NURSE ROSTERING: A TABU SEARCH TECHNIQUE WITH EMBEDDED NURSE PREFERENCES

Transcription

1 NURSE ROSTERING: A TABU SEARCH TECHNIQUE WITH EMBEDDED NURSE PREFERENCES Thesis submitted to UUM College of Arts and Sciences in fulfillment of the requirements for the degree Master of Science (Decision Science) Universiti Utara Malaysia SIT1 NURIN IMA BINTI AHMAD Copyright Cj Siti IVurin Iina Binti Ahmad, All right reserved.

2 Kolej Sastera dan Sains (UUM College of Arts and Sciences) Universiti Utara Malaysia Kami, yang bertandatangan, memperakukan bahawa (We, the undersigned, cerfify thaf) PERAKUAN KERJA TESlS 1 DlSERTASl (Certification of thesis / dissertation) SIT1 NURlN IMA AHMAD calon untuk ljazah (candidate for the degree ol) MASTER telah mengemukakan tesis I disertasi yang bertajuk: (has presented hisher thesis /dissertation of the following title): "NURSE ROSTERING: A TABU SEARCH TECHNIQUE WITH EMBEDDED NURSE PREFERENCES" seperti yang tercatat di muka surat tajuk dan kulit tesis I disertasi. (as it appears on the title page and front cover of the thesis / disserfation). Bahawa tesisldisertasi tersebut boleh diterima dari segi bentuk serta kandungan dan meliputi bidang ilmu dengan memuaskan, sebagaimana yang ditunjukkan oleh calon dalam ujian lisan yang diadakan pada : 06 Oktober That the said thesiddisserfation is acceptable in form and content and displays a safisfactoy knowledge of the field of study as demonstrated by the candidate through an oral examination held on: - October 06,2010. Pengerusi Viva: (Chairman for Viva) Assoc. Prof. Dr. Engku Muhammad Nazri Engku Abu Bakar... Pemeriksa Luar: Assoc. Prof. Dr. Abas Md Said Tandatangan (External Examiner) (Signature) Pemeriksa Dalam: (Internal Examiner) Dr. Mohd Kamal Mohd Nawawi (Signature) / Nama PenyelialPenyelia-penyelia: Assoc. Prof. Dr. Razamin Ramli (Name of Supe~isor/Supervisors) Tandatangan (Signature) Tarikh: (Date) October 06, 2010

3 PERMISSION TO USE In presenting this thesis in fulfillment of the requirements for a graduate degree from Universiti Utara Malaysia, I agree that the University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes, may be granted by my supervisor(s) or, in their absence, by the Dean of Research and Graduate Studies, College of Arts and Sciences. It is understood that any copying or publication or use of this thesis or parts thereof financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to Universiti Utara Malaysia for any scholarly use which may be made from any material from my thesis. Request for permission to copy or to make other use of materials in this thesis, in whole or in part, should be addressed to: Dean of Research and Graduate Studies College of Arts and Sciences Universiti Utara Malaysia UUM Sintok Kedah Darul Aman

4 ABSTRACT The decision making in assigning all nursing staffs to shift duties in a hospital unit must be done appropriately because it is a crucial task due to various requirements and constraints that need to be fulfilled. The shift assignment or also known as roster has a great impact on the nurses' operational circumstances which are strongly related to the intensity of quality of health care. The head nurse usually spends a substantial amount of time developing manual rosters, especially when there are many staff requests. Yet, sometimes she could not ensure that all constraints are met. Therefore, this research identified the relevant constraints being imposed in solving the nurse rostering problem (NRP) and examined the efficient method to generate the nurse roster based on constraints involved. Subsequently, as part of this research, we develop a Tabu Search (TS) model to solve a particular NRP. There are two aspects of enhancement in the proposed TS model. The first aspect is in the initialization phase of the TS model, where we introduced a semi-random initialization method to produce an initial solution. The advantage of using this initialization method is that it avoids the violation of hard constraints at any time in the TS process. The second aspect is in the neighbourhood generation phase, where several neighbours need to be generated as part of the TS approach. In this phase, we introduced two different neighbourhood generation methods, which are specific to the NRP. The proposed TS model is evaluated for its efficiency, where 30 samples of rosters generated were taken for analysis. The feasible solutions (i.e. the roster) were evaluated based on their minimum penalty values. The penalty values were given based on different violations of hard and soft constraints. The TS model is able to produce efficient rosters which do not violate any hard constraints and at the same time, fulfill the soft constraints as much as possible. The performance of the model is certainly better than the manually generated model and also comparable to the existing similar nurse rostering model.

5 ABSTRAK Tugasan membuat keputusan dalam rnenjana jadual syi f kepada kakitangan kejururawatan di dalam sesuatu unit hospital adalah sukar dan mesti dilakukan sewajamya dengan mengambil kira segala kekangan dan keperluan yang perlu dipenuhi. Jadual syif, juga dikenali sebagai jadual tugas, mempunyai kesan yang besar kepada situasi pengoperasian jururawat yang sangat berkaitan dengan tahap kualiti penjagaan kesihatan. Biasanya, ketua jururawat memerlukan masa yang secukupnya untuk menjana sesuatu jadual manual syif terutama sekali apabila terdapat banyak permintaan dan keperluan kakitangan. Namun, ada ketikanya adalah sukar untuk meinastikan yang semua keperluan dan kekangan dapat dipenuhi. Sehubungan itu, kajian ini bertujuan mengenal pasti kekangan yang berkaitan yang dikenakan dalam menyelesaikan inasalah penjadualuan jururawat (NRP) dan mengkaji kaedah yang berkesan untuk menjana jadual syif jururawat berdasarkan kekangan yang terlibat. Seterusnya, satu inodel Tabu Sear-ch (TS) dibangunkan untuk menyelesaikan satu NRP tertentu. Terdapat dua aspek penainbahbaikan dalam model TS yang dicadangkan. Aspek pertaina adalah dalam fasa pembentukan awal model TS, yang rnana kaedah pembentukan awal berasaskan separa rawak untuk menghasilkan penyelesaian awal telah diperkenalkan. Kelebihan menggunakan kaedah tersebut adalah ia dapat mengelak berlakunya pelanggaran kekangan keras pada mana-mana masa dalam proses TS. Aspek kedua adalah dalam fasa penjanaan kejiranan, yang mana beberapa jiran perlu dihasilkan sebagai sebahagian daripada pendekatan TS. Dalam fasa ini, dua kaedah penjanaan kejiranan yang berbeza dan khusus untuk NRP diperkenalkan. Model TS yang dicadangkan kemudiannya dinilai keberkesanannya, yang mana 30 sampel telah diambil untuk tujuan analisis. Beberapa penyelesaian yang sesuai (i.e. j adual tugas) telah dinilai berdasarkan kepada nilai penalti minimum. Nilai penalti diberikan berdasarkan kepada perbezaan pelanggaran kekangan keras dan lembut (kekangan yang boleh dilonggarkan). Model TS mampu inenghasilkan jadual tugas yang cekap yang tidak melanggar mana-mana kekangan keras dan pada masa yang sama, memenuhi segala kekangan lembut sebaik yang mungkin. Prestasi model tersebut adalah lebih baik daripada model yang dijana secara manual dan setanding dengan model jadual tugas jururawat sedia ada yang terhampir.

6 ACKNOWLEDGEMENTS One above all of gratitude, the omnipresent God, for answering my prayers for giving me the strength to plod on despite my constitution wanting to give up and throw in the towel, thank you so much Dear Allah. I am heartily thankful to my supervisor, Associate Professor Dr. Razamin Ramli, whose encouragement, supervision and support from the preliminary to the concluding level enabled me to develop an understanding of the subject. This thesis would not have been possible unless Mr. Abdullah, the programmer who assisted me on how to use the software needed for tny nurse roster model. It is a pleasure to thank those who made this thesis possible such as my beloved husband, Abdul Rahman Bin Ariffin who gave me the moral support I required. I am grateful to my parents and my siblings, who helped me to take care of my kids for a while to accomplish my master at UUM. I owe my deepest gratitude to my friends, Zara, Bi Lin, Kasim, Salman, Ilmi, Afidah and Uma Rani. Without their assistance, I i+~ould not have gotten any ideas for correcting my thesis. Lastly, I offer my regards and blessiilgs to all of those who supported me in any respect during the completion of this thesis.

7 TABLES OF CONTENTS PERMISSION TO USE ABSTRACT ABSTRAK (Malay) ACKNOWLEDGEMENTS TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF ACRONYMS X xi xii CHAPTER ONE: INTRODUCTION 1.1 What is nurse rostering? 1.2 Probletns in rostering Nurses' satisfaction Hospital's operating costs Quality patient care Time-consuming task Complexity of rostering process 1.3 Research questions 1.4 Research objectives 1.5 Scheduling approach The two-stage solution approach The single-stage soiution approach 1.6 Scope of the research 1.7 Contributio~ls of the research 1.8 Outline of the thesis

8 CHAPTER TWO: ASSIGNING OF NURSING STAFF THROUGH THE ROSTERING PROCEDURE: A LITERATURE REVIEW Objectives being considered 2.2 Constraintcriteria Hard constraints Soft constraint s 2.3 An overview of earlier works 2.4 Previous work 2.5 Solution approaches Cyclical and non-cyclical scheduling Optmization approach Heuristic approach Search approach Constructive heuristic approach Hybrid approach 2.6 Discussion and summary CHAPTER THREE:TABU SEARCH PROCESS AND ITS COMPONENTS 3.1 A meta-heuristic approach 3.2 Tabu search architecture Initialization of solution Fitness evaluation Neighbourhood generation Tabu list Aspiration functions Stoppiilg criterion 3.3 Summary

9 CHAPTER FOUR: NIETHODOLOGY 4.1 A case of an NRP 4.2 Research process 4.3 Data collection Number of staff Types of shifts Design of shifts Types of off days Categories of staff 4.4 Tabu search architecture Solution representation Initialization of the solution Fitness evaluation Neighbourhood generation Tabu list Aspiration functions Stopping criteria 4.5 Evaluation of schedule Coverage Quality Flexibility Cost 4.6 Summary

10 CHAPTER FIVE: IMPLEMENTATION AND RESULTS 5.1 Current problem environment Shifts and constraints Nurse category and requirement Constraiilts in consideration 5.2 Solution representation 5.3 User input data 5.4 Applying the TS approach to the NRP Initialization of solution Roster satisfactory and fitness evaluation Applying the TS mechanism to the NRP 5.5 Experiments and results The algorithm performance Comparison techniques 5.6 Summary CHAPTER SIX: CONCLUSIONS 6.1 Summary 6.2 Implications of the research Implications to the body of knowledge Implications to the nurses Implications to the field of healthcare management 6.3 Limitations of the research 6.4 Future work

11 REFERENCES APPENDIX A LIST OF PUBLICATIONS VITA

12 LIST OF TABLES Table 2.1 : Table 2.2: Table 2.3: Table 4.1 : Table 4.2: Table 5.1: Table 5.2: Table 5.3: Table 5.4: Table 5.5: Table 5.6: Table 5.7: Classification of nurse rostering models by solution approaches excerpt from Ramli (2002). 15 Classification of recent nurse rostering models by solution approaches 16 Classification of earlier scheduling models by techniques 17 Working shifts Off days Working shifts and off days Nurse level assigning for every working shift 67 The ME pattern blocks 7 5 Evaluatioil constraints for every row 7 8 Evaluation constraints for every column 7 9 TS model operation results 8 3 Results based on different criteria for models comparison 86

13 LIST OF FIGURES Figure 4.1 : Figure 4.2: Figure 4.3: Figure 4.4: Figure 4.5: Figure 4.6: Figure 4.7: Figure 4.8: Figure 5.1: Figure 5.2: Figure 5.3: Figure 5.4: Figure 5.5: Figure 5.6: Figure 5.7: Figure 5.8: Figure 5.9: Figure 5.10: Figure : The research process for the NRP The allocation of N and NO The development of TS for NRP The solution encoding and representation The possible representation of solution L*V[.l)for a partial solution The possible representation of solution kv(p)for a partial solution The process of tabu list for the TS The process of aspiration criterion Setting the nurse schedule formats Allocation of the N shifts and the NO days Allocation of the WO Allocation of the PO Rearrange heuristic (a null cell is more than six) Allocation of M and E work stretches (ME pattern block) Evaluation constraints for every row Evaluation constraints for every column The aspiration criterion process The 'best so far' schedule sample The sample grand total penalty

