Flexible batching in personalized learning: a newly developed tool for secondary education

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Flexible batching in personalized learning:

a newly developed tool for secondary education

Msc Thesis

by

J.M. Hahn

29-02-2016

Rijksuniversiteit Groningen Msc Supply Chain Management

Supervisor: Prof. dr. I.F.A. Vis Second supervisor: Dr. J. Riezebos

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ABSTRACT

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TABLE OF CONTENTS

1. INTRODUCTION ... 4

2. METHODOLOGY... 7

3. THEORETICAL BACKGROUND ... 9

3.1. Lean tools in general and their advantages ... 9

3.2. Lean tools for flexible batching among different sectors ... 10

3.2.1. Properties for distinguishing flexible batching policies ... 10

3.2.2. Existing flexible batching policies ... 12

3.3. Constraints and possibilities for using these policies in education ... 14

3.4. Comparison and selection of batching policies ... 17

3.5. Selected batching policies and their fit with personalized learning ... 20

4. NEW FLEXIBLE BATCHING POLICY ... 22

4.1. Notation ... 22

4.2. Definition of the new flexible batching policy ... 24

4.2.1. Step I: Select batches for the first half of the blocks in a schedule... 24

4.2.2. Step II: Select batches for the second half of the blocks in a schedule ... 29

4.2.3. Validation of the batching policy ... 30

5. SIMULATION STUDY ... 31

5.1. Design of the simulation study ... 31

5.1.2. Schedule ... 33

5.1.2. Simple rule of thumb ... 35

5.2. Warm-up period ... 35

5.3. Number of replications and run-length ... 38

5.4. Experiments ... 40

6. RESULTS AND DISCUSSION ... 42

6.1. First group of experiments: benchmarking the new batching policy ... 42

6.2. Second group of experiments: adjusting parameters new batching policy ... 46

6.3. Summary of the results and comparison between subgroups ... 58

7. CONCLUSION ... 60

8. LIMITATIONS AND FURTHER RESEARCH ... 62

9. ACKNOWLEDGEMENTS ... 64

10. REFERENCES ... 64

11. APPENDICES... 66

10.1. APPENDIX I: Number of replications selection ... 66

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1. INTRODUCTION

The need for more personalized learning in education is becoming increasingly important in the last decade, due to the fact that the today’s knowledge economy asks for different skills and competencies (Watson, Watson & Reigeluth, 2012). The main reason for this is that the labor market is making a sectorial shift from agriculture and manufacturing to the service-, ICT and healthcare sectors. Therefore, today’s pupils require different education and training than the pupils of the past, in order to align with the current and future labor demand (ROA, 2013).

The current educational system is organized in a way that is used for a long time and almost no changes have been made to this system throughout the years. The pupils are batched in a large class based on their age and educational level, and they basically all receive the same form of education. This means for instance that they often have the same subjects, materials and exams. This way of organizing can be compared to the organization of classic operational factories that produce in batches. Just like the teachers, also a ‘foreman’ is present in these factories, which is managing the machines and products. However, the information age society of today asks for a different educational system, in which the pupils learn through customized and personal learning plans (Watson, Watson & Reigeluth, 2012). Multiple countries (e.g., Sweden, England and the U.S.) already picked up this required change and they show better results in educational quality in comparison with other countries (Allen & van der Velden, 2012). An example of a Swedish educational system that pays a lot of attention to personalized learning is ‘Kunskapsskolan’. The ‘Kunskapsskolan’ way of teaching allows the teachers and pupils to focus on the right things and thereby improving the overall knowledge of the pupils in an efficient way.

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A part of the current educational sector does use lean management, however they use it solely in the back office. This means that, for example, the administrative tasks are subjected to the principles of lean, but the core process of teaching is not. Despite of this, there might be different types of waste in the current educational system that could be reduced by means of these principles. It might occur that multiple pupils are already ahead of others or vice versa, in terms of teaching material. In the current system the pupils are batched in classes and therefore the teacher will explain the same teaching materials to all pupils. That means that some pupils repeat materials that they already know and that some pupils do not understand the advanced materials, because they lack the basic skills. Flexible batching is one of the lean principle tools that could help with these problems. It is used in many manufacturing organizations and helped them to produce more effectively (Srivastava & Chen, 1996; Mefford, 2009). If pupils in secondary education are batched in more flexible ways, the individual pupil will receive education that will fit their specific needs. Furthermore, the teachers will also be able to focus on the relevant points and they can support and challenge the pupils in a better way. However, this might cause challenges in terms of capacity and availability of the classrooms, teachers and hours. The schools are not able to give each pupil individual education, because they simply do not have that much teachers and classrooms available.

