the maximum number of beds in both figures can not be interpreted as the maximum number of beds needed per ward. When we consider the situation in which we alter the patient-ward assignment and optimise the MSS ( ) we see interesting results (experiments described in Section When we consider to open five wards with at most 32 beds, the resource requirements are almost equal to the experiment in which we only consider optimisation of an MSS with fixed patient-ward assignment. When we consider to open three wards with at most 50 beds we see that the number of beds is reduced by one and the number of dedicated nurses by 13.6 FTE compared to the situation with five wards and 32 beds. This small reduction in the number of beds can be explained by the fact that the reduction Nurses (FTE) Experiments Expected patients ORs Wards Beds Total Dedicated Flex pool Current situation MSS Optimal ORs MSS 8 ORs MSS 5 Wards MSS 3 Wards All Patients 1 ward MSS Patients 1 ward Flex pool All combined Table 5.14: Overview of the resource usage for the current situation and the various optimisation experiments 50

61 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS in variance in demand for beds for this patient-ward combination is limited. The high reduction in FTE dedicated nurses can be explained by the fact that nurses can be better aligned with the number of patients when the patient-to-nurse ratio is high. For example, when the patient-to-nurse ratio is eight the optimal number of beds per ward is a multiple of eight. When we consider more patients per ward we have more efficient usage of dedicated nurses. Table 5.15 displays the resulting patient-ward assignment for both experiments. When we compare the results for five wards to the current situation we see that the specialties MZK, ORT, CHP and TRA are changed from location. When we allocate all patient types to the same ward, without optimisation of the MSS, we see that we require 130 beds and FTE dedicated nurses. This large reduction in the number of beds can be solely accounted for by reduction of variance in demand for beds, while the number of nurses is partly reduced by better alignment of beds with the patient-to-nurse ratios, and partly by the reduction in variance in demand. When we compare this result to the experiment in which we open three wards, we do not see this reduction in the latter. This can be explained by the fact that the total number of patient types assigned to a ward is too small, or not optimally chosen, to take benefits of variance reduction in demand for beds. Experiment Ward 0 Ward 1 Ward 2 Ward 3 Ward 4 Current URO,VAA CHI,CHIss CHI,CHIss, MZK TRA ORT, CHP MSS 5 Wards 32 URO, VAA MZK, ORT CHI, CHIss CHP, TRA CHI, CHIss MSS 3 Wards 50 CHP, ORT, VAA MZK, TRA, CHI, CHIss CHI, CHIss Table 5.15: Overview of the resulting patient-ward assignment from the ILP solution for two optimisation experiments ( ) The last optimisation stand-alone step considers the use of flex pools (experiment described in Section ). When we determine the optimal flex pool-ward combination for the current situation we can reduce the number of FTEs needed by Table 5.16 shows the optimal flex pool-ward combination for the current situation with flex pools. From the results in Appendix E.5 we see that there are large differences in reduction of FTE when we consider all flex pool-ward combinations. The reduction in nurses ranges from 0.1 to 1.8 FTE. When we consider all various optimisation steps combined we need one OR, six beds, and 4.4 FTE dedicated nurses less than in the current situation (experiment described in Section ). Figure 5.2: 90% demand for beds for ward G7 Zuid at time slot 0 for the current situation Figure 5.3: 90% demand for beds for ward G7 Zuid at time slot 0 resulting from our ILP 51

62 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS Ward Flexpool Nurses (FTE) in flex pool Dedicated nurses (FTE) Realised α Realised β G6 Zuid G7 Zuid G7 Noord G5 Noord G6 Noord Table 5.16: Best flex pool combination for each ward in the current situation with flex pools and their realised service levels ( ) Scenarios Table 5.17 presents the outcomes of the experiments regarding the two scenarios in which we alter the input distribution. We first discuss the results of the scenario considering the LOS of patients. Next, we discuss the result of alterations in the number of acute patients. Nurses (FTE) Experiments Expected patients ORs Wards Beds Total Dedicated Flex pool Current situation MSS 5 Wards All combined LOS+1 current situation LOS+1 MSS 5 Wards LOS+1 All combined LOS-1 current situation LOS-1 MSS 5 Wards LOS-1 All combined Acute-25% current situation Acute-25% MSS 5 Wards Acute-25% All combined Acute+25% current situation Acute+25% MSS 5 Wards Acute+25% All combined Table 5.17: Overview of the results for the two scenarios on the current situation and the optimisation situation When we increase the LOS of patients the resource needs in the inpatient care chain increase. When we compare scenario LOS+1 MSS 5 Wards to scenario LOS+1 current situation we need five beds and 4.4 FTE dedicated nurses less. If we compare these reductions to the same experiments in which the LOS is normal, we see that the reduction of beds is reduced by one and the total number of FTE dedicated nurses is increased by 2.1. When we look at the differences between the same two scenarios for a decrease in LOS we see that we need six beds and 0.4 FTE dedicated nurses less. If we again compare these reductions to the experiments in which the LOS is normal, we see that the reduction in beds is the same but the reduction in nurses is almost disappeared.this can be explained by the fact that When we compare the benefits of having a flex pool (All combined) to the optimisation scenario without a flex pool (MSS 5 Wards 32) we see that the benefits of having a flex pool of nurses is approximately 2 FTE for both an increase and a decrease in LOS. Table 5.17 shows that we need 17 beds and 25.3 FTE dedicated nurses less when we reduce the number of acute patients by 25% and compare this "new" current situation to the actual current situation. When we look at the differences between scenario MSS 5 Wards 32 and scenario current situation for both a reduction in acute patients and an increase in acute patients compared to the differences in the normal situation we see that our proposed solution approach for both provides higher result. If we look at the benefits of having a flex pool we notice a difference of 2.4 FTE 52