14 LIST OF ACRONYMS NSP NRP OR A 1 LP IP NLP MIP CP GP NP RM ES S A GA MA TS H CH MP GP ACO CSP CBR EA ED A M Nurse scheduling problem Nurse rostering problem Operation research Artificial Intelligence Linear programming Integer programming Non-linear programming Mixed-integer programming Constraint programming Goal programming Network programming Redundant modeling Expert system Simulated annealing Genetic algorithm Memetic algorithm Tabu search Heuristics Constructive heuristics Mathematical programming Goal programming Ant colony optinlization Constraint satisfactory problem Case-based reasoning Evolutionary approach Estimation of distribution algorithm Moi-ning shift

15 E N NO WO PO Evening shift Night shift Night off day Weekly off day Public off day

16 CHAPTER ONE INTRODUCTION Manpower scheduling (or rostering) is concerned with the scheduling of huinan resources to meet temporal operational requirements in ways that satisfy the goals and policies imposed by the management, labour union and the government (Lau, 1996). Manpower scheduling is crucial in the management of a service organisation. One example is related to the nursing services in a hospital organisation. As a rule, the nursing services in hospital wards must be available at all times with no breaks for weekends and holidays since the service is the critical type. Moreover, this job is a very high risk job because it is a difficult and tiring work, which involves patient safety and health care. In manpower scheduling, it is strongly suggested that, as the day progresses, a worker should be assigned for work no earlier than the shift he worked the day before so that he maintains a healthy biological clock (Lau, 1996). In recent developments, it is observed that the scheduling of nurses has been widely studied and there are many approaches being developed for special circumstances. A wide variety of constraints can be imposed on the rosters depending on the legal, inanagemellt and staffing requirements of individual organisations (Beddoe & Petrovic, 2005). The roster quality and optiinality are highly subjective. Therefore, it is in~possible to represent similar systems to develop the nurse roster.

17 The Operation Research (OR) and Artificial Intelligence (AI) research communities have worked hard to propose the best solution techniques to develop the nurse roster. However, there appears to be no agreement on how to come close to the problems (Millar and Kiragu, 1998). Hence, due to the challenging nature of the problems and their relevance in practice, this research is carried out to decide on the best solution techniques to overcome the nurse rostering issue. 1.1 What is nurse rostering? The nurse rostering can be defined as the problem of placing resources (nurses), subject to constraints, into slots in a pattern, where the pattern denotes a set of legal shifts defined in terms of work that needs to be done (Wren, 1996). Rostering process of nurses is actually the allocating of nurses to execute their nursing services per shift per day in a department. The nurse rostering is a set of work and rest days assigned to each nurse in the planning horizon and must also satisfy the entire problem rules and regulations that apply to the complete roster entity (Valouxis & Housos, 2000). Through this rostering process, the nursing services or operation times can be worked out as expected. Rosters are cyclic prototypes of days off and days on (or shifts-off and shifts-on) that are allocated as a whole to nurses. Burke et al. (2001) stated that the nurse scheduling problem (NSP) or the nurse rostering problem (NRP) consists of assigning varying tasks, represented as shift types, to the nursing personnel with different skills and work regulations. These rosters must ensure that there are enough nurses working at all times.

18 Nurses should be assigned to the right number of operational hours in accordance with their terms of scheduled time, i.e. there should not be over worked or underused nurses. Nurses have their individual requirements during the rostering period. Hence, due to the nurse assigning requirements, the next section is discussing detail on the NRP. 1.2 Problems in rostering It is possible that no feasible solutions to such NRP would exist because of the large number of constraints that the scheduling attempts to satisfy. For this reason, the constraints are divided into two classes; the hard constraints that must be satisfied at all times and the soft constraints that may be violated. The hard constraints are those that must be satisfied in order to have a feasible schedule due to physical resource restrictions and legislation. When the requirements are desirable but not obligatory, they are often referred to as soft constraints. The soft constraints are often used to evaluate the quality of feasible schedules (Brucker et al., 2005). The soft constraints are referred to the nurse satisfaction. The breakdown of subsection discusses more on the problems in rostering.

19 1.2.1 Nurses' satisfaction Thus, the nurses' satisfaction and quality work are important to the hospital management to take into consideration when preparing the nurse roster. The aim of rostering is to ensure that there are enough nurses on duty at all times, while taking account of individual preferences and requests for days off in a way that is seen to treat all nurses fairly (Dowsland, 1998) Hospital's operating costs An NRP is important in real life because nurses' salaries make up the largest single item in the hospital's operating cost. The nursing staffs are a larger worker in every hospital. Hence, the hospital management must make full use of the available nurses. Effective scheduling of nursing personnel is important in controlling health-care costs. To retain and recruit the nurses, hospital administrators have to pay more attention to the needs of these nurses. Thus, the assigning of nursing staff to shift duties in an efficient way is a crucial job to the hospital management and can affect the quality of patient care (Sitompul & Randhawa, 1990) Quality patient care Rosters requiring nurses to work difficult and tiring combinations of shifts can again impact on the quality and safety of patient care (Thomton, 1997). Therefore, the number and skilled level of nurses are primary detem~inants of safety and quality of patient care (Thomton, 1997). For this reason, the nurse rostering process becomes more critical because it must allocate nurses with different skills and work regulations.

20 The nurse scheduling or rostering is a crucial task that has to be undertaken properly. It has a great impact on the nurses' operational circumstances which are strongly related to the intensity of quality of health care. Improving the quality of nurses' schedules is one of the most economic ways for hospitals to create a better working environment for them (Liao & Kao, 1997). Furthermore, a manual nurse scheduling by the head nurse may turn out to be insufficient because it requires a large amount of effort and therefore is a time-consuming task (Dias, 2003) Time-consuming task Most hospital wards each have a head nurse or nurse manager or nursing officer, who is regularly responsible for ascertaining the nurse roster. Normally, this nurse manager performs her task manually. Nurse Managers usually spend a substantial amount of time developing rosters, especially when there are many staff requests. Even more time is consumed when handling ad-hoc changes to the current duty rosters (Cheang et al., 2003). The manually generated working rosters are not just inconvenient with long work stretches, but usually also with improper allocation of days off that disturb the family and social life Complexity of rostering process The decision making in the nursing staff allocation must be done appropriately. Rainli et al. (2002) have determined that the assignment of nurses is a inulti-stage decision making problenl consist of the assignment of work shifts and days off prior to the real time scheduling period. Then it continues with the fine-tuning of the schedule in real time, where there maybe unexpected changes in staffing capacity and variations in patient demand.

21 Thus, this problem is difficult and complex with many constraints to fulfil. Due to that, the NRP is considered a challenging combinatorial problem, which is also classified to be NP-hard (non-deterministic polynomial-time hard) (Davis, 1991 ; Dias et al., 2003). 1.3 Research questions Based on the problem statement discussed in the previous subsection, there are the research questions need to be answered What are the constraints being imposed in solving the NRP? What is the efficient method to generate the nurse roster based on the constraints involved? Which of the solution techniques is the best or able to show the best performance in solving the NRP? Subsequently, each of these techniques have been studied and reviewed in Chapter 2 (pg. 16). 1.4 Research objectives This section described the objectives of the research to answer the research question. This study discusses about the NRP. The main objective of the study is to develop a inodel to solve a particular lvrp that would meet the workload demand with the available resources in the respective organization while minimizing the operational costs and maximizing personnel preferences as well as the service quality. In order to develop the model, specific objectives have to be identified to ensure that the main objective of the study is met. In order to achieve the main objectives, the researcher breakdown it into the following objectives:

22 1.4.1 To minimize the imbalances of coverage over the assignment of nurses in different shifts in a day in each skill level To minimize the deviation in the number of different shift arrangements for each nurse. In other words to ensure equitable shift distributions To evaluate the proposed model to solve the NRP. 1.5 Scheduling approach In the general concept of staff scheduling, there are two main approaches for solving these scheduling and assignment decisions The two-stage solution approach The first approach is called the two-stage solution approach. In this approach, the scheduling decision is made during the first stage while the assignment decision, which is determined based upon the result of the scheduling decision, is made in the second stage. In other words, the work patterns of days off and working days are generated first. Then, shift duties are assigned based on the working day patterns as studied by Ozkarahan & Bailey (1988), Ozkarahan (1991), Chen & Yeung (1992) and Randhawa & Siton1pu1(1993) The single-stage solution approach The second approach is the single-stage solution approach where the scheduling and the assignment decisions are done siiilultaneously as explored by Baker (1976) and Rainli (2004). However, the single-stage approach is considered to be more challenging because all the rules and constraints must be taken into consideration at the same time.

23 1.6 Scope of the research This thesis is focused on the personnel scheduling which stressed on the nurses' workforce only. The research will not cover the planning stage, which is determining the number of staff available. The number of shifts involved in the rostering process is consulted from the management. The real time or ad-hoc situation is not considered in this research due to the uncertainty of the situation. 1.7 Contributions of the research The main implications or contribution to the body of knowledge is the enhancement of the TS approach in solving the problem of the nurse rostering. There are two aspects in the enhancement of the approach that has been done in the research: i. The first aspect is in the initialization phased. In this initialization phase, we used the semi-random initialization method to initial a solution. To our knowledge, this semi-random initialization method has not been used in any previous literature as described in Chapter Three (pg. 33). The advantage of using this method is that it avoids the violation of hard constraints whereas in other initialization method such as greedy initialization, it can not guarantee the violation of hard constraints The second aspect is in the neighbourhood generation phase where several neighbours are generated as part of the TS approach. This research introduced a special neighbourhood generation method, which is specific to the NRP.

24 1.8 Outline of the thesis Chapter Two discusses the overview of the past nurse scheduling solutions. It illustrates the varying trends of healthcare delivery system. A summary of the solution techniques is also presented in the form of classifications based on the types of techniques. Chapter Three examines the foundation of the approach to be undertaken in relation to NRP. Its discussions begin with the background of the tabu search algorithm, concepts and the underlying reasons for exploiting this approach. Chapter Four examines the methodology of the proposed solution semi-cyclic scheduling approach related to the NRP. Discussions on how the nurse scheduling constraints are met and the objective functions applied are presented. The criteria for the evaluation of the solutions are also presented in this chapter. Chapter Five describes the implementation of the proposed scheduling algorithm to solve the NRP. This implementation was then tested on a real life extensive NRP at the general hospital in Malaysia. An evaluation of the model was done using both the quantitative and qualitative methods. Chapter Six as the final chapter presents the conclusion to the thesis. It summarizes the proposed tabu search strategy to the problem, discusses its implications, limitations and how further and work future in the area might best be directed.

25 CHAPTER TWO ASSIGNING OF NURSING STAFF THROUGH THE ROSTERING PROCEDURE: A LITERATURE REVIEW The issue of rostering problems is treated as an optimisation problem by the Operation Research (OR) coinmunity. The scheduling technique or algorithms related to the NRP are of various kinds representing the Operation Research (OR) or Artificial Intelligence (AI) models. The final output of these models is normally in the form of rosters. A roster is a set of work and rest days assigned to a nurse in the planning horizon and must also satisfy the entire problem rules and regulations that apply to the complete roster entity (Valouxis & Housos, 2000). In the recent decade, many variations of the scheduling processes and techniques have been developed. This thesis attempts to review some of the earlier works that had been done since 2001 to date. These literatures are extension nurse rostering models by approaches reported by Razamin (2004) from the 1990s to These previous works are studied in relation to the objectives employed, the constraint criteria and the approaches or techniques adopted in assigning shift duties to the nursing staff. 2.1 Objectives being considered The objectives being considered in previous studies vary in complexity depending on the problenl at hand, and can be simple or complex (Nonohe & Ibaraki, 1998). In previous studies, Li et al. (2003), Oughaliine et al. (2008) and Kundu & Acharyya (2008) stated the main objective in solving the NRP is to produce a schedule which satisfies all hard and soft constraints as many as possible.