This is the reason that this research focuses on the possibilities of flexible batching in secondary education. The objective of this research is to formulate a new flexible batching policy that helps deciding in what way it is possible to make group compositions that fit the needs of implementing personalized learning effectively in secondary education.

In order to be able to formulate this new flexible batching policy, a main research question and multiple sub questions are proposed. These are displayed below.

Main research question (MRQ): How can a new flexible batching policy be formulated to make group compositions in an effective way that fit the needs of a personalized learning system in secondary education?

Sub questions (SQ):

SQ1: What lean tools could provide guidance with making group compositions?

SQ2: Which of these tools are already used/implemented in other sectors and how are they used in these different industries?

SQ3: What kinds of constraints are important and need to be taken into account for the potential application of these tools (from SQ2) in the educational sector?

SQ4: How do these tools (from SQ2) correspond with the needs of a personalized learning system and what needs to be changed in such a way that they can be implemented? SQ5: Which performance measures are important for flexible batching in personalized

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2. METHODOLOGY

To find answers to the sub questions and therewith the main research question, different types of research methods are used. First, an extensive literature research is executed to provide an overview of what lean tools could be beneficial for making group compositions (SQ1). Mainly review articles that deal with different tools of lean management are used. These articles help with selecting the most important tools for making flexible group compositions. Furthermore, by means of this literature review, the practical implications of these lean tools are discussed. This results in a list of multiple lean/batching policies, which will serve as a starting point for SQ2. Subsequently, previous literature that conducted research concerning these policies for other organizations in different sectors, is used to display the challenges and implications that arose in those sectors (SQ2). As mentioned before, examples of sectors that benefited from lean management are manufacturing, healthcare and logistics sectors, among others. Therefore, these are examples of sectors that are reviewed in the literature review. Besides the insights of existing literature and research, the ideas and insights of stakeholders of the current educational system are important. Their initial insights will have an exploratory purpose and they can reflect on the current list of batching policies (SQ3). To get these insights, semi-structured interviews will be held with different stakeholders within the participating schools. Examples of stakeholders that will be interviewed are schedulers and teachers. Furthermore, different expert sessions will be held to brainstorm on the constraints that exist and implications that might occur when the batching policies are translated to fit the educational sector. The results of SQ2 and SQ3 will help in selecting the most appropriate batching policies for the educational sector and this serves as a base for the remaining questions.

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an important role within the models and the input data is subjected to different probabilities. Therefore, dynamic and stochastic simulation models are used (Robinson, 2004).

When it is clear which batching policies are selected and will be used in the experiments, it is also possible to define the performance measures that will be helpful in the simulation study (SQ5). Nevertheless, with selecting the right performance measures, the expert sessions will be an important source for input as well. Namely, these experts work with the main challenges from the educational sector on a daily basis and that is why their insights might generate valuable input. The before mentioned methods are summarized in a visual representation in Figure 1 below.

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3. THEORETICAL BACKGROUND

This section will review the current literature of flexible batching and will provide an overview of the potential useful batching policies for personalized learning in the educational system. The results of this section will serve as a base for developing a new flexible batching policy that fits the needs of personalized learning.

3.1. Lean tools in general and their advantages

Lean management origins from the Toyota Production System (TPS), which was founded by Taiichi Ohno’s from the Toyota Motor Company (Sha & Ward, 2007). This evolved and ultimately received the name of lean management. However, because existing research was not able to distinguish one conceptual definition of the principle, Sha and Ward (2007) tried to do so. They defined lean management as an integrated socio-technical system whose main objective is to eliminate waste by concurrently reducing or minimizing supplier, customer and internal variability.

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3.2. Lean tools for flexible batching among different sectors

To find out which batching policies fit the needs of personalized learning in secondary education, previous batching literature of different sectors is analyzed. As described before, the main focus of this literature review is on flexible batching. Flexible batching is defined as the degree of flexibility within a batching policy, which enables an organization to make flexible groups based on different characteristics (e.g., size or composition) (Sun, Fan, Shao & Zhou, 2015). Multiple properties are important within this concept of flexible batching, which need to be considered in selecting the right policies for this research. The next paragraph will expound on these properties.