63 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS dedicated nurses for a decrease of acute patients and 2.7 FTE dedicated nurses for an increase in acute patients Sensitivity analysis In Table 5.18 we present the results of the experiments in which we vary several parameters. We first discuss the outcomes of adaptations of the weighing factors. Next, we explain the outcomes related to changing the service levels. We end this section with a discussion of the results regarding an alternative patient-to-nurse ratio. Nurses (FTE) Experiments Expected patients ORs Wards Beds Total Dedicated Flex pool MSS 5 Wards ILP Exp ILP Exp Current situation Flex pool optimisation Serv lvls (0.80,0.80,0.66) no flex Serv lvls (0.80,0.80,0.66) flex Serv lvls (0.80,0.70,0.66) no flex Serv lvls (0.80,0.70,0.66) flex Serv lvls (0.80,0.70,0.50) no flex Serv lvls (0.80,0.70,0.50) flex Patient-nurse ratio no flex Patient-nurse ratio flex Table 5.18: Overview of the results for the sensitivity experiments When we adapt the weighing factors of the ILP (experiments described in Section ) we see no difference in number of beds needed between the three experiments. However, a difference in total number of dedicated nurses can be noticed. The first experiment ( ILP Exp 1) requires 1.1 FTE dedicated nurses more and the second experiment ( ILP Exp 2) requires 1.3 FTE of dedicated nurses less compared to our initial ILP solution ( MSS 5 Wards 32). Table 5.19 shows the resulting patient-ward assignment for both experiments. When we compare this assignment to the initial experiment from Section we see some interesting results. In all experiments patient types URO and VAA are allocated to the same ward, and the same holds for patient types CHI and CHIss. The patient types MZK, ORT, TRA and CHP are assigned in a different manner in each experiment. The patient-ward assignment differs between all scenarios because due to these changing weights CPLEX chooses an alternative solution. We can explain the difference in total number of nurses for both experiments by differences in alignment of the demand for beds to the patient-to-nurse ratio. Experiment Ward 0 Ward 1 Ward 2 Ward 3 Ward MSS 5 Wards 32 URO, VAA MZK, ORT CHI, CHIss CHP, TRA CHI, CHIss Exp 1 CHI, CHIss TRA CHI, CHIss MZK, URO, VAA CHP, ORT Exp 2 CHI, CHIss ORT URO, VAA CHI, CHIss CHP, MZK, TRA Table 5.19: Overview of the resulting patient-ward assignment from the ILP solution for experiments with different weight factors When we alter the nurse service level requirements (experiments described in Section ) we see that lowering the overall service level and minimum service level both have a positive effect on the total number of nurses needed. When we set the overall service level equal to the minimum service level the benefits of having a flex pool are reduced to zero FTE reduction ( Serv lvls (0.80,0.80,0.66)). This is to be expected because both service levels are equal and the method 53

64 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS proposed by Burger uses the difference between both service levels to determine the optimal flex pool configuration. In the situation that we have an overall service level of 0.80, a minimum service level of 0.70, and we require the minimum fraction of dedicated nurses per ward to be 0.66 we have an reduction of 0.9 in FTE compared to the same situation without flex pools. When we consider the situation in which the minimum fraction of dedicated nurses per ward is lowered to 0.50 we have less nurses in the flex pool yielding an additional reduction of 2.4 FTE. In the experiment in which we adapt the patient-to-nurse ratios based on the values given by California legislation (experiments described in Section ) we need FTE dedicated nurses compared to FTE dedicated nurses in the current situation Additional interventions Table 5.19 and Figure 5.5 display the results of the various additional interventions performed. We structured this section as follows. First, we discuss the results of the experiments in which we change the length of the MSS. Next, we present results of the interventions in which we increase the number of blocks that need to be scheduled. We continue with experiments in which we allow weekend surgery, and more detailed information of surgery blocks to obtain an MSS with a higher reduction of the resource requirements. We conclude this section with results of changing the admission and discharge processes. Experiment Expected patients ORs Wards Beds Nurses (FTE) Current situation MSS length 1 week MSS length 2 weeks Steady cyclic demand % more surgery blocks % more surgery blocks extra surgery per block increase demand extra surgery per block normal demand Short/long surgery blocks Weekend surgery Admission/discharge Table 5.20: Overview of the resource needs for the various intervention steps When we change the length of the MSS to one week or two weeks, we see that we two need two beds less compared to an MSS of four weeks ( MSS Optimal ORs, Table 5.14). Our ILP solves the block scheduling assignment to optimality, for one week within one second. We would expect that an MSS of four weeks requires at least the same number of beds or less because we have more flexibility in assigning blocks. We have two possible explanations for this difference. First, our MSS of one week is solved to optimally and our MSS of two weeks has a smaller integrality gap (Appendix E) than an MSS of four weeks. Second, because we level the expected number of beds it could occur that due to stochastic nature of the actual bed occupation the total number of beds differs. The number of patients treated is higher than considering an MSS of four weeks due to the rounding of the number of blocks that need to be scheduled. The number of dedicated nurses that we need is less than in our optimal situation with an MSS of four weeks. This means that we can treat more patients and require less beds and nurses. The difference in number of nurses is caused by the fact that these MSSes (one and two weeks) better align the demand for beds with the patient-to-nurse ratios. When we base the total demand on the yearly average we require 16 beds, and FTE dedicated nurses less compared to our model of the current situation. When the number of surgery blocks is increased, the number of nurses needed per patient decreases ( , Table 5.20). Again, we can explain this by the fact that the the demand for care is better aligned with the patient-to-nurse ratio. When we compare the results of an increase of surgery blocks by 40% to an additional surgery performed per surgery block, we notice that the total 54

65 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS expected number of patients treated is equal, while the required number of resources is lower for the situation in which we have more surgery blocks. We can explain this as follows. When we have more surgery blocks to schedule with fewer surgeries we are more capable to level the required number of beds and thereby reduce the variance in demand for care. When we consider the output of the scenarios with long surgeries and short surgeries ( , Table 5.20) we would expect that the demand for beds and nurses is lower compared to the situation in which we consider only one generic surgery block per patient type. However, the results are comparable and not lower. This difference can be explained by the increased number of surgeries that are scheduled. Another reason may be that the differentiation between surgery blocks is not detailed enough. The scenario in which we consider weekend surgery ( , Table 5.20) reduced the number of ORs used by three and the number of beds by twelve compared to the current situation. The total number of dedicated nurses needed is only slightly affected. Figure 5.4 shows the 90% demand for beds when we allow surgery during weekends. We can see that our ILP solution improves the leveling of number of occupied beds compared to the 90% demand for beds in Figure 5.2 and Figure 5.3. When we adjust the admission and discharge process of elective patients ( , Table 5.20), we can conclude that we need three beds and 2.6 FTE dedicated nurses less than in the current situation. Figure 5.4: 90% demand for beds for ward G7 Zuid at time slot 0 for the scenario in which we allow weekend surgery Comparison of all interventions In this section we determine which interventions from Section and the additional interventions from Section have the most potential to improve the resource utilisation compared to our model representation of the current situation (Section 5.4.1). We determine the efficiency of an intervention by calculating the efficient frontier, in which we divide output by input. We choose to compare all outcomes of the interventions based on the resources nurses and beds because these are directly comparable to our model of the current situation. We do not choose to base the outcomes on total costs because the obtained cost figures are relatively rough estimates. We do not consider the total number of ORs required because this is in practice also influenced by surgery requests for other divisions. Figure 5.5 shows the efficient frontier for all interventions. The interventions in which we allocate all patients to the same ward are excluded because these interventions are of an exploratory nature. On the horizontal axis we show the number of expected patients per year (output) divided by the total number of beds needed (input). On the vertical axis we show the expected number of patients per year (output) divided by the total FTE of nurses needed (input). The dotted line is the efficient frontier, which denotes the interventions that are 100% efficient. We use the term 100% efficient because it is not possible to improve on one indicator without worsening the other. Figure 5.5 shows that our model of the current situation (located in the figure on the bottom left) scores worse compared to almost all interventions. The only intervention that scores lower on the number of nurses per patient is the scenario in which we consider a steady cyclic demand and less 55