26 Bard & Purnon~o (2005) and Vanhoucke & Maenhout (2005) goal on the other hand, is to satisfy the coverage requirements at illinimum costs while taking into account the nurse preferences. In constructing a nurse roster under a pre-defined scheduling horizon, Oughalime et al. (2008), Kundu & Acharyya (2008) Maenhout & Vanhoucke (2008), Bellanti et al. (2004) and Dias et al. (2003) assigned the nurses to shifts in order to meet the minimal coverage constraints to maximize the constructed timetable, they also took into consideration all the nurse preferences. Hence, the current work attempts to develop a computerized nurse scheduling system that utilises effectively the nursing personnel. The system also relies on fairness bases among nurses and considers the nurse preferences to maximize their satisfaction. This helps them provide a better quality of service (Oughalime et al., 2008; Kundu & Acharyya, 2008; Bester et al., 2007; Bellanti et al., 2004 and Dias et al., 2003). The objective function adopted by Azaiez & Sharif (2005) and Dias et al. (2003) is to minimize in particular the unnecessary overtime undertaken by the nurses. Oughalime et al. (2008), Bester et al. (2007) and Dias et al. (2003) also improved the nursing staff efficiency by minilnizing the overstaffing or understaffing that are expected to happen if the generating schedule is done manually. The primary objective of the NRP by Parr & Thompson (2007), Kellogg & Walczak (2007) and Bester et al. (2007) are io ensure there are sufficient nurses on each shift. The main objective is to ensure there is sufficient staffing to provide adequate patient care and ensure service continuity among the nurses.

27 2.2 Constraint criteria The rostering of full time staff can be a very complicated problem because this is a highly constrained task. A set of nurse rostering constraints must be satisfied as much as possible to get a consistent assignment of different shift duties, for a set of nurses, over a fixed period of time. Some constraints are hard, whereas others are soft (Thornton, 1997). The hard constraints must be respected at all costs while the soft ones are only desirable (Hertz, 1991). Classifying constraints into hard and soft constraints as well as assigning important weights have been made through consultation with the head nurses who are in charge of the nurse scheduling (Azaiez & Sharif, 2005). The results are applicable to real life problems because both the hard and soft constraints are considered (Oughalime et al., 2008; Kundu & Acharyya, 2008; Bester et al., 2007; Bellanti et al., 2004 and Dias et al., 2003). The constraints of the NRP make it unique within the field of staff scheduling. The situation is further complicated by the existence of different policies and circumstances within different hospitals and in different wards (Thornton & Sattar, 1997) Hard constraints The following hard constraints are considered in the previous studies: i. Each nurse cannot work more than k consecutive days (Bellanti el al, 2001) Assign each week a weekly off-days (Kawanaka et al., 2001; Azaiez & Sharif, 2005; Bard & Purnomo, 2007; Dias et al., 2003; Brucker et al., 2007).

28 Avoidance of working shift pattems that rnight adversely affect nurses' health (Ikegami & Niwa, 2003). iv. Minimum nurse skill level required for each working shift (Bellanti et al., 2001; Ikegami & Niwa, 2003; Bard & Purnorno, 2007; Azaiez & Sharif, 2005). v. Each nurse takes on only one working shift (Li et al., 2003; Azaiez & Sharif, 2005; Moz & Pato, 2005; Brucker et al., 2007; Kundu & Acharyya, 2008). vi. Minimum number of night shifts (Kawanaka et al., 2001; Azaiez & Sharif, 2005; Brucker et al., 2007). vii. Every nurse is not permitted to work night shifts followed by morning shifts or evening shifts (Kawanaka et al., 2001; Li et al., 2003).... vrr~. The roster must follow the biological clock. For example night shift should follow after evening shift and evening shift should follow after morning shift (Kundu & Acharyya, 2008; Kawanaka et al., 2001) Soft constraints The following soft constraints are considered in the previous studies: i. The nurse preferences for longer off days (Li et al., 2003; Kundu & Acharyya, 2008) Avoid assigning a morning shift or evening shift followed by a night shift (Azaiez & Sharif, 2005). iii. Avoid isolated working shifts or off days (Azaiez & Sharif. 2005; Bard & Pumomo, 2005). iv. A balanced number of morning, evening and night shifts must be made available to all the nurses (Kawanaka et al., 2003; Bellariti et al., 2004).

29 2.3 An overview of earlier works Up to date, many enhancements have been made to solve the NRP but it is still hard to obtain the best nurse duty roster. Nevertheless, many researchers are still trying hard to overcome these challenges. We feel the same way especially when dealing with special and complex problem situations or environments. Further discussions on the NRP are presented in this section to study on earlier work conducted, thus gaining some insights into the issue. Next, the existing approaches to nurse rostering are considered in more detail, through a review of the current literature. The optimal or near-optimal solutions are acquired through various appropriate solution approaches. Some previous works done (Ramli, 2004) has been classified in Table 2.1. The literature reported is from the 1990s to The classifications are optimization, search, constructive heuristics, and hybrids approaches. Ramli (2004) has concluded that the more recent researches are focused on the search and the hybrid techniques. This is due to potential enhancements in the techniques and their efficient performances.

30 Table 2.1: Classification of nurse rostering models by solution approaches excerpt from Rainli Authors Optimization Isken & Hancock ( 1990) Ha~~neier (I 991) 1. P Hung (1991) * Kostreva &.lenings (I 991 ) M IP Ozkal-ahan (1 991) Chen & Yeung (1992) Khoong & Lau ( 1992) * Okada ( 1992) Randhawa & Sitompul(l993) Dannoni et al. (1 994) Hung (1 994a) * Hung ( l994b) Lazaro & Aristondo (1995) Weil et al. (1995) Berada et al. ( 1 996) Kusumoto (1 996) Venkataraman & Brusco (I 996) Van Wezel & Jorna (1 996) Bailey et al. (1997) Cheng, Wu & Lee (1997) Hare ( 1997) # Liao & Kao (1 997) Chan et al. (1998) Dowsland ( 1998) Jaulnard ct al. (1998) Millar & Kiragu (1998) # Nonobe & lbaraki (1998) Abdennadher & Schlenker (1999) Burke et ai. ( 1999) Huarng (1 999) Aickelin & Dowsland (2000) Nonobe & lbaraki (2000) Valouxis & Housos (2000) Burke et al. (200 1 ) MIP (2) MIP Search SAiGA CP C I' TS Constructive Heuristics c 1 CH Hybrids - LP+H+SA CP+RM Note: * represents models with structures built for cyclic schedule only. # represents models built with both cyclic and non-cyclic structures. Key to abbreviations: LP = Linear Progra~nlning IP = Integer Programming MIP : Mixed-Integer Programming CP = Constraint Programming GP = Goal Programming NP = Network Progra~n~ning RM : Redundant Modeling ES = Expert System SA = Simulated Annealing GA = Genetic Algorithm MA = Memetic Algorithm TS = Tabu Search H = Heuristics CH - Constructivc Heuristics MP = Mathematical Programming GP = Goal Programming

31 2.4 Previous work In consequent, this thesis attempts to review some of the previous works with the similar classifications due to solution approach. The solution approaches adopted in previous works are classified into four main types of approaches. The types are optimization approach, search approach, constructive heuristic approach and hybrid approach. 2.5 Solution approaches Based on Table 2.2, the classification of recent nurse rostering models by solutions approaches which had been done since 2001 to date. There are many recent researches still focused on the search approach and hybrid approach. Details of the every solution approaches are discussed in the following subsection. Table 2.2: Classification of recent nurse rostering models by solution approaches Authors Optimization Searcl~ Constructive Hybrid Heuristic Bellanti et al. (2001) TS Kawanaka et al. (200 I ) Dias et al. (2003) Li et al. (20031 lkegalni & Niwa (2003) Bcllnnti et al. (2004) Bu1-k~ et al. (2004) Aickelin & White (2004) Aickelin & Dowslantl (2004) Li gl AicLeli11 (2004) Gu,ja111. gl Rauner (2004) Raramil~ (2004) Mot gl Pato (2005) Bard 6r Pu~liomo (2005) Azaiez & Al Shn1il'(2005) GP ACO MA G A Mat~li~.ws (2005) 1.P Beddoe gl Petlovie (2005) Zasegawa & Kosugi (2006) Pan- gl Thompson (20071 Aickelin & Li (2007) Bester et al. (2007) Bard gl Pumomo (2007) Kul~tlu & Acha~yya (2008) Oughalilne et al. (2008) Ohki el al. (2008)

32 2.5.1 Cyclical and non-cyclical scheduling Both heuristic and optimization approaches have been developed for the cyclical and non-cyclical cases of the NRP. A cyclic schedule consists of a set of work patterns which is rotated among a group of workers over a set scheduling horizon (Nlillar & Kiragu, 1998). In the non-cyclical scheduling, the scheduler generates a new schedule for each scheduling period throughout the planning horizon with available resources and policies (Ramli, 2004). As shown in Table 2.3, most of the heuristics model focused on solving the cyclical scheduling process. Table 2.3: Classification of earlier scheduling models by techniques Types of Scheduling Procedures -- scheduling Cyclical Heuristic -- Optimization Li & Aickelin (2004) Azaiez & A1 Sharif (2005) Moz & Pato (2005) Aickelin & Li (2007) Bard & Pumomo (2007) Kundu & Acharyya (2008) Non-Cyclical Bellanti et al. (2001) Ikegami & Niwa (2003) Gutjahr & Rauner (2004) Bellanti et al. (2004) Cyclic schedules require relatively less scheduling effort than non-cyclic schedules. The main disadvantage of a cyclical schedule is its rigidity and general inability to adapt to changes in scheduliilg demands. Non-cyclical schedules address this major shortcoming of cyclic schedules by frequently rescheduling the workforce (Millar & Kiragu, 1998).

33 Ramli (2004) claims that the major advantage of non-cyclical schedules is flexible, which can overcome the inflexibility of cyclic schedule. However, it is timeconsuming and typically produces unstable schedules with imbalanced coverage. However, Ramli (2004) used the combination of both benefit with semi-cyclic scheduling approach to the NRP Optimization approach Optimization approach is a mathematical programming approach, which some objectives to minimized or maximized subject to a set of constraints on the possible solution sets (Sitompol & Randhawa, 1990). These include the linear programming (LP), the integer programming (IP), the nonlinear programming (NLP) and the goal programming (GP). Several of these approaches had been employed in earlier works on the NRP. The linear programming (LP) uses a linear equation as the model to obtain a maximum or minimum outcome based on a series of constraints. This optimization approach is used by Mathews (2005) to solve the nurse's capacity problem. They used the LP as a tool for determining the effective combination of nurses that allow for all weekly tasks to be covered while providing the lowest possible cost.

34 The goal programming (GP) is a form of integer and linear programming where more than one objective (or goal) is optimized in the objective function. The GP is the optimization approach that had been used by Azaiez & A1 Sharif (2005). They developed a model for nurse scheduling using the 0-1 LGP approach to improve the manual schedule at the Riyadh Al-Kharj Hospital. They considered the hospital objective, that is to reduce overstaffing (and hence overtime cost), as well as incorporating the nurses' preferences and establishing fairness bases among nurses. However, this approach does not always work well. Some problems are so complicated that it may not be possible to solve with an optimal solution. In such situations, it is still important to find a good feasible solution that is at least reasonably close to being optimal. Search approaches are coininonly used to search for such a solution Heuristic approach A heuristic approach is a procedure that is likely to discover a very good feasible solution, but not necessarily an optimal solution. This approach can be sufficiently efficient to deal with very large problems. Two main reasons to justify the use of heuristics in this context are since the problem is NP-hard and heuristic approaches are more flexible, they enable one to more easily handle specific features to be found in real cases, besides the sporadic alterations in these features (Moz & Pato, 2005). Much research work has been con~pleted to solve the NRP heuristically. Heuristic approaches are quiet often used because they tend to be faster and in most practical settings, a feasible schedule is usually acceptable (Millar & Kiragu, 1998). Heuristics can conduct the search through the search space.