3.2.1. Properties for distinguishing flexible batching policies

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Table 1: overview of batching policies in existing literature

Properties Sector # of products # of machines Static vs. dynamic Deterministic vs. stochastic Rework considered Scrap considered

Batch size Batch composition

(a) Manufacturing Multi Multi Dynamic Stochastic No No Fixed Flexible

(b) Manufacturing Single Single Dynamic Stochastic Yes No Flexible Fixed

(c) Manufacturing Single Single Dynamic Deterministic Yes No Flexible Fixed

(d) Manufacturing Single Multi Dynamic Stochastic Yes No Flexible Fixed

(e) Manufacturing Single Multi Dynamic Deterministic Yes No Flexible Fixed

(f) Manufacturing Single Single Static Deterministic Yes Yes Flexible Fixed

(g) Manufacturing Multi Multi Dynamic Stochastic No No Flexible Flexible

(h) Manufacturing Multi Single Dynamic Stochastic No No Fixed Flexible

(i) Automobile painting shops

Multi Single Dynamic Stochastic No No Flexible Flexible

(j) Automobile painting shops

Multi Multi Dynamic Stochastic No No Flexible Flexible

(k) Medical teaching facility

Single Multi Dynamic Stochastic No No Flexible Fixed

a) Srivastava & Chen (1996)

b) Jamal, Sarker, & Mondal (2004) policy 1

c) Jamal, Sarker, & Mondal (2004) policy 2

d) Sarker, Jamal & Mondal (2008) policy 1

e) Sarker, Jamal & Mondal (2008) policy 2

f) Biswas & Sarker (2008)

g) Van der Zee (2007)

h) Tang (1993)

i) Sun, Fan, Shao & Zhou (2015)

j) Wang, Li, Arinez & Biller (2012)

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3.2.2. Existing flexible batching policies

In the batching policy ofSrivastava and Chen (1996), each batch that is selected may contain several part types and the entire batch needs to be processed before the next batch can arrive. When this is finished the machines will be set up for the next batch. Therefore, they focus on the flexibility of the batch composition. The batch size is fixed. Furthermore, the types of machines each have a tool magazine with a limited capacity and the size of a tool is measured with the number of tool slots it has. Only one machine of each type is considered for tool assignment and therefore the total processing time that is available depends upon the number of units of each machine type, time for preventive maintenance, likelihood of machine breakdowns, among others. Subsequently, Jamal, Sarker and Mondal (2004) proposed two other batching policies (policy 1 and 2) in the manufacturing sector, which calculate an optimal batch size and in both the rework of defective items are considered. In the first policy the defective items are produced in every batch and they are reworked within the same cycle. This means that also new processing and inventory costs are included for the defective items. Scrap that may be produced during the process is not considered. In their second policy the defective items from each cycle are accumulated until x cycles are completed. Only after that the defective items are reworked. Again, scrap is not considered. In policy 1 and 2, Jamal et al. (2004) focused on the flexibility of the batch sizes instead of on a flexible batch composition. This means that the batch exists of the same products every process, but that the sizes of these batches can differ.

Sarker, Jamal and Mondal (2008) (policy 1 and 2) repeated the two policies from Jamal et al. (2004) (with rework consideration), but then customized them to fit a multi-stage production system instead of a single-stage system. They considered a multi-stage production system as a system with multiple machines. Therefore, a single-stage system is considered as a system with a single machine. Moreover, Jamal and Mondal (2008) also focused on the flexibility of the batch sizes. They considered the batch composition to be fixed, but the batch size was adjustable. This is because they calculated an optimal batch size for both policies, which can be changed according to different input variables like processing costs, demand rate and number of production cycles. A policy that differs from the previous policies is from Biswas and Sarker (2008). They proposed a policy in a single-stage production process with in-cycle rework and scrap that is inspected for 100%. The defective items are reworked within the same cycle and the scrap production and detection corresponds to the general objectives of a ‘lean production system’. Subsequently, they focus on calculating an optimal batch size, which may differ for specific variables. They do not consider flexible batch compositions, but they focus on in what way different types of scrap detection influence the batch size.

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Another policy that mainly focused on the flexibility of the batch composition is the policy of Tang (1993). They presented a class of batch composition problems for the process of printed circuit boards and integrated circuits. Furthermore, they developed different procedures for determining batch compositions considering six different performance measures (e.g., target rate, kit production, product ratio mix) and tried to find an optimal batch composition with the aid of two heuristics.