66 5.5. RESULTS CHAPTER 5. COMPUTATIONAL RESULTS Figure 5.5: Overview of the efficient frontier for the various interventions and optimisation experiments patients are treated. Figure 5.5 shows that there are three most promising interventions. Most beneficial for efficient alignment of nurses to demand for care is to consider three wards with a maximum bed size of 50. Due to the larger ward sizes and better alignment with the patient-tonurse ratio less nurses can treat more patients. Most beneficial for reducing the number of beds is considering an MSS with a length of 1 week. The reason that this intervention performs better than for example an MSS of two weeks is twofold. First, when we consider an MSS with a length of 1 week we have a lower complexity of scheduling of surgery blocks. Because we manually stop our ILP after 30 minutes it could occur that our 1 week MSS solution is solved to optimality but our 2 weeks MSS solution is not yet solved to optimality. Second, we had to round the total number of surgery blocks that need to be scheduled, which also explains a difference. When we increase the number of patients per surgery block while considering the same amount of surgery blocks, we notice that this intervention gives the best results in both directions. From this figure, we also conclude that all interventions in which the number of patient is increased are beneficial for the number of nurses in FTE and beds per patient required. Table 5.21 shows the improvement potential for the three most promising interventions compared to our model of the current situation. When we consider 3 wards with 50 beds we can treat 11.1% more patients with the same number of FTE nurses or decrease the total FTE of nurses by 11.1% while treating the same number of patients. We conducted an additional experiment that consists of a combination of the interventions MSS length 1 week and MSS 3 wards 50 beds. Table 5.21 shows that this scenario even performs better, yielding a reduction of 12.0% in expected patients per nurse in FTE and a reduction of 11.8% in expected patients per bed. 56

67 5.6. CONCLUSIONS CHAPTER 5. COMPUTATIONAL RESULTS Improvement potential % Experiment Patients/FTE nurse/year Patients/bed/year Patients/nurse/year Patients/bed/year Current situation MSS length 1 week % 8.0% MSS 3 Wards % 4.7% extra surgery per block a % 7.2% MSS length 1 week, 3 wards % 11.8% Table 5.21: Improvement potential of the best interventions compared to our model of the current situation 5.6 Conclusions In this chapter we conducted several experiments to show the performance of our solution approach and to determine profitable interventions to improve the resource usage in the inpatient care chain. Based on our approach we conclude that our ILP to create an MSS, as expected, minimises the maximum number of beds required per ward and thereby levels the workload. When we consider the ILP to make the patient-ward assignment decision we see that reducing the number of wards does not necessarily require a lower total amount of beds. However, a reduction of wards combined with an increase in total number of patients per ward is beneficial for the total number of nurses required, due to better alignment of demand care with the patient-to-nurse ratio. A shortcoming of our ILP solution approach is that, due to the large amount of variables, it is not possible with our set-up to take the nursing requirements into account when creating the MSS, and deciding upon the patient-ward assignment. Therefore, we only focus on minimisation of the maximum number of beds. When considering the use of flex pools, we conclude in most experiments a reduction of two FTE is achieved. Furthermore, we have noticed that the block scheduling decision and the flex pool-ward assignment influences the added value of a flex pool between two wards in terms of total FTE nurse reduction (experiment ). Based on the scenario analysis, we conclude that our solution approach outperforms our model of the current situation for changes in the LOS of patients or in the total number of acute patients. When the LOS is changed we see that the difference in resource needs between our solution approach and our model of the current situation is reduced when the LOS is decreased, and increased when the LOS of patients is increased. When the number of acute patients changes the difference in resource needs between our model of the current situation and our optimisation methods remains almost the same. After conducting a sensitivity analysis of the weighing factors we conclude that the resource requirements of our ILP hardly changes, but that these weights do influence the patient-to-ward assignment and the block scheduling decision. Reducing the service level requirements of nurses is beneficial for reducing the total number of nurses needed, but could endanger the quality of care. From the additional interventions conducted we conclude that a higher amount of surgery blocks in combination with more detailed information about these surgery blocks yields opportunities to increase the utilisation of beds and nurses. When considering an alternative discharge and admission procedure additional savings can be achieved in a rather easy manner. The reductions that we obtain in each intervention have the same level of quality of care as in our model representation of the current situation, due to the predefined target service levels. When we compare the improvement potential of our model of the current situation to the interventions conducted, we see that three wards with at most 50 beds reduces the total number of FTE nurses required by 11.1%. When we want to reduce the number of beds, an MSS with a length of one week is most benefical, yielding a reduction of 8.0% in beds. When we consider an intervention that consists of three wards with at most 50 beds and an MSS length of one week, we require 12.0% less nurses and 11.8% less beds or we can increase the number of patients by these percentages, while keeping the same number of FTE and beds as in our model of the current situation. 57

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69 Chapter 6 Conclusions & recommendations In this chapter we reflect on our approach to reach our research objective. Section 6.1 contains the conclusions we draw from this research. In Section 6.2 we discuss our proposed model and the applicability of the results. Section 6.3 states the recommendations we give to the AMC to improve utilisation of resources in the inpatient care chain. Section 6.4 concludes with suggestions for further research. 6.1 Conclusions In this research we provide insight in the complex relations between resource capacity planning decisions in the OR department and the clinical nursing wards to support decision making by health care managers and to make efficient use of resources. The objective of this research is: To develop a method which determines the best combination of patient case mix, OR capacity, care unit and nurse staffing decisions in such way that total cost margins are maximised while satisfying production agreements and resource, capacity, and quality constraints. To reach our research objective we conducted a literature study in Chapter 3 to find methods and models related to resource capacity planning decisions. In Chapter 4 we defined our research objective as a mathematical optimisation problem, and we proposed a decomposition approach to solve this problem. Our approach consists of the following six steps: 1. Set the desired patient case mix and the length of the MSS. 2. Solve an ILP to create a master surgical schedule and assign elective and acute patient types to wards, while minimising the number of ORs, wards, and the expected number of nurses and beds required. 3. Evaluate the access time service level of the created block schedule with the model of Kortbeek et al. [19]. 4. Determine the number of beds required per ward while satisfying target rejection and misplacement rates with the model of Smeenk et al. [29]. 5. Iteratively use the model of Burger et al. (Step 6) to determine the best flex pool-ward combination. 6. Determine the optimal number of dedicated nurses per ward and the total number of nurses in a flex pool given various target service levels with the model of Burger et al. [9]. 59