35 2.5.4 Search approach A search approach is used in implementing the general solving method based on real-world problem. There are several types of specific approaches for searching for the efficient nurse schedule. These approaches are adopted from the heuristic mechanism. The search approach consists of meta-heuristics and constructive heuristics. In recent years, there is a class of general heuristics strategies called meta-heuristics, which have been implemented by many researchers and have proven to be successful in their experiments. Meta-heuristics is also called the generic heuristic methods or general local search methods (Taillard et al., 2001). This method enables the solving of large scale scheduling problems with signiilcant savings in computational time. The recognition that most real life rostering problems are NPhard has led researchers to explore the meta-heuristics approach. Various metaheuristics approaches have been developed for nurse rostering with some success (Beddoe & Petrovic, 2004). Meta-heuristics often use randomized searches including the tabu search (TS), the genetic algorithm (GA), the simulated annealing (SA), the ant colony optimisation (ACO) and the memetic algorithm (MA) have been proven to be very efficient in obtaining near optimal solutions for the NRP (Cheang, 2003). These approaches are different for every work; depending the on users' opinions and requirements.

36 i. Tabu search In its most basic form tabu search (TS) can be described as a local search procedure that allows uphill moves by selecting the best move in the current neighbourhood even if it is non-improving (Dowsland, 1998). It consists of building blocks such as neighbourhood, short term memory (embodied in the tabu lists), long term memory and aspiration criteria (Nonobe & Ibaraki, 1998). TS is an enhancement of the basic local search concept that employs memory to avoid getting trapped in local optima (Michalewicz & Fogel, 2000). The memory contains a list of recent moves within the search space (e.g. swapping the assignment of two nurses) that may not be performed again for a set number of iterations. This concept of tabu moves gives the potential for short term degradation of solution quality allowing algorithms to explore new areas of the search space. Glover (1997) has found the idea of basic tabu search. The philosophy of TS is to derive and exploit a collection of principles of intelligent problem solving. TS is a meta-heuristic method that has been proved successful in solving many practical optimization problems (Glover, 1989; Glover, 1995; Nonobe & Ibaraki, 1998; Cheang et al., 2003; Dias et al., 2003). TS is a different and always better procedure to use, involving the use of possible non-monotonic moves in an attempt to stay away from a possible local minimuin and thus achieve a better overall solution (Voiouxis & Housos. 2000). Indeed, only a few exchanges are required then; hence a small computation time is required (Berrada et al., 1996).

37 A practical NRP was considered by Bellanti et al. (2001), which arose at the ward of a hospital located in Turin, Italy. They developed the nursing software to improve the manual schedule as well as considered the coverage of the working shifts with particular emphasis on the night shifts. They proposed the TS approach based on the neighborhood that operates on partial solutions in order to avoid the generation of unfeasible solutions. Dias et al. (2003) developed and implemented the automated nurse schedule to consider a large facility at the School Hospital of the University of Campinas, Brazil. They considered minimizing the extra working hours. Experimentally, the TS technique implementation used to be more time efficient in most cases. In previous work, Bester et al. (2007) adopted in their paper the well known TS technique. They constructed a nurse schedule for Stikland Hospital. This is a psychiatric facility in the South Afi-ican Western Cape with the objective of ensuring that there is always sufficient staff on duty, while attempting to accommodate individual work pattern preferences and financial restrictions. The TS approach used achieved the best results to solve the NRP. Kundu & Acharyya (2008) wanted to create instances of the IURP that were realistic and indicative of real life situations and they collected data from the well-known Peerless Hospital in Kalkata. They applied several search approaches and investigated the best method that could achieve the niirse schedule with a high quality.

38 The latest work by Oughalime et al. (2008) presented three stages of the TS heuristics for NRP. The problem is approached via the goal programming (GP) method. The GP model focuses on both hospital objectives and the nurses' preferences. The combined use of many neighbourhoods and the improvement of each solution by using a smart intensification had obtained a faster, more fitting and more precise automatic tool for this problem. The result of this paper was the ability to construct a feasible solution. ii. Genetic algorithm The genetic algorithm (GA) is an abstraction of the natural genetic evolution containing simple operations like selection, crossover and mutation (Davis, 1991). In the GA, the basic idea was to find a genetic representation of the problem so that the "characteristics" could be inherited. Starting with a population of randomly created solutions, better solutions were more likely to be selected for recombination into new solutions. In addition, new solutions may be formed by mutating or randomly changing the old ones (Cheang et al., (2003). In a previous work, Aickelin and Dowsland (2004) applied the GA approach to a manpower scheduling problem arising at a major UK hospital. Although the GA has been successfully used for similar probleins in the past, they always had to overcome the limitations of the classical GA paradign in handling the conflict between objectives and constraints. The approach used indirect coding based on pennutations of the nurses and a heuristic that built schedules from these permutations. The author found that, there were still some problems appearing,

39 where the algorithm had difficulty in finding feasible solutions because it was too disruptive to make any minor changes necessary in improving the algorithm. Dias et al. (2003) investigated experimentally the result that using genetic approach produced less time efficient compared with the TS approach. Experimentally, the TS technique implementation used to be more time efficient in most cases. The approach is proposed to minimize the extra working hour. The research reported by Moz & Pato (2005) is part of the design to develop a system for the management of nurse schedules for implementation in Portuguese public hospital. The specific problem of rebuilding nurse schedules was addressed when unexpected staff absences arose. They proposed several versions of the GA, whose difference lay on the encoding of permutations and in the genetic operators used for each encoding. The results analyzed were that most of the GA versions are time consuming tasks. Kundu & Acharyya (2008) investigated whether the GA or other local search performed the better solutions to consider the constraint satisfaction problem (CSP), involving a set of constraints. The experimental results showed that the GA approach failed to satisfy all the soft and hard constraints in most of the cases. iii. Simulated annealing The SA mimics the physical process of annealing, which refers to a prcrcess of heating a material to a temperature. Then it is cooled very slowly in order to reach the desired low-energy equilibrium (Bailey et al., 1997).

40 Parr & Thompson (2007) had proposed the SA approach to overcome the NRP. Their primary objective is to ensure that there are sufficient nurses on each shift. Their survey showed that as the importance of constraints differ between hospitals, the weights placed on them will differ also and this may have a significant effect on solution quality. The final experimeiits showed that it was difficult to find the feasible solution and the processed run times were so lengthened. Kundu & Acharyya (2008) investigated either the SA or other local search which performed the better solution to consider the CSP. The experimental result also showed that the SA approach failed to satisfy all the soft and hard constraints in the most of the cases. iv. Ant colony optimisation The ant colony optimisation (ACO) is a recent proposed meta-heuristic that was inspired by the behaviour of real ant colonies which enabled ants to find the shortest paths between food sources and their nests. While walking fi-om food sources to the nests and vice versa, ants deposit s substance called pheromone on the ground. When they decided about a direction to go, they chose with higher probability the paths that were marked by the stronger pheromone concentrations. This basic behaviour is the basis for a cooperative interaction which leads to the emergence of the shortest paths (Dorigo & Blum, 2005). Guijahr & Rauner (2004) applied the ACO to the NRP, analyzing a dynamic regional problem which is currently under discussion at the Vienna hospital compound. Each day, a pool of nurses had to be assigned for the following days to

41 public hospitals while taking into account a variety of soft and hard constraints regarding working dates and times, working patterns, nurses' qualifications, nurses' preferences and hospitals' preferences as well as costs. Extensive computational experiments were used and the results achieved highly significant iinprovements compared to a greedy assignment algorithm. However, there are no standard solving techniques with reasonable computation times at present. v. Memetic algorithm The memetic algorithm (MA) is a hybrid of the GA and local search, which was first inspired by Dawkins (1976; 1989). Ramli (2004) designed the MA mechanism to readily tackle each conflicting instance in modular evolutionary approach (EA). In a way, the whole original problem is facilitated into smaller sub-problems by the evolutionary operators, which searched the sub-solution space sequentially. The whole original problem is in Malaysian general hospitals. The initialization solution used in this approach can fulfil the hard constraints. The limitation of this approach generated the unwanted solution repetitively Constructive heuristics approach A constructive heuristic approach began with nothing, which was no initialization of solution. Then personnel or other variabies were added in an iterative manner until all personnel requirements were satisfied (Brusco & John, 1995). This heuristic is based on the classical assignineilt problem (Moz & Pato, 2005).

42 The case-based reasoning (CBR) is a reasoning paradigm in which new problems were solved using the solutions to similar problems that had been previously encountered. Previous problems and their corresponding solutions were stored as cases in a database called case-base. New problems were coinpared to the cases in the case-base and the most similar was retrieved. The solution to the problem from the retrieved case was then adapted to the context of the new problem. If the new solution could be useful for fhture problem solving then it was stored in the casebase, thus increasing the total knowledge held (Kolodner, 1993). Beddoe & Petrovic (2005) described a method for the automated selection and weighting of features for a CBR approach for the NRP. The comparatively large number of cover violations reflects their prevalence within the rosters used to train the case-base. This approach uses a large run time to reach the feasible solution. Another research paper that used an estimation of the distribution algorithm (EDA) was proposed by Aickelin & Li (2007) for the NRP, which involved choosing a suitable scheduling rule from a set for the assignment of each nurse. The EDA was applied to implement such explicit learning by building a Bayesian network of the joint distribution of solutions. However, many of constructive heuristic approaches are time consuming to set for the assignment of each nurses.

43 2.5.6 Hybrid approach A search-based hybrid begins the scheduling procedures with any of the search techniques and then continues to improve on the solutions using other techniques (Cheng et al., 1997; Rainli, 2004). The nurse rostering problems have been solved using a variety of different mathematical programming and a number of metaheuristic approaches have been explored including the investigations of different approaches. In the research paper by Kawanaka et al. (2001) studied the method of the coding and the genetics operations with the absolute constraints for the NRP. They proposed mathematical method that could reduce the searching area of the GA drastically. They modified the solution that causes the useless search in genetic operations. The results showed that the nurse scheduling table satisfied the absolute constraints. Another research paper by Li et al. (2003) presented in their paper a hybrid A1 approach for a class of over-constrained NRP. Their approach came in two phases. The first phase solved a relaxed version of problem which only included the hard rules and part of the nurses' request for shifts. This involved using a forward checking algorithm with non-binary constraint propagation, variable ordering, randoin value ordering and compulsory back jumping. In the second phase, adjustments with descend local search and the TS were applied to improve the solution. This was to satisfy the preferences as far as possible. Experiments showed that their approach was able to solve this class of problems well.

44 However, another research paper by [kegami & Niwa (2003) introduced a mathematical programming fonnulation of the NRP in Japan, and developed a TS approach to solve the problein. This scheduling problein is a hard combinatorial problem due to tight constraints involving such factors as the skill level of a team, the need to balance workload among the nurses, and the consideration of the nurses' preferences. Both a TS and an iterated local search procedure are proposed by Bellanti et al. (2004). They considered a practical NRP, which arises at a ward of an Italian hospital. It is required to consider both holidays planning and parametric contractual constraints, but no cyclic schedules and corresponding weekly patterns are involved. The proposed approach is now currently used on site. A hybrid TS approach is presented Burke et al. (2004), that has been developed for a commercial nurse rostering system (Plane). They want to satisfy as many constraints and personal preferences as possible while constructing a schedule which meets the personnel requirements of the hospital over a predefined planning period. Another research paper by Aickelin & White (2004) modeled and solved a complex NRP with an integer programming (IP) fonnulation and evolutionary algorithm. Secondly, they set to detail a novel statistical method of comparing and hence building better scheduling algorithms by identifying successful algorithm modifications. The proposed method of analysis is not a methodological search of all possible algorithms or improveinents, but can assist in the intelligent improvement of parameter estimates and assess the effect of innovative alterations to an algorithm.