The policies described before are mainly focusing on the manufacturing sector. However, because the focus of this research is on the educational sector, other sectors with different properties for their batching policies may also be valuable. For example, Sun et al. (2015) present two policies, which focus on automotive painting shops. Their first policy is an arraying heuristic for car storage with the intention of batching colors in which more cars are painted. Their second policy is a shuffling heuristic, which attempts to consecutively release car bodies of the same colors given a mapping of specific car bodies. This means that they try to find the best way of shuffling the parts among the different batches, but also try to calculate optimal batch sizes. Therefore, they use flexibility for both the batch size and batch composition. Furthermore, the described situation of this automotive painting shop is considered to be static. Another policy that contributes to this is the policy of Wang, Li, Arinez & Biller (2012). They also executed their research in the automobile painting shops sector, which is in line with the research of Sun et al. (2015). Their policy tries to find out in what way different sizes, compositions and sequences of batches influence the overall quality in this sector. Hence, they also consider flexibility in the batch size and in the batch composition. However, they consider a dynamic situation, in which the previous policy considered a static situation.

A completely different sector is the focus for the policy of Dobson, Lee, Sainathan and Tilson (2012). They focus on a medical teaching facility in which resident physicians take exams. They created an analytical model of a tandem queue with a finite buffer to analyze the impact of different policies that prioritize the work on the throughput and flow times of patients in the facility. With this, they mainly consider flexible batch sizes and try to find out which batch size would be optimal. However, they also use a model with fixed batch sizes and analyze the differences. Moreover, the compositions within the batches stay the same.

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into account in the next paragraphs, in which the practical implications and the constraints for using these policies in the educational sector will be analyzed and compared.

3.3. Constraints and possibilities for using these policies in education

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Table 2: properties and conditions for a personalized educational system

Properties Conditions

# of pupils Multi, a lot of different pupils from different levels are present in schools. This differs per type of school.

# of teachers Multi, different teachers with different skills. Not every teacher is able to teach all type of lectures and topics. Teachers might be specialized to teach a specific level of pupils.

Demand for lectures Stochastic demand, because randomness is present in the demand for lectures and different courses.

Static vs. dynamic Dynamic situation, because time plays an important role in the schools. Lectures are scheduled for a specific period of time and this can be different among different types of lectures.

Examination Fixed dates for final exams, not adjustable by the schools (government). However, all the other moments throughout the year are adjustable and could be determined by the schools. Pupils can fail and need to rework the specific topics, but repeating an entire year is not necessary anymore. The pupils can for instance lack behind on one specific topic, but just follow the normal lectures on another topic.

Group size Minimum of 5 pupils, maximum of 80 pupils (depending on type of classroom and school). The limited available space in the classrooms needs to be considered.

Group compositions Flexible group compositions are possible. Pupils from different levels (VMBO, HAVO, VWO) can be grouped together and work on the same modules. Classrooms Size of classrooms matters. It is possible to divide the classrooms in a flexible

way (e.g., using a chemistry classroom for an English lecture). This is only possible if the tools needed are easy to store, but most of the time there are possibilities to do so. Furthermore, different sizes of classrooms are convenient, because they can be used for different purposes.

Scheduling Partly fixed schedules. It might be the case that certain lectures are the same for everyone and that more specialized lectures are different among pupils. This requires a flexible schedule.

Lecture duration Flexible. In the old system the lecture duration was fixed (for instance 40, 45, 50, 60, 70, 80, 90). The new system might require more flexible lecture durations. An example is that theory can be explained in 25 minutes and thereafter the pupils can start working individually in other classrooms. Organization of this has consequences for the schedules among other things. Type of teaching method Different types of teaching methods are possible and this also has a connection

with the lecture duration (time), because it might be that specific types of teaching methods just take 25 minutes and others take 45 minutes. Examples of the different types of teaching methods are: workshops, instruction lessons, practical’s, independent working hours etc.

Teacher utilization A prescribed percentage by the collective labor agreements in the Netherlands of 45% for giving lectures (VO-Raad, 2014).

Effective hours needed per chapter

The amount of effective hours that pupils need to finish a chapter might differ per pupil and this is where a personalized learning system wants to be flexible. This is because every pupil needs a specific amount of hours to complete a chapter and the system needs to allow this, so that slower pupils can take more time and faster pupils have the opportunity to work ahead.

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pupils need to be taken into account and what level of education they need. The number of teachers shows how much teachers are available every day to teach lectures and what courses they master. There might be teachers available that can only teach one specific course, but others might be able to teach multiple courses. A connection is recognizable with different machines that are able to execute specific tasks in the production process and with their utilization. Furthermore, this can also be compared to the specific skills of different employees within the organization. Moreover, the type of teaching methods corresponds to this, because different types of teaching methods might contribute to the expertise of pupils in different ways. An example is that workshops or instruction lessons contribute twice as much to the overall progress of pupils as an individual working hour, because the teachers explain the difficult material with examples.