70 6.2. DISCUSSION CHAPTER 6. CONCLUSIONS & RECOMMENDATIONS The approach we present supports hospital managers with a tool to determine and optimise the resource requirements given a patient case mix and the length of the MSS. Furthermore our tool can be used to (re)design, evaluate and improve current hospital processes. In Chapter 5 we demonstrated the strength and diversity of our solution approach by conducting several experiments on data obtained from the surgical division in the AMC. Based on these results we draw the following conclusions: Our solution approach reduces high fluctuations in demand for beds and care. The MSS that we create reduces the maximum number of beds required. When we consider an MSS of one week we need 5.3% beds less and 2.1% FTE nurses less to treat 2.8% more patients compared to our model representation of the current situation. To improve the alignment of nurses to demand for care it is beneficial to reduce the number of wards and increase their size. When we consider three wards with at most 50 beds we can reduce the number of nurses in FTE by 11.1% compared to our model representation of the current situation. Having a flex pool of nurses can save an additional 1.7% in total FTE nurses required, but our experiments show that this reduction highly depends on the organisation of the MSS, the flex pool-ward assignment and the chosen service levels. Admissions occur in the morning while discharges occur in the afternoon. Adjusting these processes in such way that admissions occur in the afternoon and discharges take place in the morning leads to a reduction of 1.3% in beds and 1.7% in FTE of dedicated nurses. When we compare the improvement potential of the intervention in which we consider three wards with 50 beds and an MSS with a length of one week, we require 12.0% less nurses and 11.8% less beds, compared to our model reprensentation of the current situation, or we can increase the number of patients by these percentages while keeping the same number of FTE and beds as in our model of the current situation. 6.2 Discussion In this section we discuss our solution approach and its applicability. First, we reflect on our decomposition approach to answer our research objective. Next, we elaborate on the effect of the assumptions made compared to reality. We conclude this section with a brief discussion on the quality of data. Our research objective consists of nine resource capacity planning decisions and encompasses both the nursing wards and the OR department. Due to the problem size, the non-linear relations and interdependence between the planning decisions, the objective of this research was very ambitious. Our solution approach integrates all described planning decisions but does not provide one optimal output. We created a good initial solution for the development of a cyclic master surgical schedule, but due to the complexity of the problem we could not experiment by matching the demand for beds with the total number of nurses required. Because we choose for an ILP based solution approach we could not take the stochastic nature of the various patient types into account, when considering the patient-ward assignment. By considering this stochastic nature a larger reduction in variance in demand for care can be achieved when these patient types are combined in an intelligent manner. A possible solution to improve our MSS and the patient-ward assignment is to use a local search heuristic, but the relative long evaluation time of two minutes to determine the required number of beds and nurses per combination, makes it difficult to obtain a good solution fast. Our proposed solution approach can be used to determine the resource needs for a given patient case mix, but to determine the optimal case mix additional steps are necessary. Evaluation of all possible case mix combinations is time consuming and therefore we recommend to develop a separate model to compute the desired patient case mix based on simple, representative parameters. The resulting 60

71 6.3. RECOMMENDATIONS CHAPTER 6. CONCLUSIONS & RECOMMENDATIONS case mix can then be imported in our developed model to obtain a more detailed view of the actual resource needs. To reduce modelling complexity we made several assumptions. These assumptions are necessary to capture the real situation in a model. In the real situation a lot of variation in the daily processes of patients is present. For example, some patients undergo more than one surgery or visit multiple wards during their hospital stay. It is undesirable and not possible to capture all these variations in an analytical model. Therefore, our modelling approach is not a tool that mimics the real situation in exact detail, but a tool that can be used to study the effect of various resource capacity planning interventions on the resource requirements. Our solution approach consists of an exact model that requires reliable historic data. The quality of the data available in the AMC was sufficient to demonstrate our solution approach and to see the influence of various interventions on the total resource requirements. However, more reliable results can be achieved when the quality of data is improved. 6.3 Recommendations In Chapter 2 we identified possible causes that yield a low bed occupation, a high number of nurses staffed and high variance in number of patients that undergo surgery within a surgery block. Based on the results of the experiments we conducted, we give the following recommendations to the AMC to make more efficient use of resources in the inpatient care chain: Reduce number of wards and increase their size: Reduce the total number of wards by assigning multiple patient types to the same ward. When we consider less wards with more beds we reduce the variance in demand for beds and care that results from the MSS. Thereby we increase the bed occupancy and require smaller total numbers of beds and FTE dedicated nurses. When we consider an intervention in which we consider three wards with a maximum of 50 beds, we save 11.1% FTE compared to our model of the current situation. Another advantage of reducing the number of wards is the possibility to save additional costs by integration of several overhead functions. For example, the secretary staff of both wards can be combined and less square meters are required. When considering a combination of multiple patient types into the same ward it is important to research whether nurses require additional training. We recommend that the size of a ward is not increased too much, which could result in patients dissatisfaction, safety concerns and managing difficulties. Opportunities MSS: The solution approach we present consists of a method to create a cyclic master surgical schedule while lowering the maximum number of beds required. We show that we can reduce the total number of beds required by 5.3% compared to the model of the current situation. Another advantage of a cyclic MSS is that it provides the possiblity to predict the demand for beds and care, and thereby determine the number of nurses to be staffed per shift in advance. Additional reductions can be achieved when more detailed information on the number of patients in a surgery block is known or when less acute patients are considered. We recommend to perform additional research into the development of an MSS that focuses on better alignment in the demand for beds and care in the nursing wards. Flex pool of nurses: A flex pool of nurses reduces the total number of nurses required, while maintaining predefined service levels. The total reduction in FTE is rather limited in our experiments due to the MSS and the chosen service levels, so considering a flex pool of nurses mainly depends on the additional costs required, such as extra training to treat more than one patient type. If these additional costs are low compared to the reduction in FTE, we would recommend to use a flex pool of nurses. Perform additional experiments with misplacements: In our experimental design we did not conduct experiments regarding the possibility of misplacing patients between two wards, because the realised rejection and misplacement rates of the model of Smeenk et al. 61