45 In the research paper by Bard & Purnomo (2005) present several ideas for maximizing the use of the available staff and to quantify the resultant benefits. The problem is modeled as an IP and solved with a column generation technique that relies on intelligent heuristics for identifying good candidates' schedules. Next, Bard & Purnomo (2007) formulated the NRP as an IP and decomposed using Lagrangian heuristic to develop the preference schedule of NRP. Solving the NRP by 1P based local searches by Hasegawa & Kosugi (2006) was the operation that corresponded to the neighbourhood search in the local search algorithm, applicable to work tables with periods of four weeks or longer. Results from simulations on benchmark problem sets for a typical hospital in Japan shows that this algorithm facilitated in creating schedules in a short time with constraints substantially satisfied. Bai et al. (2008) proposed a hybrid algorithm for the NRP that combined the GA and the SA hyper-heuristic. In this algorithm, a stochastic ranking method was used to improve the constraint handling capability of the genetic algorithm while simulated annealing hyper-heuristic procedure was incorporated in order to locate local optima more efficiently. Two key components in the hyper-heuristic layer are the heuristic selection mechanism and the SA acceptance criteria. However, another research paper by Ohki et al. (2008) proposed a mountainclimbing operator to the cooperative GA to create and optimize the nurse schedule. The problem is the complex task to create schedule because have to assigned nurses to many sections.

46 Many researchers proposed deviation of the hybrid approach that is suitable for the particular cases. However, this approach needs a large amount of effort because it uses a combination of many approaches. Therefore, this approach is a time consuming to achieve the feasible solution. 2.6 Discussion and summary The solution approaches are normally different for every work depending on the users' opinions and requirements. Many approaches have been accomplished in past decades. However, the recent trend can be viewed in optimization, search, constructive heuristics and hybrids, as compared to earlier classifications of cyclical, non-cyclical, optimization and heuristic approaches. Optimization approaches produce optimal solutions. I-lowever, they are timeconsuming if they are involved with large-sized decision variables. The heuristic approaches are quite often used because they tend to be faster and in most practical settings, a feasible schedule is usually acceptable and sufficient. The most successful approach so far is the TS since it is also able to find feasible solutions. The TS is quite established because of its solution quality and its algorithm is more efficient. The TS approach has always been presented as an open technique that can include components from various fields. The TS is only one that has been explicitly developed with a memory (Dowsland, 1998).

47 The memory property of the TS is distinct and has proven useful for many difficult combinatorial problems. The memory based penalty hnction of this paper encourages search within both the feasible region and promising areas of the infeasible region. A short term memory is the most important idea on the TS, known as the tabu list. The previous proposed approaches cannot prevent the unwanted repetitive solution in their search process. However, the TS is composed of various other principles. One of the principles is intensification and diversification phases. These phases are the neighbourhood generation processes that are often implemented by repeatedly building new solutions before running the basic TS for a given number of iterations. The new solutions built can be similar to the best solutions found by the search so far (intensification) or different fi-om the solutions visited (diversification). The neighbourhood process is to generate the current solution to search for the feasible solution. The work normally attempts to design and implement a nurse rostering system that reduces scheduling or processing time. Therefore, the appropriate initial solution is chosen to reduce the nurse rostering titne and also relate to problem environment in Malaysia. Hence, the best initial solution that can fulfil the Malaysian general hospital rule is selected from the research paper by Ramli (2004).

48 CHAPTER THREE TABU SEARCH PROCESS AND ITS COMPONENTS This chapter pursues some fundamental concepts of the Tabu Search (TS) approach. The discussion begins with the background of the TS and its concepts. The discussion continues with the procedure of TS. Finally, the chapter ends with a conclusion on the potential variant of the TS that can be examined. 3.1 A rneta-heuristic approach A wide variety of different methodologies and models have been developed to deal with different problem circumstances in the NRP during the years. These include mathematical programming, meta-heuristic methods and constraint satisfaction techniques or approaches. However, as has been identified in Table 2.2, most of the recent work on the NRP attempted various meta-heuristic approaches. Many researchers have developed the meta-heuristic procedures to solve a NRP instance heuristically in an acceptable time liinit (Vanhoucke & Maenhout, 2005; Oughalime et al., 2008). 117 recent years, meta-heuristics including TS, GA and SA have been proved to be very efficient in obtaining near-optimal solutions for a variety of hard coinbinatorial problems including the NRP (Cheang et al., 2003). This means that the existing solutions to the problem have not yet been widely applied (Sitompul, 1992).

49 Among the meta-heuristic approaches which give the best feasible solution and has proven to be very efficient on a variety of problems is the Tabu Search (Glover, 1990; Dias et al., 2003). The Tabu search technique is very efficient in dealing with various coinbinatorial probleins, which can be adopted for single or multi-objective problems. The TS approach has proven successful in solving inany practical optimization problems (Nonobe & Ibaraki, 1998; Dias et al., 2003). Glover & Laguna (1997) define the most important distinguishing property of the TS as the exploitation of adaptive form of memory. These take the form of shortterm memory strategies (i.e., tabu list and aspiration criteria) and long-term memory strategies (e.g., intensification and diversification). 3.2 Tabu Search architecture As a meta-heuristic approach, the TS technique comprises several sub-techniques or sub-heuristics which are special to the approach. In other words, the TS technique consists of building blocks which are identified as neighbourhoods, short term memory (embodied in the tabu list), long term memory and aspiration criteria (Nonobe & Ibaraki, 1998).

50 In its most basic form, the TS can be described as a local search procedure that allows up-hill moves by selecting the best move in the current neighbourhood even if it is non-improving. The use of a tabu list prevents the search returning too soon to previously visited solutions. The short nurse tabu list was vital in stopping small cycles. This short memory is effective at preventing cycling through regions of the solution space (Dowsland, 1998) Initialization of solution In general, the TS starts with an initial solution which has been previously constructed, and runs through an iterative process by means of which it seeks to improve the objective function value of subsequent solutions (Dias et al., 2003). i. Random solution Burke et al. (2004) obtained a feasible initial solution using three possible strategies; (1) use of current schedule when urgent changes in the schedule are required to avoid any drastic changes of the schedule for other nurses; (2) use of previous schedules when the constraints on the current and the previous planning periods are similar; and (3) use of random initialization.

51 Dias et al. (2003) generated a randoin initial solution. The initial solution is generated line by line and the initial solution is implemented in matrix solution. However, using a random solution is easy, but may make the early stage of the search wasteful (Oughalime, 2008). ii. Greedy algorithm Bellanti et al. (2001) generate an initial solution by means of a greedy algorithm. The greedy algorithm firstly allocates night shifts, then morning and afternoon shifts, in order to guarantee that shifts are evenly assigned. Kundu & Acharyya (2008) applied the greedy local search procedure assigns through values to variables in an effort to satisfy all the clauses in a given set. Then, Oughalime et al. (2008) also applied a greedy algorithm to get a good initial solution. To remedy this, a greedy method on four steps which starts from the solution that no variable is assigned a value is used. The four steps in the initial solution of the greedy method consists of assignments of; (1) evening shifts for each nurse that satisfy personnel requirement; (2) afternoon shifts for each nurse while trying to satisfy the soft constraints; (3) afternoon shifts for each nurse; however, if there is an isolated afternoon shift assigned for a nurse, it would be gathered with the other afternoon shifts; and (4) morning shifts for each nurse. It s!~ould be noted that all the days 11~~st be covered while trying to satisfy the soft constraints.

52 3.2.2 Fitness evaluation The evaluation function is similar with Burke et al. (2001), where each function is directed towards each type of violation. The generation of the solution is based on the objective function or fitness value. This problem takes into account the soft and hard constraints. Nonobe & Ibaraki (1998) introduce the penalty function after modifying violation in initial solutions. The appropriate weights must be determined for all of the constraints. This is an important issue because the performance of the search highly depends on the constraints. The cost of a solution is the sum of the penalties associated with the solution. The objective functions that sums up penalties that are attributed to the rule violation, this is the minimization problem (Dias et al., 2003). The roster quality is determined by the satisfaction of coverage constraints and shift pattern penalties which are in turn based on the notions of pattern quality and individual nurse preferences (Beddoe & Petrovic, 2005) Neighbourhood generation The neighbourhood generation is performed tc improve the initial solution. This is an interactive procedure moving from a current feasible solution to mother feasible solution that selected from a set of i~eighbouring solutions. Each element of neighbourhood is generated by exchanging some of the positions of the schedule.

53 i. Swapping procedures The generation of neighbour solutions are obtained by simply swapping a generic shift with another leads almost always to an unfeasible solution (Glover, 1989; Bellanti et al., 2001). A neighbour generated by Bellanti et al., (2001) applies one of four operations; (1) a new night shift is assigned to a nurse, (2) the first night of a set is moved from one nurse to another, (3) the last night of a set is moved from one nurse to another and (4) one night shift is assigned to a nurse as first or last night shift of a set, with respect to the contractual constraints. Another version of neighbourhood is presented by Dias et al. (2003), implementing two steps of the neighbourhood. In the first step, search a line from the left to right to find the maximum consecutive working shifts. Then, reposition the day off besides one of the working shifts that had been searched for in the first step. Even so, the restriction on the minimum number of working shift was not satisfied by the work schedule associated with two of the staff members. In the second step, search line by line over the matrix that encodes the current solution. Dowsland (1998) overcomes an intelligent approach of actively generating chains of moves. All pairs of moves is between nurses. A chain of move is to increase the covering of any under-covered shift, decrease the covering an over-covered shift, and leave the remaining coverage unchanged. However, this covering constraitit is not reasonably tight.

54 However, Ikegami & Niwa (2003) decide that the neighborhood around the current solution is a set of schedules that can be obtained by exchanging shifts between any two nurses for a single day. The resulting schedule using a set of actual data satisfied a vast majority of the constraints, but this algorithm could not completely remove the violations. Then, they found the feasible shift patterns for each nurse by replacing some of the nurse's day shifts with days-off for each nurses. Initially, Oughalime et al. (2008) used a combination of three types of neighbourhood. The first neighbourhood N (S) is based on stints. A stint is defined as a work-stretch or off-stretch. Once the different stints are assigned to the nurses, a partial solution to the problem is obtained. The second neighbourhood N (A) is based on afternoon shifts. The approach is to try to assign for nurses a pattern equal to the minimum request of afternoon shifts per schedule. 'The partial solution is improved by gathering afternoon shifts for each nurse. The third neighbourhood N (M) involves moving from schedule (full solution) to schedule by changing the morning shift patterns of a single nurse or more than one nurse by a single shift or more. ii. k-opt procedures Valouxis and Housos (2000) optimized the initial solutions using a local search with a 2-opt neighbourhood and a tabu search. Their inethod coinpared very favourably with a coilstrained prograinining approach.

55 The 'best' neighbourhood solution is selected as the next iterate. At each iteration, the method searches for the best candidate in some neighbourhood of the current solution. In order to avoid cycles, the algorithm keeps a dynamical list of prohibited moves and at some point, migrate to a solution that is actually worse than the current solution. The iteration proceeds until some halting criterion is met (Dias et al., 2003) Tabu list Nonobe & Ibaraki (1998) defined tabu list as the set of solutions obtained from the current solution by the moves that use some attributes in iteration. This inemory forbids solution attribute changes recorded in the short-term memory to be reused. Hence, a cycling or repeating in using the current local optimum schedule is prevented (Berrada et al., 1996; Nonobe & Ibaraki, 1998; Ikegaini and Niwa, 2003). Tabu tenure is a parameter that describes how much of the past that should be remembed (Nonobe & Ibaraki, 1998; Cheang et al., 2003). Glover & Laguna (1997) define the most iniportant distinguishing property of TS as the exploitation of adaptive forms of memory. These take the form of short term memory strategies (i.e., tabu list and aspiration criteria) and long tenn lnemory strategies (e.g., intensification and diversification). The feasible solution by using this approach provides an acceptable time limit.