The static vs. dynamic situation of the different policies and sectors is a property in Table 2, which also can be recognized in the educational sector. Therefore, it is important to know whether a policy is used in a situation in which ‘time’ is an important factor or if it is used in a steady state. For the educational system ‘time’ is an important factor, because the schedule determines where the pupils or teachers need to go. This is the reason that the lecture duration needs to be considered. This duration might differ among different types of lectures. In addition to this, time is important because the pupils move from period to period with partly different characteristics (e.g., their progress in the school material) and partly the same characteristics (e.g., their learning speed). Subsequently, the demand for lectures can be compared to the deterministic vs. stochastic property from Table 1. The demand of the specific used policies might differ and randomness might be present. To be able to analyze the policies and know whether these are in line with the educational sector, this needs to be identified.

The examination property in Table 2 shows similarities with the considerations of rework and scrap in Table 1. Namely, the machines might need to do some rework of specific products and this might also be the case when specific pupils fail an exam. It can also occur that pupils leave the school and this can be compared to the scrap that is considered in for instance manufacturing organizations. However, for the new educational system this might be less important, because the schools themselves are able to determine at what moment the pupils take a test. Therefore, repeating a whole school year will typically not be the case, because the pupils can work on the subjects by means of specific modules. This means that they are not restricted to a specific year, but to a specific module.

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progress through the chapters. Finally, the properties of the classrooms and schedules are also in line with this. For instance, the classrooms might be used in a flexible way (e.g., the chemistry room with specific tools for this course, can also be used for normal lectures) and the schedule might also differ per pupil based on their choices.

3.4. Comparison and selection of batching policies

By means of the previous paragraphs, it is possible to compare the characteristics of the different policies with the characteristics and conditions of the new educational system. This has the purpose of selecting a couple of the most interesting policies for using in the development of a new batching policy. The experiments that will be executed will be used to validate the new policy and thus trying to find out if the policy fits the needs of a personalized learning system.

If we look at Table 2, it can be noticed that a personalized educational system is classified as a dynamic situation with stochastic demand. When comparing this to the policies in Table 1, policies (c), (e), and (f) do not meet these requirements. In order to be able to implement the potential policy in an educational setting, these properties are important and need to be in line. This is because the factor ‘time’ and the factor ‘type of demand’ need to be considered in the educational setting. Hence, these policies will not be considered in the development of the new batching policy.

Subsequently, to be able to compare the policies, the use of flexibility in the size or composition of a batch is an important property as well. It needs to be clear for developing and implementing a new potential educational system with personalized learning, in what way flexibility can be used (e.g., in the batch size or in the batch composition). Therefore, looking at Table 1, different policies need to be chosen which differ on these properties. For instance, policy (a) and (h) both focus on the flexibility of the batch composition. However, both policies do not consider the flexibility of the batch size. That is an important factor that needs to be considered in the new batching policy, because a personalized learning system wants to be able to reduce the large batches of pupils of the current educational system when this is necessary. This is why policy (a) and (h) both will not be used in the development of the new flexible batching policy. Furthermore, three policies that focus on the flexibility of the batch size instead of the batch composition are policies (b), (d) and (k). When comparing these policies, is can be seen that they all consider a single product, but that policy (d) focuses on a multi-machines setting. That is another interesting focus point for the new educational system, because multiple teachers that can teach different topics and types of lectures exist and need to be taken into account. Moreover, policy (d) also considers the possibility to rework the products. This is also important in the educational setting, because it might be necessary that pupils need to rework different modules or exams. That is why policy (b) and (k) will not be taken into account.

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Table 3: selected batching policies for the simulation study

Properties Sector # of products # of machines Static vs. dynamic Deterministic vs. stochastic Rework considered Scrap considered

Batch size Batch composition

(d) Manufacturing Single Multi Dynamic Stochastic Yes No Flexible Fixed

(g) Manufacturing Multi Multi Dynamic Stochastic No No Flexible Flexible

(j) Automobile painting shops

Multi Multi Dynamic Stochastic No No Flexible Flexible

d) Sarker, Jamal & Mondal (2008) policy 1 g) Van der Zee (2007)

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3.5. Selected batching policies and their fit with personalized learning

The three policies that originate from other, selected sectors to serve as a base for developing a new batching policy for a personalized educational system, display specific similarities in their properties with an educational batching policy. However, this does not mean that these policies are completely duplicable for the educational sector. Policy (d) focuses on the flexibility of the batch size and considers rework. These two factors can be used in the new batching policy for a personalized learning system, because this policy should create flexible batch sizes and also provide the pupils the possibility to redo a specific module of a course. However, policy (d) considers a system with single products. This needs to be changed in the new batching policy, because there are a lot of different types of pupils that possess different learning styles and amounts of learning speed. Furthermore, policy (d) incorporated multiple types of costs as performance measures. This is not important to consider in the new batching policy, because the costs are less valuable for the schools and pupils in a secondary educational system than they are for manufacturing organizations.