72 6.4. FURTHER RESEARCH CHAPTER 6. CONCLUSIONS & RECOMMENDATIONS [29] showed some inconsistencies. We recommend to perform additional experiments when these inconsistencies are resolved and to see how this influences the resource requirements. Structure admission and discharge process: From our data analysis it became clear that the admission and discharge process of patients is intertwined. Admissions occur in the morning, while discharges take place in the afternoon. We would recommend to discharge patients in the morning and to admit patients in the afternoon, which leads to a reduction of 1.3% in total number beds and 1.7% in FTE of dedicated nurses compared to our model of the current situation. Financial monitoring and control: When we gathered data regarding the cost parameters that we need for our model, limited information was available in terms of resource costs and patient revenues. We recommend more financial monitoring and control, and encourage decision making based on the real costs. More reliable information will provide the AMC with the possibility to compare several alternative case mix configurations. Improve data monitoring and consistency: The data we derived from the management systems showed some inconsistencies. The number of patients that could not be matched to a scheduled surgery was more than half of the total number of patients in our case study. Furthermore, we had to manually combine two different databases to obtain the appropriate distributions required for our model. We encourage the AMC to continuously improve their quality of data and recommend one integral system in which a patients stay in the hospital can be completely tracked. 6.4 Further research We point out the following directions for further research: Improve our solution approach: Our solution approach provides a good initial solution for the creation of a MSS and the assignment of patient types to wards. We suggest improvement of our solution approach in better alignment of the demand for beds with the patient-to-nurse ratio. Furthermore, we suggest to perform additional research to optimise the patient case mix decision. Integration of outpatient clinics: In this research we neglected the outpatient clinics. It is interesting to integrate the outpatient clinics in our solution approach and to see how the opening hours affect the demand for surgeries and influence the need for care. Additional value can be obtained when the scheduling of surgery blocks is aligned with the clinic hours a specialist needs to perform. More detailed surgery scheduling: Our proposed surgery method consist of the assignment of surgery blocks, that consist of a stochastic distribution in number of patients that undergo surgery, to days. We would recommend to create a more detailed surgery scheduling approach in which the number of patients that are present in a surgery block is exactly known. When more detailed information is known it is possible to better align the demand for care with the required number of nurses per ward. From one surgery block to multiple wards: In our modelling approach we make the assumption that patients from a scheduled surgery block need to go to the same ward. It would be valuable to research the possibilities of allowing patients, that are present in the same surgery blocks, to go to different wards. This provides us with the possibility to study alternative types of ward configurations. For example, the creation of a surgical short stay ward. 62

73 Bibliography [1] Groot personeelstekort zorg verwacht in [2] Kwaliteit en proces innovatie. 1 [3] I. Adan, J. Bekkers, N. Dellaert, J. Jeunet, and J. Vissers. Improving operational effectiveness of tactical master plans for emergency and elective patients under stochastic demand and capacitated resources. European Journal of Operational Research, 213: , , 34 [4] L.H. Aiken, D.M. Sloane, J.P Cimiotti, S.P. Clarke, L. Flynn, J.A. Seago, J. Spetz, and H.L. Smith. Implications of the california nurse staffing mandate for other states. Health Research and Educational Trust, [5] J. Beliën and E. Demeulemeester. Building cyclic master surgery schedules with leveled bed occupancy. European Journal of Operational Research, 176: , , 22 [6] J. Beliën, E. Demeulemeester, and B. Cardoen. A decision support system for cyclic master surgery scheduling with multiple objectives. Journal of Scheduling, 12: , [7] J.M. Bosch. Better utilisation of the or with less beds: A tactical surgery scheduling approach to improve or utilisation and the required number of beds in the wards. Master s thesis, University of Twente, [8] A. Braaksma and N. Kortbeek. Adviesreport: Herziening indeling beddenhuis divisie b. AMC. 9, 14, 43 [9] C.A.J. Burger. Flexible nurse staffing: Determining staffing levels for nursing wards in the amc. Master s thesis, University of Twente, i, ii, 1, 2, 3, 5, 22, 32, 37, 38, 59 [10] E.K. Burke, P. De Causmaecker, G. Vanden Berghe, and H. Van Landeghem. The state of the art of nurse rostering. Journal of Scheduling, 7: , [11] B. Cardoen, E. Demeulemeester, and J. Beliën. Operating room planning and scheduling: A literature review. European Journal of Operational Research, 201: , , 20 [12] Y. Carson and A. Maria. Simulation optimization: Methods and applications. In Proceedings of the 1997 Winter Simulation Conference, [13] J.K. Cochran and A. Bharti. Stochastic bed balancing of an obstetrics hospital. Health Care Management Science, 9:31 45, , 21 [14] D. Cornelisse. Ok-reglement 2011: Operatiecentrum (klinische ok en dagcentrum), divisie h, amc. AMC. 11, 12 [15] F. Guerriero and R. Guido. Operational research in the management of the operating theatre: a survey. Health Care Management Science, 14:89 114, , 20 [16] E.W. Hans, M. Van Houdenhoven, and P.J.H. Hulshof. A framework for health care planning and control. Handbook of Healthcare System Scheduling. International Series in Operations Research & Management Science, 168: , , 5 63

74 BIBLIOGRAPHY BIBLIOGRAPHY [17] P.J.H. Hulshof, N. Kortbeek, R.J. Boucherie, and E.W. Hans. Taxonomic classification of planning decisions in health care: a review of the state of the art in or/ms. Center for Healthcare Operations Improvement and Research, -:, i, 1, 5, 19 [18] J.B. Jun, S.H. Jacobson, and J.R. Swisher. Application of disrete-event simulation in health care clinics: A survey. Journal of the Operational Research Society, 50: , [19] N. Kortbeek, E.M. Zonderland, R. J. Boucherie, N. Litvak, and E.W. Hans. Desidesigning cyclic appointment schedules for outpatient clinics with scheduled and unscheduled patient arrivals. Memorandum 1968 December i, 35, 59 [20] A.M. Law. Simulation modeling and analysis: Fourth edition. McGraw-Hill International Edition, [21] G. Ma, J. Beliën, E. Demeulemeester, and L. Wang. Solving the strategic case mix problem optimally by using branch and price algorithms. In Proceedings of the 35th International Conference on Operational Research Applied to Health Services, [22] M. Maiza. Heuristics for solving the bin-packing problem with conflicts. Applied Mathematical Sciences, 5: , [23] J.H. May, W.E. Spangler, D.P. Strum, and L.G. Vargas. The surgical scheduling problem: Current research and future opportunities. Production and Operations Management, 20: , [24] M.W. Mullholand, P. Abrahamse, and V. Bahl. Linear programming to optimize performance in a department of surgery. The american college of surgeons, published by Elsevier, [25] A.E.F Muritiba, M. Lori, E. Malaguti, and P. Toth. Algorithms for the bin packing problem with conflicts. INFORMS Journal on Computing, 22: , [26] J. Needleman, P. Buerhaus, V.S. Pankratz, C.L. Leibson, S.R. Stevens, and M. Harris. Nurse staffing and inpatient hospital mortality. New England Journal of Medicine, 364: , [27] J.M. Nguyen, P. Six, D. Antonioli, P. Glemain, G. Potel, P. Lombrail, and P. Le Beux. A simple method to optimize hospital beds capacity. International Journal of Medical Informatics, 74:39 49, [28] P. Santibáñez, B. Mehmet, and D. Atkins. Surgical block scheduling in a system of hospitals: an application to resource and wait list management in a british columbia health authority. Health Care Management Science, 10: , [29] H.F. Smeenk. Predicting bed census of nursing ward from hour to hour. Master s thesis, University of Twente, i, 1, 2, 3, 5, 12, 21, 22, 36, 37, 59, 62 [30] H.F. Smeenk and C.A.J. Burger. Bed capacity management and nurse scheduling: Process description g6. AMC. 9, 10, 12, 14 [31] V.L. Smith-Daniels, S.B. Schweikhart, and D.E Smith-Daniels. Capacity management in health care services: Review and future research directions. Decision Sciences, 19: , [32] A. Testi, E. Tanfani, and G. Torre. A three-phase approach for operating theatre schedules. Health Care Management Science, 10: , [33] J. M. Van Oostrum, M. Van Houdenhoven, J. L. Hurink, E.W. Hans, G. Wullink, and G. Kazemier. A master surgical scheduling approach for cyclic scheduling in operating room departments. Operation Research Spectrum, 30: , , 22 [34] J.M. Van Oostrum, E. Bredenhoff, and E.W. Hans. Suitability and managerial implications of a master surgical scheduling approach. Annals of Operations Research, 178:91 104,