56 The tabu list is initially einpty and is implemented as a FIFO queue (Kundu & Acharyya, 2008). Dowsland (1998) proved through experiments suggested that a short list gave the best results and a length of six has been implemented. Bellanti et al. (2004) identified various cases consisting of the tabu list, and an incorporation of an objective function with the automatic control mechanism based on weight. The tabu list in the neighbour generation is used to avoid the cycle in generating schedule. It is a short memory and some research experiments that the best tabu list length is six to seven. The iteration is stopped until the feasible solution is found. Oughalime et al. (2008) used three tabu lists associated with the three types of neighbourhood. Each time a diversification on the each neighbourhood is done, each of the tabu lists is emptied. This is to ensure it will not contain any future works Aspiration functions Aspiration function allows the search to override the tabu status of the solution and also provides backtracking of recent solutions as they lead to a new path towards a better solution (Oughalime et al., 2008). If a tested solution is better than the current solution, then it is selected to be next current solution. This step is caiied the aspiration function (IVonobe & Ibaraki, 1998). Bellanti et al. (2004) used a standard aspiration criterion. The unwanted schedule in the tabu list is accepted if it is inore feasible than the current best schedule. Oughalime et al. (2008) identified the aspiration criterion is used when no new solutions is found for a number of successive iterations.

57 3.2.6 Stopping criterion Oughalime et al. (2008) developed two stopping criteria. The first, if after r diversification on the stint neighbourhood there is no improvement (for the full solution), and then the algorithm is stopped. The second, the algorithm will be stopped as well if it reaches the maximum number of iteration given by the user. The maximum number of iteration is the number of visited full solution either the solutions are improved or not. The stopping criterion by Bellanti et al. (2004) is set to stop after 50 iterations without improvement. 3.3 Summary In developing the TS model to solve the NRP, it must be based on its principles and concepts. This chapter sets to highlight those fundamental as the basis for supporting this research. In consequence, the nurse scheduling model can developed with the idea of the TS approach that has been discovered before and give the advantage in searching for the solution.

58 CHAPTER FOUR METHODOLOGY This chapter is concerned with the proposed solutions that are necessarily needed to solve the problems. This chapter presents the design methodology for achieving the specific research objectives as laid down in the initial schedule. It sets to (1) develop a model to solve a particular NRP, (2) minimize the imbalances of coverage over the assignment of nurses in different shifts in a day in each skill level, (3) maximize the nurses preferences, so that job satisfaction among the nurses is increased, (4) to minimize the deviation in the number of different shifts arrangement for each nurse, (5) to evaluate the proposed model. To achieve the specific research objectives, this chapter will take a look at the design methodology. The fundamental of the scheduling technique used in this study is the TS technique. In the real life situations, it is crucial to find a feasible nurse schedule. This chapter discusses the problem environment, the TS concept and the proposed TS solution framework. The chapter ends with the evaluation of the proposed TS technique. 4.1 A case of an NRP This research is about the case study that has been discovered before. In this case, the solution of the NRP is related to the nurse scheduling environment in any general hospital in Malaysia. Every general hospital in Malaysia has the same constraints or objectives to develop the nurse schedule. For this reason, one of the

59 general hospitals selected in this research to find the best schedule that will satisfy the objective or the constraint of the NRP. The objective is to meet the nurse roster that satisfies the nurse preferences as much as possible so the job satisfaction among the nurses is increased. Currently, the TS approach is used to the IVRP. The search process is one of the TS algorithm purposes that must consider all of the NRP constraints. The solution algorithm is developed to solve the NRP. The nurse scheduling system is programmed by the Sharp Develop System. #develop (short for Sharp Develop) is a free IDE for C# and VB.NET on Microsoft's.NET platform. It is an open-source, and you can download both source-code and executables from this site. The TS technique is a very practical technique for searching the feasible nurse roster that must consider a large number of constraints (Cheang et al., 2003). This is a heuristic local search procedure whereby a sequence of the potential nurse schedule is updated iteratively. The iteration in TS algorithm is repeatedly applying modifications to attribute of the previous schedule in the sequence. The two weeks nurse schedule is adopted by Berrada et al. (1996) to obtain the quality solution and used for computational efficiency. Most current literature shows that the schedule is presented for one or two weeks or even a one-month schedule. However, in the real world, it shows that the schedule is presented in two weeks be-.,.. ~aclse a one-week schedule is unworkable to create in a time frame. The one- month schedule is also impossible to create because it is timeless and difticult to make a change to the schedule.

60 4.2 Research process Figure 4.1 below shows the research process for the NRP that is considered in this chapter. There are four phases for achieving the research objectives. It consists of data collection, tabu search architecture, model evaluation and implementation and the results. The details of these phases are discussed in the following section. Data collection Phase 2: Tabu search architecture Phase 3: Model evaluation Phase 4: Implelnentation and result Figure 4.1 : The research process for the NRP 4.3 Data collection Data source is a secondary data, which is in tenns of constraints about a particular problem environment. These constraints were excerpted from existing literature (Ramli, 2004). The constraints taken into account in this data source can be classified as the hard constraints, which nlust not be violated, and the soft constraints, which would be pleasing to satisfy but can be violated, if necessary. It is difficult to strictly follow the classification and reach the satisfactory schedule manually.

61 4.3.1 Number of staff There is no specific number of staff in each ward in the hospital. The number of staff considered in this schedule is 39 nurses, which is a large ward at a Malaysian general hospital Types of shifts There are three types of shifts in a day, such as the morning shift (M), the evening shift (E) and the night shift (N), as shown in Table 4.1. The M and E shifts are usually regular shifts, which carry seven or eight hours of nursing duties each. The N shift can be varied with equal to or longer hours than the M or the E shifts. For example, the M shift can start from 7.00 am to 2.00 pm, the E shift can start from 2.00 pm to 9.00 pm, and the night shift can be from 9.00 pm to 7.00 am. Table 4.1 : Working shifts A p e Symbol Working shifts Moniing M Evening E N&t N -- A nurse can be assigned only one shift per day and normally the overtime shift is not allowed. In this environment, it is possible that there is no official allocation of rest meal breaks during work shifts. Nurses can take their meals in their units only and within that shift hours depending on allowable situations. Due to that, we do not consider the meal breaks allocation in our system.

62 Details of the further specifications are discussed below (Ramli, 2004). i. Mandatory workdays constraints The fulltime nurses must get one day off per week. Therefore, they need to work six days in a week. Other special days off also occurred that followed after the existence of N shifts. ii. Covering constraints The required number of nurses for each shift and for each skill level in a day has to be met. This is really a hard constraint, and a unit would operate inefficiently without it. iii. Work stretch constraints In the nurses' personal schedules, consecutive work days must not exceed certain maximum days and must not be less than the minimum days limit. If this constraint is violated, the maximum work stretch will exhaust the nurses and the minimum work stretch will create uneasiness and reluctance on their part to go to work. iv. Ordering constraints The quality of the schedule is reflected by the assigning of the shifts in a particular order. Therefore, the nurses' well-being can be assured. The sequence of shifts in a day is in this order, which is M + E + N. It means M must occur before E and E must ciccur before N. It is considered forbidden if the sequence is reversed because we must respect the monotonic or circadian rhythm rule.

63 v. Pre-assigned constraints Three N shifts start the assigned shifts in cyclic order. Then two nights off (NO) days are assigned following the three night shifts, as shown in Figure 4.2. Note that a nurse must have a day off after working for three consecutive night shifts (Valouxis & Housos, 2000). This pre-assigned constraint is in staggered and rotation manner. Therefore, every nurse gets an equal number of N shifts, in such a way that they are equitably distributed throughout the planning horizon. N N &, N - N N N O N O ns N N N NO NO I I ~~~~~ Figure 4.2: The allocation of N and NO vi. Split days off constraints Split days off is defined when any two off days are separated by a single working day. This fonnation is forbidden and needs to be avoided because it promotes negative effects on the nurses, such as feelings of unfairness, laziness and absenteeism (Rarnli, 2092).

64 4.3.3 Design of shifts The N shifts has assigned in cyclic order in the beginning of the scheduling process, so the M and E work stretches are assigned together, followed by this assigning process. Therefore, it must observe the distributions of M and E shifts across a scheduling period. The M and E shifts are expected to be equivalent for each nurse Types of off days Table 4.2 shows the types of off days such as night off day (NO), weekly off day (WO) and, public off day (PO). The two NO is automatically followed by the three N shifts for every cyclic order. WO is one day off weekly for each nurse. Other types can be PO (holidays), which generally are distributed in a year or some special occasion holidays. It shows that consecutive days off are much preferred by every nurses whenever possible. Table 4.2: Off days A p e Symbol Off days Night off day NO Weekly off day WO Public off day PO Categories of staff The ward represents a typical scheduling problem evident in many large hospitals throughout Malaysia, including privately operated and teaching hospitals. The ward is determined to improve the quality of services provided through the increase in the number of higher skill grade staff and the freezing of lower skill grade staff intake. Therefore, the whole staff nurses is re-categorized into experienced and less experienced sub-groups or also known as senior and junior level (Ramli, 2002).

65 4.4 Tabu search architecture In order to adopt the TS methodology, an initial solution is required. To generate that initial solution, some heuristics will be used to develop flexible and reliable solutions relevant to an actual hospital situation. A framework on the whole process of the organization of the TS is shown in Figure 4.3. c Construct initial schedule 1 Create neighbourhood = Ai 1 Choose best neighbourhood = min Ai 4 Keep unwanted schedule in tabu list = Ai ' Yes 1 Print Report 1 I (3 Exit Figure 4.3: The developme~lt of TS for the NRP

66 4.4.1 Solution representation In order to construct the feasible nurse roster, some solution framework must be represented appropriately. Due to its complexity and relevance, the nurse rostering system is implemented. This nurse programming can be found in Appendix A. The model is based on the tabu search (TS) technique. The construction of the nurse scheduling is a two-dimensional matrix configure, in which the rows represent the nurse' personal schedules (ni) and the columns represent the days (d;) in the scheduling period (Figure 4.4). The row is divided into two levels; consisting of the junior and senior levels. The number of nurses can be adjusted. The dimensions of the matrix depend on the number of nurses and the number of days. The elements assigned in the matrix cells are representation of working shifts and the appropriate off days. Figure 4.4: The solution encoding and representation

67 4.4.2 Initialization of the solution The first task is setting the starting point for the schedule, which is their initial schedule in matrix form. The initial matrix is generated line by line. There are some choices that a user has for the starting point. The semi-random initialization start is to give the nurses exactly the schedules they want. In this work, a semi random initialization is applied to get a high-quality of initial solution to maximize the satisfaction of nurse preferences (Burke et al., 1998). This indicates the nurses' assigned to work in their rotations and any shifts that they requested. Therefore, it can improve the job satisfaction among the nurses and it can also meet the hospital's requirements. Firstly, the three N shifts start with the assignment in cyclic order. Then the assignment of two NO days follow the three N shifts. Then, randomly assign one WO day for every week. Put in front of the WO day with one PO day for every two weeks. Therefore, it can generate longer consecutive days off such that it fulfills nurse preferences. The other working shifts are M and E shifts which are assigned together for each row in the schedule. They fill up the empty cells raildomly with M and E work stretches based on circadian rules obtained from a list. This is also depends on the available cell blocks and work stretch constraints (Ra~nli, 2004). Every nurse is assigned the working shifts with satisfying working shift type (Valouxis & Housos, 2000). The patterns of M and E shifts must satisfy the nurses preferences. Furthermore, the numbers of M and E shifts are also equivalent for every nurse.