The second policy focuses on the flexibility of both the batch size and the composition. Therefore, both factors from policy (g) can be incorporated in the new batching policy and especially the way they put together different product families within a batch in an oven is interesting for the new batching policy. That is because the flexibility of the secondary educational system will increase a lot, if pupils of different levels can be batched together when they require the same lecture at a specific point in time. Subsequently, this policy looks at the required processing time and the waiting time of the different products and product families. These performance measures need to be incorporated in the new batching policy, because they can be compared to the total time a pupil needs to finish secondary school and the waiting times before they can start with a new module or chapter. Nevertheless, policy (g) considers two different triggers for the planner to load a machine (the moment a product arrives and the moment a batch job finishes). This is not in line with what should happen in a personalized learning system. Namely, when a single pupil finishes a chapter, he or she immediately wants to continue with the next chapter. So the new batching policy needs to focus on the individual level of a pupil and make a decision when this pupil is ready to start with the next chapter.

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4. NEW FLEXIBLE BATCHING POLICY

This section will present a new batching policy that is based on the previous literature review (section 3.2), the properties and conditions from the current and new educational system (section 3.3), and the expert opinions from different schools (section 3.3). To guarantee a clear understanding of the sets, indexes, parameters and the performance measures that are used in the model and batching policy, first the notation is explained and displayed. The notations used are partly based on the notations of Van der Zee (2007), but because the research of Van der Zee (2007) was executed in the manufacturing sector a large part of the needed notations are made up for this research. Subsequently, the definition of the new batching policy is displayed.

4.1. Notation

Two important assumptions that are made in the scope of the system in which the new batching policy is used, are that (1) the pupils (l) that are added to another batch (b) with another module (j) for a specific course (i) can start working by themselves with another module for a course within this classroom (c), (2) or they can participate in the lecture (w1) from the other batch. This means that they can do self-study (w2)within a classroom with another batch that is following a lecture. These assumptions are based on multiple expert opinions from teachers and principals from different schools that participate in the ‘Zo.Leer.Ik!’ project in the Netherlands. The policy uses two different types of teaching methods (w), because these two are the most common in an educational system in the Netherlands. At each school lectures are taught and it often happens that pupils can work for themselves during an hour of self-study. These two teaching methods are used to get an idea of the flexible batching possibilities of pupils and to keep it clear not more types of teaching methods are considered. Furthermore, the policy considers two levels of pupils (k): HAVO and VWO. The educational system in the Netherlands has three different levels of pupils, but for this policy two of them are selected because HAVO and VWO are commonly clustered at school in the Netherlands. This is therefore in line with a real life educational system. Besides different levels of pupils, the policy also considers two levels of teachers (t). A distinction is made between level 1 teachers and level 2 teachers. This has to do with their skills and competencies to teach specific levels of pupils, just like in real life.

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is the maximum teacher t utilization rate (Utmax), which can be set at a specific rate that is commonly used in a school. A parameter that might differ per pupil is the learning speed per course i (Sli), which means that it is possible that a pupil is good at Math, but that he or she thinks English is difficult. This might contribute in different ways to the progress of the pupil. The batch composition (Bbcitw) might also differ per batch b and per course i and is assigned to teacher t, classroom c and working method w. That is the reason this variable is denoted in the policy.