75 BIBLIOGRAPHY BIBLIOGRAPHY [35] P.T. Vanberkel and J.T. Blake. A comprehensive simulation for wait time reduction and capacity planning applied in general surgery. Health Care Management Science, 10: , [36] P.T. Vanberkel, R.J. Boucherie, E.W. Hans, J.L. Hurink, and N. Litvak. A survey of health care models that encompass multiple departments. International journal of health management and information, 1:37 69, , 19, 22 [37] P.T. Vanberkel, R.J. Boucherie, E.W. Hans, J.L. Hurink, W.A.M. Van Lent, and Van Harten W.H. An exact approach for relating recovering surgical patient workload to the master surgical schedule. Journal of the Operational Research Society, 62: , , 3, 21, 22 [38] S. Villa, M. Barbieri, and F. Lega. Restructuring patient flow logistics around patient care needs: implications and practicalities from three critical cases. Health Care Management Science, 12: ,

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77 Appendix A Mathematical optimisation problem and ILP This appendix summarises the notation used for the mathematical optimisation problem defined in Section 4.1 and the ILP defined in Section We define the sets, parameters, distributions and decision variables for the mathematical optimisation problem in Section A.1. Section A.2 contains the sets, parameters, distributions and decision variables of the ILP. A.1 Notation mathematical optimisation problem Sets J E J A J R W F Set of elective patient types Set of acute patient types Set of all patient types, J E J A Set of resources in the OR Set of wards inside the hospital Set of flex pools Parameters General parameters: D Number of working days in a week T Number of time slots T Number of shift types bigm Some large number µ j,d Average number of appointment request for patient type j on day d, j J E, d {0,..., D 1} λ j,d,t Poisson arrival rate of patient type j on day d during timeslot t, j J A, d {0,..., D 1}, t {0,..., T 1} 67

78 A.1. NOTATION APPENDIX A. MATHEMATICAL OPTIMISATION PROBLEM AND ILP Performance indicator targets: (A j, θ norm (A j )) Desired access time service level for patient type j, A j denotes the maximum access time, θ norm states the fraction of patients that should have an access time lower than A j, j J E χ Desired rejection probability, equal for all patient types η α β γ Cost parameters: Fr nonrenewable F F T Ed F F T Ef Fw ward F bed F patient j Desired misplacement probability, equal for all patient types Desired overall service level, equal for all wards, the fraction of the time that the number of patients on a ward does not exceed the number of nurses staffed times the patient-to-nurse ratio Desired minimum service level, equal for all wards, the minimum fraction of patients per time slot that is covered by the staffed number of nurses times the patient-to-nurse ratio Desired minimum fraction dedicated nurses of the total nurses assigned to a ward, equal for all wards Fixed costs of non renewable resource r, r R Cost of one dedicated nurse per year (one FTE) Cost of one nurse in a flex pool f per year (one FTE) Fixed costs for opening ward w, w W Fixed costs per bed Costs* per patient of type j, j J *We assume that the costs include both costs and profits per patient of type j. In case the costs are higher than the profits, costs will be positive. In case the costs are lower than the profits, costs will be negative. Moreover, costs per patient should only include patient-specific costs. Solution space constraints Ω max r Maximum number of non renewable resource r, r R Ψ max Maximum number of wards possible Mw max Maximum number of beds possible inside ward w, w W Xj min Minimum number of patients of type j to treat per year, j J E Xj max Maximum number of patients of type j to treat per year, j J E RE j,r Amount of resource r patient type j requires in a full surgery block, r R, j J E REj,r M Amount of resource r patient type j requires in a morning surgery block, r R, j J E REj,r A Amount of resource r patient type j requires in an afternoon surgery block, r R, j J E NF f,j 1 if nurses inside flex pool f are not allowed to treat patient type j, f F, j J NP i,j 1 if patient type i and j may not be allocated together, i, j J AS Maximum number of patient types that may be assigned to the same ward rs,τ w Patient-to-nurse ratio for ward w on day s during shift τ, w W, s {0,..., S 1}, t {0,..., T } Maximum length of stay for patient type j, j J L max j Distributions C j (k) Probability of k surgeries performed in one OR-day assigned to type j, j J E Cj M (k) E Probability of k surgeries performed in a morning block assigned to type j, j J Cj A(k) E Probability of k surgeries performed in an afternoon block assigned to type j, j J E j (n) probability that patient type j is admitted on day n, j J E, n { 1, 0} Wn(t) j Probability that patient type j admitted on day n is admitted during time slot t, j J E n { 1, 0}, t {0,..., T 1} P j (n) Probability that patient type j has a length of stay of n days, j J, n {0,..., L max j } Mn(t) j Probability that patient type j is discharged on day n during time slot t, j J, n {0,..., L max j }, t {0,..., T 1} Zs,t(x) w Probability that x patients are present in ward w on day s during time slot t, w W, s {0,..., S 1}, t {0,..., T 1} 68