68 Next, the fitness evaluation is used to evaluate the completed nurse schedule, as discussed in the following subsection. The components count, record and give penalty values or weights to the respective violations. The initial schedule is considered the hard violations and the weights are given to the soft violation that occurs in the solution Fitness evaluation The penalty levels set depend on the aspiration of the whole workable schedule. However, this aspiration level can be appropriately determined through the scheduler's vast experience (e.g. the head nurse). This objective function can be presented in two aspects in which the targeted hard requirements and soft preference constraints violation are merged. The function can be viewed as the vertical and horizontal checks of a schedule (Ramli et al., 2002). The general objective function for the schedule f (s) is shown below, and the weight penalty constraint type q is MI, with the number of constraints violation for type q in the schedule, where s becomes: c, (5) is tine overall function in 4.2. The function in 4.2 is embedded in 4.1. The number of constraints violation for type p in the schedule is zero, c, (5) = 8 if there is no violation constraint type q. The description of function 4.2 is given as follows:

69 Decision variables: 51 if nurse iof skill level y werks shift pattern rr. bprm - (0 nth~r~vse c,:c 1 if constraint vpe 4 for skill level p in each day j exists = ( 0 nth~npise - $1 if constsainttype qexists for each nurse A ';" -0 otherwise Parameter: I = number of nurses P = number of skill levels I = number of days in the scheduling period Q = number of constraint types hf = number of possible shifts patterns cv,:, = weight or penalty cost for the relative decision variable b,,,,? w,!, = weight or penalty cost for the relative constraint c,,,, ir1;, = weight or penalty cost for the relative constraint.rl,

70 Objective functions: The fitness function describe above is reflected in the following three objectives as follows: Objective I : To minimize the imbalances of coverage over the assignment of nurses in different shifts in a day in each skill level. In other words, to maximize the number of nurses assigned to each shift (nurse coverage) in each skill level. Objective 2: To minimize the nurses' dislike (negative preferences) over shifts arrangement. Objective 3: To minimize the deviation in the number of different shifts arrangement for each nurse. In other word, to ensure equitable shifts distribution. Hence, the overall objective function 2, of a solution s is the sum of three sub- functions as follows: To minimize

71 Based on the intuitive judgment by Ran~li (2004), the relative penalty costs or values are chosen in such a way that they reflect the relative importance of dissatisfying different kinds of constraint. The idea behind this is that, the higher penalty cost, the higher is the pressure for the TS operator to remove the constraint, or even discards the affected individual Neighbourhood generation Two different types of swapping (neighbourhood) were considered in the TS algorithm consisting of the nurse skill level swap and shift pattern swap. i. Nurses skill level swap Nurse skill level swap k%'(t) is by swapping a row in junior skill level with a row in senior skill level, which at the same time does not violated night shift constraint, a row represent a particular nurse roster throughout the scheduling period of two weeks. This is based on assignment of at least 3 nurses for each shift type, for each nurse skill level on each day. The nurse skill level consists of both the senior level and junior level. The possible problem which occurs in the two weeks of NRP is not enough shift types for the nurse skill level. In each day, there must be at least 3 nurses for every shift types (morning, evening and night) in each day. The probleln occurs because the assigning work days and off days are in row by row, considering the hard and soft constraints. The night shift type is enough for each day because the three night shifts are assigned in cyclic

72 order. The other types of work days such as the morning and evening shifts are not enough for a few days, mainly for the senior nurse level only because the number of senior nurse level is smaller than the number of junior nurse level. In order to execute enough shift types for the senior nurse level, swapping the arrangement of work days and off days is done. The swapping is done between the senior nurse level with the senior nurse level and both nurses have the same cyclic night shift order, as shown in Figure Figure 4.5: The possible representation of solution 1.2'(1) for a partial solution

73 ii. Shift pattern swap The morning and evening shifts are assigned together in the schedule. The pattern of shift pattern swap L%F('~) is based on the nurse preferences. The In this swapping method we exchange shift patterns with equivalent length. The shifts types involve in the pattern are only M and E shifts. The problem occurs because there are many available patterns of morning and evening shifts which are assigned to a null cell in the initial schedule. The total number of morning and evening shifts in a few personal rosters is not roughly equivalent. The way to execute this problem is by moving between the two patterns of morning and evening shifts that is the same number of morning and evening shift blocks and different personal roster, as shown in Figure 4.6. The moves may be equivalent to the total number of morning and evening shifts. two nurses I Exchanged the pattern shifts between them. Figure 4.6: The possible representation ~f solutioil W (p) for a partial solution All the hard and soft constraints are considered at all moves (neighbourhood). The number of shift types (morning? evening and night) inay be roughly balanced in each persolla1 roster (nurse) and in each day.

74 4.4.5 Tabu list A tabu list of the most recent exchanges used is updated by each iteration to forbid inoving back to the most recent solutions encountered in the search process (move sequence). This was guided by a short term memory structure, where the reversal of recent moves was prohibited in order to prevent cycling during the search. This prohibition was enforced by maintaining a list and use comparing process of the last inoves performed during the search with the unused schedule in the list, length of tabu list called the tabu tenure, as shown in Figure 4.7. Start after neighbourhood Tabu list = t 1 FIYO schedule until t = 6 Keep in garbage < Go to the next iteration 3 Figure 4.7: The process of tabu list for the TS

75 Figure 4.7 shows the process of tabu list in the TS algorithm. Once the iteration started, a sequence of several schedules was generated by the neighbourhood process. Then, the next iteration was continued by generating the best schedule from the current neighbourhood process. The empty tabu list, t will keep the unwanted schedule in the iteration. The length of suggested list (tabu tenure) is six lists (Bellanti et al., 2004). Therefore, the sequence of unwanted schedule was saved a while (short term memory) by first in and first out (FIFO) keeping processes. Then, the unwanted schedule will keep in the garbage if the tabu list is full. The unwanted schedule in the tabu list can be used to the next iteration if it is more feasible compared to the current schedule, and this process is called the aspiration criterion Aspiration functions An important criterion is the aspiration function. This criterion also as a rule to decrease the length of tabu lists or calls as tabu tenure with the unfeasible solution (Bellanti et al., 2004: Oughalime et al., 2008). This is effective and efficient way to minimize the time consuming to get the feasible solution. Hence, the time can be minimized and the TS algorithm is reasonable to the NRP.

76 (Start once the iteration is started \ Minimum violated schedule in the tabu list = rnin Ai ' If rnin A, ' < rnin Ai No v Yes Continue v using min A; Interchange rnin Ai with nzin Ai ' Continue neighbourhood Process Figure 4.8: The process of aspiration criterion Figure 4.8 shows the process of aspiration criterion in the TS algorithm. The aspiration criterion process is started once the iteration process is set. This process begins with choosing the minilnuln unwanted schedule in the tabu list (min A,'). Then, compare the ~ninimum unwanted schedule (min A,') with the current best schedule (min A,). If the unwanted schedule in the tabu iist is feasible than the current best schedule (min A,,' < min A,,), the rnin A,' schedule is be chosen to create the neighbourhood process and vice versa. This procedure is adopted from Nonobe & Ibaraki, 1998).

77 4.4.7 Stopping criteria This operation will stop with the number of iteration requested. The number of iteration can be adjusted depending on no more than what the current best schedule can find in the iteration process. There is no inore improvement to the current local optimum schedule, after a number of consecutive iterations. 4.5 Evaluation of schedule In any case of approach used to create a nurse schedule, the completed schedule should be checked and evaluated. Early works have been given different considerations when evaluating their models. Obviously, different quantitative and qualitative criteria have been proposed for evaluating the scheduling methodologies in hospitals. Evaluation criteria has been described by Warner (1976) and have been further discussed in Sitompul & Randhawa (1990) and Ramli (2004). In considering the approaches to the scheduling decision, the author identified the following characteristics Coverage Coverage refers to the extent to which a schedule meets the niinimum coverage requirements and provides a balanced coverage. It is the primary concern of any hospital service. Minimum coverage requirements are usually the hard constraints.

78 4.5.2 Quality Quality refers to the perceived value of the schedule for the nurses in terms of work stretch length, days off, weekends, equalization of rotation and so on. The perceived fairness in shift distributions can also be included for quality checks. If the schedule is of low quality or seems unfair, it can occur resulting in lost productivity, bad staff morale and possible strike actions Flexibility Flexibility refers to the extent to which the scheduling systein can adapt to changes in different staffing level environment. It may also be viewed from the aspect of how scheduling methodology provides the output, i.e. whether it is limited or with a choice Cost Cost is in ternls of resources consumed in making the scheduling decision. Soine of the scheduling systein includes salary and expenses as their costs, while some use the tiine coilsumed as a ineasureinent of cost.

79 4.6 Summary Due to its complexity and relevance, a nurse rostering model will be implemented. The nurse scheduling model is programmed by the Sharp Develop system. A semi random initialization is applied to get a high-quality of the initial solutions to maximize the satisfaction of the nurses' preferences. Afterwards, the TS technique successively generates the solutions. The combination of some neighbourhood generation process can improve the searching process to achieve the feasible solutions. Two different types of moves are implemented in this research. In summary, this chapter has answered the problem environment, TS concept and the proposed TS solution framework. The chapter ends with the evaluation of the proposed TS technique.

80 CHAPTER FIVE IMPLEMENTATION AND RESULTS This chapter is demonstrating a nurse scheduling model. The work is involved some amount of computations because it's potential for solving hard problems in a short time. Hence, this chapter is producing a system which could match the quality of the manual schedules that suits our problems or are even better. With powerful technology, of course it can help save the computational time. To implement the nurse scheduling system, an appropriate framework has been designed in the previous chapter. The implementation of the system is to achieve the research objective. To achieve the objective, the nurse scheduling system is developed and can generate the diversification of the nurse schedule improvement. In this chapter, the combination two different neighbourhoods has been implemented to improve the generation of the nurse schedule diversification. 5.1 Current problem environment The number of staff considered in this schedule is 39 nurses from large ward at a Malaysian general hospital. The construction of the nurse scheduling is a two dimensional matrix for two weeks schedule configure, that is assigning the w ~rk shifts and off days which take into account the hard and soft constraints.

81 The row is divided into two levels; consisting of junior and senior levels. The number of junior or senior level nurses can be adjusted. The number of nurses depends on the total number of nurses in the department. One department in the hospital has been preferred. The numbers of senior nurses are eighteen and the number ofjunior nurses is twenty one. Each solution is represented by the number of columns or the number of days is fourteen days or two weeks Shifts and constraints Table 5.1 shows working shifts and days off together with their related durations and selected symbols. If other types of days off exist, they are not considered here because it is not a consistent event (Arrange off day). The explanations of the shift types are described in previous chapter. Table 5.1 : Working shifts and off days Schedule Shift Duration Symbol Working shifts Morning 07:OO - 14:OO M Evening 14:OO - 21:OO E Night 21 :00-07:OO N Days off Night off day Nil NO Weekly off day Nil WO Public off day Nil PO Nurse category and requirement The nurse schedule must consider the nurse experience. There are different nurses with different experience. The experience nurses have been working for a long time. This car, be identified with the nurse levels consisting of senior level and junior level. The list of senior nurse level is under the list of junior nurse level in the schedule. There must be enough senior nurses for every working day to handle each type of expertise, as S~C)WII in Table

82 Table 5.2: Nurse level assigning for every working shift Shift Levels Minimum Nurses for Number of Every Level Nurses M E N Senior Junior Total The main objective is to build the nurse schedule that considers the nurse's satisfaction as much as possible. Hence this nurse satisfaction is considered as soft constraint because this constraint can be violated. The nurse schedule must follow all the rule or priority as the hospital administration arranged. This is considered as hard constraint, and it cannot be violated. The soft and hard constraints have been studied in one of the general hospitals by Ramli (2004) Constraints in consideration The constraints that cannot be violated consist of the hard and soft constraints. The constraints taken into account in this data source can be classified as the hard constraints, which must not be violated, and the soft constraints, which would be pleasing to satisfy but can be violated, if necessary. Details of the hard and soft constraints will be explained in the following parts.