Specific performance measures will follow from the decisions in the model, which are based on these sets, indexes, parameters and variables. For instance, the number of self-study (Zt), teaching (Ot) and preparation (Vt) hours for teacher t are useful performance measures to analyze the teachers’ behavior. These insights will result in the utilization rates of the teachers t (Ut). Moreover, the utilization rates of classrooms (Uc) will show in what way the capacity of classrooms c is used in the policy. Finally, the processing time per module j per course i for pupil l (Tijl) and the waiting time for a module j per course i for pupil l (Wijl) will show how long it takes a pupil to finish a module and how long he or she has to wait until he or she can start with a new module. The waiting time is calculated from the point a pupil finished a module until the point in time he or she follows a lecture for the new module. This explanation results in the following notation for the sets, indexes, parameters, variable and performance measures: Sets: B set of batches C set of classrooms I set of courses L set of pupils T set of teachers M1 set of modules VWO M2 set of modules HAVO

Indexes:

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w teaching method identifier = 1, 2. 1 = lecture 2 = self-study q teacher level = 1, 2. 1 = high competency 2 = low competency Parameters:

Al probability (%) that pupil l is present

At probability (%) that teacher t is present

Cbcw allowed batch size per batch b per teaching method w, i.e. the cumulative size of pupils that fit in a classroom at the same time, this can differ per classroom c

Cbw minimum required batch size per batch b per teaching method w, i.e. the minimum amount of pupils that need to be batched in order to start a module j

Gk probability (%) of pupil l at level k

Hij number of effective hours needed to complete a module j per course i, ∈ M.

N number of new pupils that start each year

pi probability (%) of pupils to pass a test per course i

Sli pupil learning speed (%) per course i

Tqi number of teachers per teacher level q per course i

Utwmax maximum teacher t utilization rate per teaching method w

Variable:

Bbcitw batch composition per batch b per course i assigned to teacher t to classroom c and to teaching method w

Performance measures following from decisions in model: Ot teaching hours for teacher t

Tijl required processing time per module j per course i for pupil l

Ut teacher t utilization rate (%) Uc classroom c utilization rate (%)

Vt preparation hours for teacher t

Wijl waiting time for a module j per course i for pupil l

Zt self-study hours for teacher t

4.2. Definition of the new flexible batching policy

In order to select the right batches of pupils in the policy, two steps are described in detail below. First an explanation of the steps in words is described. Thereafter the steps are described with the proper notations from section 5.1.

4.2.1. Step I: Select batches for the first half of the blocks in a schedule

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priority in step I. At step II the minorities have priority, in order to prevent that specific groups of pupils will only receive one type of teaching method (i.e., lecture or self-study). In this step the policy looks at the first hour in the schedule and defines which pupils are participating in the scheduled course. Then it generates a list of the amounts of pupils per module and ranks the modules according to their frequency of pupils (with the module with the highest amount of pupils first). This is adapted from the research of Van der Zee (2007), because his policy first sorted and ranked the products according to their type. After this ranking process, the different batches are composed. The policy first searches for the biggest group of pupils that need to start with the same module and batches this group. When this is done, it searches for a classroom with sufficient capacity in which this batch fits. The biggest classrooms are chosen first, where after the smaller classrooms are picked. Finally, it searches for a teacher with the right skills and allocates him/her to the class. The decision to search for a classroom prior to the decision to search for a teacher is based on the availability of classrooms. When there are not any appropriate classrooms available, the pupils will be assigned to do self-study. In this case it is less important that a teacher of that same course guides the hour of self-study than with a lecture. So in the case that there is not an available classroom, no teacher is assigned to this batch of pupils. However, when the decision to search for a teacher is prior to the decision to search for a classroom, a teacher has already been assigned to a batch when it turns out that no appropriate classroom is available. Therefore, this teacher will guide the self-study hour, while he or she might be needed to give a lecture to another batch of pupils. That is the reason that searching for a classroom has priority over searching for a teacher. This is not adapted from any of the three policies from section 3.4., but is based on the observations at different schools and the exploratory insights of teachers from those schools.

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pupils from completely different modules or courses, so in order to improve the flexibility the flexible composing of batches from Van der Zee (2007) is used. With this adding process, the policy takes into account that the pupils of this batch can only be placed in a classroom with pupils that need to work on a module that is two more or two less than the module this batch need to work on. An example is that pupils that need to work on module 12 for a specific course, can only be placed in a classroom with pupils that need to work on module 10, 11, 13 or 14. This enables the pupils to ask questions to the right teachers or to their fellow pupils, but the diversity within one classroom will still be manageable for the teachers. This is in line with the policy of Wang et al. (2012), because their policy pays attention to the product sequence in a process of painting cars. This policy takes into account that specific colors of paint need to be scheduled in front of others. When looking at the batching of pupils with different modules, this sequencing is also important. Namely, the diversity within a batch cannot be too high, otherwise it might not be manageable for the teacher.