79 A.2. NOTATION APPENDIX A. MATHEMATICAL OPTIMISATION PROBLEM AND ILP Decision variables S Length of the MSS b j,s Number of complete OR-days assigned to type j on day s, j J E, s {0,..., S 1} b M j,s Number of morning blocks assigned to type j on day s, j J E, s {0,..., S 1} b A j,s Number of afternoon blocks assigned to type j on day s, j J E, s {0,..., S 1} ass j,w 1 if type j is assigned to ward w, j J, w W mis j,w 1 if type j may be misplaced at ward w, j J, w W flex f,w 1 if ward w makes use of flex pool f, w W, f F x j Number of type j patients to treat per year, j J E Ω r Total number of non renewable resources r to be used, r R Ψ Total number of wards to open n ded Total amount of dedicated FTE (nurses) needed n flex Total amount of flexible FTE (nurses) needed n ded w,s,τ Total amount of dedicated nurses assigned to ward w on day s during shift τ, w W, s {0,..., S 1}, τ {0,..., T 1} n flex f,s,τ Total amount of nurses assigned to flex pool f on day s during shift τ, f F, s {0,..., S 1}, τ {0,..., T 1} Number of beds on ward w, w W m w Auxiliary variables: u w 1 if one or more patient types are allocated to ward w (i.e. ward w is opened), w W Assignment procedure: φ Assignment procedure that determines the sequencing and control rules for misplacing patients A.2 Notation ILP Sets J J E J A R W Set of patient types Set of elective patient types Set of acute patient types Set of non renewable resources inside the OR Set of wards Parameters S Length of the MSS in days T Number of shift types L max j Maximum length of stay of patient type j κ nonrenewable r Weight factor of the non renewable resource r, r R κ wards w Weight factor of a ward w,w W κ beds Weight factor of nursing beds κ nurses Weight factor of nurses ɛ Stochastic capacity scale factor rs,τ w Patient-to-nurse ratio for ward w on day s for shift type τ, w W, s {0,..., S 1}, τ {0,..., T 1 69

80 A.2. NOTATION APPENDIX A. MATHEMATICAL OPTIMISATION PROBLEM AND ILP Bj norm Number of full blocks that need to be scheduled for patient type j, j J E B norm,m j Number of morning blocks that need to be scheduled for patient type j, j J E B norm,a j Number of afternoon blocks that need to be scheduled for patient type j, j J E RE j,r Amount of resource r that patient type j requires during surgery, j J E, r R REj,r M Amount of resource r that patient type j requires during morning surgery, j J E, r R REj,r A Amount of resource r that patient type j requires during afternoon surgery, j J E, r R Maximum number of non renewable resource r, r R 1 if patient types i and j may not be allocated together, i, j J AS Maximum number of patient types that may be assigned to the same ward C j (k) Probability of k surgeries performed in one OR-day assigned to type j, j J E Cj M (k) E Probability of k surgeries performed in a morning block assigned to type j, j J Cj A(k) E Probability of k surgeries performed in an afternoon block assigned to type j, j J G j (n) Probability that patient type j is in the hospital n days after surgery, j J, n { 1,..., L max j } λ j,s Poisson arrival rate of acute patient type j on day s, j J A, s {0,..., S 1} Ω max r NP i,j Decision variables Ω r Number of non renewable resources r, r R Ψ Number of wards to open m µ w Expected number of beds in ward w, w W n µ w,s,τ Expected number of nurses needed in ward w on day s during shift τ, w W, s {0,..., S 1}, τ T b j,s,w Number of full blocks of patient type j scheduled on day s and assigned to ward w, j J E, s {0,..., S 1}, w W b M j,s,w Number of morning blocks of patient type j scheduled on day s and assigned to ward w, j J E, s {0,..., S 1}, w W b A j,s,w Number of afternoon blocks of patient type j scheduled on day s and assigned to ward w, j J E, s {0,..., S 1}, w W ass j,w 1 if patient type j is assigned to ward w, j J, w W 1 if one or more patient types are allocated to ward w (i.e. ward w is opened), w W u w 70

81 Appendix B Data analysis We have gathered data from two separate systems: Locati and OK-Plus. Locati contains all the admission and discharge data of the patients that reside on a ward. OK-plus contains all data on the planned and performed surgeries. There is no direct link between both databases so we have to perform several steps to create the relevant data sets. We define a surgical specialty as an unique patient type. We use the following procedure to extract patient arrival and discharge distributions, and the accompanying surgical distributions. 1. Select all patients with the specialty codes (Chi,Chp,Vaa,Uro,Ort,Tra,Mkz) that reside on the wards (G5NO,G5ZU,G6NO,G6ZU,G7ZU,G7NO) in Division B. Patients with a specialty code of another division are neglected, because it is not clear if they are designated to the wards of Division B. We use data obtained from January 2010 till December Determine the Length Of Stay (LOS) for the obtained patient set from step 1. We assume that when a patient arrives at the hospital this is day 0. When a patient leaves the hospital this is considered day N. A patient has thus a length of stay of N days. When a patient resides at more than one ward during his hospital stay, we assume that this patient stays on his designated ward only. We remove patients with a length of stay of 0 days. Most of these patient visits are short and do not require a bed. 3. We select all planned surgeries and operating theatres with urgency code N for the specialty codes of Division B (Chi, Chp, Uro, Ort, Tra, MKZ). Code N indicates that a surgery is planned more than one week in advance. The other surgical procedures are planned in a shorter time period and consists of acute patients. 4. We combine the patient set from step 2 with the surgery information obtained in step 3. We match patients on patient id. We remove the following surgeries: Surgeries with no resulting admission at a ward in Division B. Surgeries of patients that are discharged from a ward before the surgery starts. All except the first surgery of patients that undergo more than one surgery during their length of stay. 5. The surgeries of specialty vascular surgery (VAA) are planned on the operating room days assigned to the specialty General Surgery (CHI). Based on the profile code (Dutch: Profiel code) we make an extra distinction between VAA surgery blocks and general surgery blocks, to obtain the surgeries of specialty VAA. If general surgery patients are present in a surgery block of VAA we discard the surgeries of these patients and hence assume that these are non elective. We update the output of step 4. 71

82 APPENDIX B. DATA ANALYSIS 6. Next, we determine the preoperative stay of patients that undergo surgery. We remove all surgical procedures patients with a preoperative stay longer than five days from the set in step 5 and assume that these patients are non elective. For patients with a preoperative stay of at least one day and shorter than five days, we assume that these patients arrive on the day before surgery. 7. We generate a set of elective patients and acute patients. In step 2 we have obtained a set of all patients. In step 6 we have matched elective surgeries with an admission. We call this our elective patient set. The acute patient set is then build by removing all elective patients from the set of all patients. 8. Next, we determine the distribution of the number of patients in a surgery block. For each day a specialty uses the OR we count the number of performed surgeries in the same OR. This gives us the distribution for the number of patients in a surgery block. Furthermore, we extract the first four weeks of September for the used Master Surgical Schedule from this step. Below we give numeric values for the different steps of Division B for the year admissions counted admissions left after removal of admissions that contain multiple following numbers. A following number is created each time a patient enters or leaves a ward surgeries selected that are planned. 4. After matching the surgeries based on patient id and the removal of surgeries according to the three points mentioned in step 4 there are 2784 left surgeries left after removal of General Surgeries performed in a VAA surgery block. 74% of the surgeries of a VAA surgery block are dedicated to the specialty VAA surgeries left after removal of patients with a preoperative stay longer than 5 days. 7. Total number of elective patient admissions: 2674; Total number of acute patient admissions: Table B.1 shows the number of patients per type for elective and acute patients. Specialty Total patients Elective patients Acute patients CHI CHP MZK ORT TRA URO VAA Total Table B.1: Overview of the total number of patients per type for the year 2010 in Division B 72