83 i. Hard constraints 1. As a general rule, nurses are required to work 6 days a week. That is the reason for the entitlement of a one off day in each week. (Mandatory Workdays Constraint) 2. Consecutive work days must not exceed 6 days and must not be less than 2 days. Hence, split of days off or single work day is disallowed. (Work Stretch Constraint) 3. There must be at least 3 nurses assigned for each shift type, for each nurse skill level on each day. (Covering Constraint) 4. A nurse is allowed to work on only one shift a day. (Work Requirement Constraint) 5. N shifts are assigned in blocks of 3 shifts according to turns and rotations. (Pre-assigned Constraint) 6. Two days off must follow the third N shift of the block. (Pre-assigned Constraint) 7. For each nurse, it is forbidden to have a formation of N-M shifts in any adjacent work shifts. (Ordering Constraint) 8. The assigning of shifts must respect the forward clockwise direction rule or circadian rhythm, i.e. the M-+E-+N. (Ordering Constraint)

84 ii. Soft constraints 1. Days off are strongly preferred to be consecutive in the arrangement. (Days Off Arrangement Constraint) 2. In a stretch of six work days with no N shifts, it is preferred that the combination of M's and E's be four M's two E's, three M's three E's or two M's four E's. Therefore, the ~naxitnum number of M is two. It is vice versa with E. That is, 2 < M 5 4 and 2 I E I 4. (Shift Arrangement Constraint) 3. Silnilarly, for 5 days work stretch the preferred combinations of M and E are 2 I M 1 3 and 2 I E I 3. (Shift Arrangement Constraint) 4. It is also preferred that for a 4 days stretch, the combination be 2M's and 2E's. (Shift Arrangement Constraint) 5. For 2 and 3 days stretch, it is preferred that it either be 2M's or 2E's and 3 M' s or 3 E' s, respectively. (Shift Arrangement Constraint) 6. The total number of M and E shifts in a personal roster of the nurses (the row) should roughly be equivalent. The acceptable criteria is (( M + E) 1 2) f (1 or 2). (Shift Arrangement Constraint) 5.2 Solution representation The model of nurse scheduling is a two-dimensional matrix configure, in which the rows represent the nurse personal schedules and the column represent days in the scheduling period. The dirnelision of the matrix depends on the nurnber of nurses, and the number of days, 11. The elements assigned in the matrix cells are representations of working shifts and the appropriate off days. If a designated shift is allocated in cell a,,, then the interpretation is nurse, i works in that shift on day, j.

85 5.3 User input data Before the system can be run, some information has to be given or made available for the system to absorb it. Firstly, the number of nurses needed to be scheduled in a roster needs to be set. These are the nurses assigned to the particular ward consisting of the number of senior nurse level and the number of junior nurse level. Then the number of days needed to be scheduled is set. The number of public off day is also set if there are public holidays in the scheduling days. Figure 5.1 below shows the input data before the nurse scheduling system is started. Davs Publlc Offs I Figure 5.1 : Setting the nurse schedule formats 5.4 Applying the TS approach to the NRP The previous chapter has shown the TS process in Figure 4.3. This approach is applied in this research process. The TS technique is successively developed in this research. The following sub-topics are explained in the research development Initialization of solution The initial matrix is generated line by line. There are some choices that a user has for the starting point. The semi-randoin initialization start is to give the nurses exactly the schedules they want. All of these allocations are based on the constraints as discussed earlier in section 4.3.

86 To generate that initial solution, some heuristics will be used to develop flexible and reliable solutions relevant to an actual hospital situation. It consists of small functions as listed below and is expressed through their algorithms as explained in the following sections. At first, the three N shifts and two NO days are allocated to the schedule. This is the pre-assigned constraint and its output is shown in Figure 5.2. Figure 5.2: Allocation of the N shifts and the NO days i. Night shift allocations I. Firstly, allocation of the first nurse is randonlly carried out, in order to determine who will be assigned the N shifts first. 2. Next, allocate the N shifts 3 coluinn in a row. 3. Then three N shias with the same slots (column 1 to column 3) are allocated to the two nurses immediately after the randomly chosen nurse.

87 4. Repeat the allocation of N shifts for the next three nurses but starting from column 4 to column 6 and so on, until all the columns are tinished. ii. Allocation of the NO 1. Take each row in a roster; insert two NOS after each set of N shifts until all the columns are finished. iii. Allocation of the WO This algorithm takes care of the mandatory workday's constraint and thus allow for one compulsory off day in each week as shown in Figure Take each row in a roster of 7 columns, allocate one WO randomly in null (empty). >, Figure 5.3: Allocation of the WO

88 iv. Allocation of the PO This algorithm is optional and is activated when the PO day or holidays fall in that particular scheduling period, as shown in Figure Take each row, identify the first WO, and then insert the PO next to the WO. 2. Else, insert the PO in front of the first WO. 3. Else, insert the PO after the second WO. 4. Else, insert the PO in front of the second WO. - / t Figure 5.4: Allocation of the PO Public holidays are enjoyed by public and private sectors workers alike. The difference with the nurses is that they cailrlot expect to enjoy these holidays on exact dates. At the same time, this algorithm fulfills the nurses' preferences for off days arrangements.

89 v. Rearrange heuristics (a null cell is more than six) This requires the common sense of arranging the schedule which happens when considering the ME patterns block in the next sub-topics, as shown in Figure If there are seven null cells in between two off days, break it down into three null cells each with the middle cell being inserted with the AO. Then, insert the appropriate pattern blocks of the ME in the null spaces. 2. If there are eight null cells in between two off days, break it down into three and four null cells with the fourth cell inserted with the AO. Then, insert the appropriate pattern blocks of the ME in the null spaces. 3. If there are nine null cells in between two off days, break it down into four null cells with the middle one inserted with the AO. Then, insert the appropriate pattern blocks of the ME in the null spaces. 4. If there are ten null cells in between two off days, break it down into four and five null cells with the fifth one inserted with the AO. Then, insert the appropriate pattern blocks of the ME in the null spaces. 5. If there are 1 1 null cells in between two off days, break it down into five null cells with the middle one inserted with the AO. Then, insert the appropriate pattern blocks of the ME in the null spaces.

90 Figure 5.5: Rearrange heuristic (a null cell is more than six) vi. Allocation of M and E work stretches (the ME pattern block) This rule takes care of the work stretch constraint and the ordering constraint of the work stretch, as shown in Figure 5.6. The following allocation steps are from the pattern generator, as shown in Table 5.3. Table 5.3: The ME pattern blocks Number of The available ME pattern Blocks null cells 6 "MMMMEE","MMMMMM","EEEEEEE","MMEEEEEE,"MNIMEEE" 5 "MMMEE","MMMMM","MMEEEE","EEEEE'. 4 "MMMM","EEEE","MMEEE' 3 "MMM","EEE" 2 "MM","EE" 1 - "M","E" -

91 Take each row in order, identify blocks of consecutive null cells situated in between any two off days, with sizes larger or equal to two but less than seven. Get the ME work stretch pattern blocks from the pattern generator. The ME pattern blocks are available in the generator. Insert pattern blocks in an interchangeable manner with a random start, without completing the row even if there are still null cells blocks. If the first chosen block is from group M, then the next one should be from group E and so on. Then, move on to the next row and repeat until all designated null cells are filled. If you reach the end of the rows, then go back to the first row. Identify the null cells block and sizes. Insert the ME pattern blocks until all the rows and column are finished. If there are more than six consecutive null cells, then do Rearrange Heuristics (as explained below). Else, identify any single null cell situated in between any of the two off days. If there is none, then insert the single M or E at random in between the two off days.

92 ' t Figure 5.6: Allocation of M and E work stretches (ME pattern block) Roster satisfactory and fitness evaluation To reach the research objective, it is important to decrease the number of violated constraints. The violated constraints have provided a high penalty and the satisfy constraint has given a small penalty. Hence, this is a minimization problem. The components count, record and give penalty values or weights to the respective violations (Ramli, 2004). The first objective of the algorithm is to minimize the number of hard and soft constraint violations for each nurse, as shown in Table 5.4. This is the penalty values to evaluate constraints for every row (nurse). The total number of the M al~d E shifts in a personal roster of the nurses (the rows) should roughly be equivalent. Moreover, the days off are strongly preferred to be consecutive in the arrangement. 77

93 Table 5.4: Evaluation constraints for every row Number of constraint Evaluation Constraints Penalty Values 1 Different number of M and E is zero 0 2 Different number of M and E is 1 3 Different number of M and E is 2 4 Different number of M and E is 3 or more After NO is off day 0 6 After NO is E 7 After NO is M 8 Before N is off day (WO, A0 or PO) 0 9 Before N is M Before N is E 50 Figure 5.7 shows the nurses' evaluation constraint for nurse is row number 6. Different number of morning shift and evening shift for the nurses is 2, so the penalty value given is 20. Then, after the night off is the morning shift, so the penalty value given is 20. Hence, the total penalty value for the nurse is 40. Figure 5.7: Evaluation constraints for every row 7 8

94 The second objective of the algorithm is to minimize the number of hard and soft constraint violations for each day, as shown in Table 5.5. The following is the penalty values to evaluate constraints for every column (day). There must be at least 3 nurses assigned for each shift type, for each nurse skill level on each day. Hence, the high penalty is given if there is less shift type in each day. Table 5.5: Evaluation constraints for every column Number of constraint Evaluation Constraints Penalty Values 1 Number of M in the column is more than Number of M in the column is Number of M in the column is Number of M in the column is Number of M in the column is zero Number of E in the column is more than Number of E in the column is Number of E in the column is Number of E in the column is Number of E in the column is zero 100 Figure 5.8 shows the day evaluation constraints for day is column number 6. The number of morning shift in the column is 3, so the penalty value given is 10. Then, the number of evening shift in the column is 3 also, so the penalty value given is 10. Hence, the total penalty value for the day is 20.

95 Figure 5.8: Evaluation constraints for every column Applying the TS mechanism to the NRP The TS procedure is moving from a feasible solution of a current best schedule to another feasible solution selected from a set of a current neighbouring process. Two different types of swapping (neighbourhood) were considered in the TS algorithm consisting of the nurse skill level swap 1 ;*\I) and the shift pattern swap II7\p). Then, two neighbourhoods, -4. are created from the current best schedule, indicating when i is the neighbor in a neighbourhood. Next, choose the best neighbor which is the minilnuin toial penalty of neighbourhood, i?: l i; -4, which neighbourhood that is the minimum total grand penalty. The unwanted schedule,.-1.' in the iteration will keep in the empty tabu list, t. The length of suggested list (tabu tenure) is six lists. Therefore, the sequence of

96 unwanted schedule save for a while (short term memory) by first in and first out (FIFO) keeping process (Bellanti et al., 2004). Then, the unwanted schedule will keep in the garbage if the tabu list is full. The unwanted schedule in the tabu list can be used to the next iteration if it is more feasible compared to the current schedule. This process is called aspiration criterion. The aspiration criterion process is started once you start the iteration process. This process begin with choosing the minimunl unwanted schedule in the tabu list, mi ~1-4;. Then, compare the minimum unwanted schedule, inin A; with the current best schedule, mi?: -4:. If the unwanted schedule in the tabu list is feasible than the current best schedule (in!?; -4, < vzln _;I'), the mi^ -4'' schedule is be chosen to create the neighbourhood process and vice versa. Figure 5.9: The aspiratioil criterion process

97 Figure 5.9 shows the aspiration criterion process. At the iteration 98, the penalty of neighbourhood generated is 1630 and The unwanted penalty in tabu list is minimum than the neigbourhood schedule is Therefore, the iteration 99 used the schedule in the tabu list to generate other neighbourhoods because the unwanted schedule in the tabu list is minimum than both neighbourhoods. This operation is stopped after doing the requested iteration, k. The number of iteration can be adjusted depending on no more than the minimum current best schedule one can find in the iteration process. There is no more improvement to the current local optimum schedule. 5.5 Experiments and results The whole algorithm has been applied and tested on real data as presented in Table 5.2. This is to determine its effectiveness and performance. In this section we emphasize on the performance of the system. Initially, the performance of the algorithm is compared with the usage of different number of iterations. Then, in order to further provide a reasonable base for comparison, these TS outputs were coinpared with the human generated output The algorithm performance The TS model is tested for the 30 sample of different initial schedules. This model searches for the most feasible solution. The operation stopped until there is no more improvement to the current local optimum schedule, after a number of consecutive iterations. Table 5.6 shows a set of 30 samples in the TS model operation.

98 Table 5.6: TS model operation results Sample Best schedule Stop at iteration The 'best so fix' schedule is the lowest grand total penalty in the experiment using the TS model. Sample 18 is the -best so far' schedule in the experiinent with the grand total penalty is 1040.

99 The 'best so far' schedule sample in the experiment is shown in figure Its initial solution grand total penalty is 1230; the best schedule grand total penalty is 1040 at iteration and there is no more improvement after iteration t Figure 5.10: The 'best so far' schedule sample The initial solution is improved through the neighbourhood generation process for a number of iteration. The initial solution, 1230 is improved to the best solution grand total penalty is Figure shows the grand total penalty from starting initial solution (1230) to the best solutioil (1040) and there is no more improvement after 111 iterations.