83 Appendix C Financial parameters This appendix is confidential. 73

84

85 Appendix D Class diagram Delphi Figure D.1: Class diagram of the developed Delphi model 75

86

87 Appendix E Detailed results In this section we present detailed information of the experiments conducted in Chapter 5. Table E.1 shows the outcome of the patient-ward assignment for the experiments in which we make the patient-ward decision. Table E.2 shows the outcome of our experiment to determine the run time of our ILP. A run time longer than 2 hours is with our ILP solver not possible due to memory allocation problems. Table E.3 contains the run time for the models of Smeenk and Burger for four experiments. We choose to show the run time of these experiments, because these are representative for all our experiments. Table E.4 contains the ward specific results for all experiments conducted. Table E.5 shows the benefits in FTE reduction of having a flex pool between two wards in the current situation. Table E.6 presents the managerial output for the ILP experiments. We want to place one remark regarding this table that concerns the high integrality gap of the experiment that contains five wards and 32 beds. Within the ILP relaxation of this experiment there exists a non integer solution that considers a solution with one ward less. However, when we consider the experiment in which we only allow one ward to be open, we see that the minimum number of expected beds is 92, which does not fit in four wards. We conclude with an overview of the 90% demand for beds during time slot 0 for each day of the MSS per ward in Figures E.1, E.2, and E.3. Experiment G6 Zuid G7 Zuid G7 Noord G5 Noord G6 Noord MSS 5 Wards 32 URO, VAA MZK, ORT CHI, CHIss CHP, TRA CHI, CHIss MSS 3 Wards 50 CHP, ORT, VAA MZK, TRA, CHI, CHIss CHI, CHIss Exp 1 CHI, CHIss TRA CHI, CHIss MZK, URO, VAA CHP, ORT Exp 2 CHI, CHIss ORT URO, VAA CHI, CHIss CHP, MZK, TRA Table E.1: Results of the resulting patient-ward assignment for various experiments Runtime 30 min 60 min 2 hours 4 hours 8 hours Integrality gap 0.5% 0.4% 0.4% n/a n/a Table E.2: Results of the integrality gap versus run time for experiment MSS Optimal ORs Runtime in seconds Model Smeenk Model Burger no flex Model Burger with flex Current situation n/a MSS 5 Wards n/a Patient one ward n/a Flexpool optimisation Table E.3: Results of the run times of the model of Smeenk and Burger for some experiments. 77

88 APPENDIX E. DETAILED RESULTS Min number of beds (90% demand) Max number of beds (90%) demand Average bed utilisation Total dedicated nurses (FTE) Flex pool assigned Average overall service level per year Average minimum service level per year Min number of beds (90% demand) Max number of beds (90%) demand Average bed utilisation Total dedicated nurses (FTE) Flex pool assigned Average overall service level per year Average minimum service level per year Min number of beds (90% demand) Max number of beds (90%) demand Average bed utilisation Total dedicated nurses (FTE) Flex pool assigned Average overall service level per year Average minimum service level per year Min number of beds (90% demand) Max number of beds (90%) demand Average bed utilisation Total dedicated nurses (FTE) Flex pool assigned Average overall service level per year Average minimum service level per year Min number of beds (90% demand) Max number of beds (90%) demand Average bed utilisation Total dedicated nurses (FTE) Flex pool assigned Average overall service level per year Average minimum service level per year G6 Zuid G7 Zuid G7 Noord G5 Noord G6 Noord Current situation Optimisation MSS Optimal Ors MSS 8 Ors MSS 5 Wards MSS 3 Wards Patient one ward MSS One ward Flex pool optimisation All combined Interventions MSS length 1 week MSS length 2 weeks Steady cyclic demand Increase 20% Increase 40% Increase one patient increased demand Increase one patient normal demand Size surgery blocks Weekend surgery Admission/Discharge MSS length 1 week 3 wards Scenarios LOS+1 current situation LOS+1 MSS 5 Wards LOS+1 All combined LOS-1 current situation LOS -1 MSS 5 Wards LOS -1 All combined Acute-25% current situation Acute-25% MSS 5 Wards Acute-25% All combined Acute+25% current situation Acute+25% MSS 5 Wards Acute+25% All combined Parameters ILP Exp ILP Exp Serv lvls Exp 1. no flex Serv lvls Exp 1. flex Serv lvls Exp 2. no flex Serv lvls Exp 2. flex Serv lvls Exp 3. no flex Serv lvls Exp 3. flex Patient-to-nurse ratio no flex Patient-to-nurse ratio flex Table E.4: Detailed managerial information per ward for each experiment 78

89 APPENDIX E. DETAILED RESULTS Flexpool combination Nurses required when considering flex pools Nurses required without flex pool Reduction (FTE) Ward Ward Ded nurses (FTE) Ded nurses (FTE) Nurses (FTE) in Flex pool Total with flex Ward1 (FTE) Ward 2 (FTE) Total Table E.5: Overview of the reduction in FTE for all possible flex pool combinations of two wards for the current situation compared to the situation in which no flex pools are used. 79

90 APPENDIX E. DETAILED RESULTS Expected beds per ward Experiment ORs Wards Beds Runtime Integrality gap G6 Noord G7 Zuid G7 Noord G5 Noord G6 Zuid MSS Optimal Ors % MSS 8 Ors % MSS 5 Wards % MSS 3 Wards % MSS 1 Ward % MSS length 1 week % MSS length 2 weeks % Steady cyclic demand % % more surgery blocks % % more surgery blocks % extra surgery per block increased demand % extra surgery per block normal demand % Short/long surgery blocks % Weekend surgery % ILP Exp % ILP Exp % MSS length 1 week 3 wards % Table E.6: Overview of the managerial output for the ILP solutions 80

91 APPENDIX E. DETAILED RESULTS (a) G6 Zuid (b) G7 Zuid (c) G7 Noord (d) G5 Noord (e) G6 Noord Figure E.1: 90% demand for beds for each day of the MSS at time slot 0, experiment Current situation 81

92 APPENDIX E. DETAILED RESULTS (a) G6 Zuid (b) G7 Zuid (c) G7 Noord (d) G5 Noord (e) G6 Noord Figure E.2: 90% demand for beds for each day of the MSS at time slot 0, experiment MSS 8 ORs 82

93 APPENDIX E. DETAILED RESULTS (a) G6 Zuid (b) G7 Zuid (c) G7 Noord (d) G5 Noord (e) G6 Noord Figure E.3: 90% demand for beds for each day of the MSS at time slot 0, experiment weekend surgery 83