If this batch cannot be placed within another classroom or within a free classroom, they are assigned to do self-study. For this self-study, the policy searches for a random classroom. It also needs a teacher to supervise the self-study and therefore the policy first looks for a teacher that teaches the same course as the block in the schedule prescribes. If this is not possible, it selects a random teacher from another course. This process continues until all modules are checked. For the modules that have less than five pupils, the policy looks for classrooms in which these pupils fit and otherwise they are allocated to do self-study. With this adding process the product sequencing of Wang et al. (2012) is considered again, as is described before. When this entire cycle is finished, the policy continues with the next block in the schedule and starts over again. This is repeated until all the blocks in the first half of the schedule are finished. So, with repeating this process for each block in a schedule the flexibility of the size and composition of a batch differ each block. It might be the case that in the first block ten pupils need to work on a specific module for a course, but that in the same classroom eight pupils are working on another module for that course. This is where all three policies from 3.4. came in. They all consider flexibility in the batch size or the batch composition. This is especially the case with the policies from Van der Zee (2007) and Wang et al. (2012), because these policies consider both types of flexibility. The decision to compose new batches every block of a schedule is therefore based on these policies. This differs a lot from today’s batching process at secondary schools in the Netherlands. Therefore, this policy increases the flexibility of composing batches, which is in line with the requirements of a personalized learning system.

The previous steps are displayed in Figure 2 and described with the proper notations from section 4.1. below:

1. Select from the schedule for a specific course Ii, year and pupil level k the next block hour

(set h = 1)

a. Output: course type i for hour h of day d

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4. For the first j in the sorted list do: BEGIN ‘Case module size’ END 5. ‘Case module size’ distinguishes 4 cases:

a) j size = >20 Ll:

- Form b with Bbcitw with all Ll

- Search for c with sufficient capacity (first biggest c, else second biggest c, etc.).

- If no c with sufficient capacity available, then 𝑏

2 .

- If j size still = >20 Ll, return to step 5. a). - Else, continue to step 5.b).

b) j size = 10< Ll < 20:

- Form b with Bbcitw with all Ll If all c = > 1:

- Place Ll in c in which it (partly) fits, search c with Ll for the same course I, but different j (with j max <= 2 or j max => 2).

- Else, set w = 2.

If 1 or more c = 0:

- Place Ll in c in which it fits.

2 .

- If j size still =10 < Ll > 20, return to step 5.b). - Else, continue to step 5.c).

c) j size >5 Ll <10,

- Place Ll in c in which it (partly) fits, search c with Ll for the same course I, but different j (with j max <= 2 or j max => 2).

- Place Ll in c in which it fits. - Else, set w = 2.

d) j size <5 Ll,

- Place Ll in c in which it (partly) fits, search c with Ll for the same course I, but different j (with j max <= 2 or j max => 2).

6. Find and assign Tt to b.

- For step 5. (a, b, c, d), if Tt with the right level q for course i and sufficient Ut is available, then assign Tt to b and set w = 1.

- Else, search for alternative i in schedule and return to step 12. - If not possible, set w = 2 for initial i.

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4.2.2. Step II: Select batches for the second half of the blocks in a schedule

For the second half of the blocks in the schedule the process is comparable to the process in the first half of the blocks. However, the sequence differs. This is because the smaller groups have priority. This means that, in the second half of the blocks, the tool starts allocating the smallest groups and moves up in Figure 2, instead of down (as step I did). These steps are described with the proper notations from section 4.1. below:

8. Select from the schedule for a specific course Ii, year and pupil level k the next block hour (set h = 1)

a. Output: course type i for hour h of day d

9. Generate a list of frequencies per module number j with the amount of pupils Ll. 10. Sort j according to highest frequency Ll(for VWO 1 to 48, for HAVO 1 to 36)

11. For the first x modules j in the sorted list do: BEGIN ‘Case module size’ END 12. ‘Case module size’ distinguishes 4 cases:

a) j size >5 Ll <10,

- Search for classroom c with sufficient capacity (first biggest c, else second biggest c, etc.).

- Place b in c in which it (partly) fits, search c with Ll for the same course I, but different j (with j max <= 2 or j max => 2).

b) j size with >10 Ll <20,

- Place b in c in which it fits.

2 .

- If j size still =10 < Ll > 20, return to step 12.b) - Else, return to step 12.a).

c) j size with > 20 Ll,

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- Place bin c in which it fits.

2 .

- If j size still =10 < Ll > 20, return to step 12.c) - Else, return to step 12.b).

d) j size <5 Ll,

- Place b in c in which it (partly) fits, search c with Ll for the same course I, but different j (with j max <= 2 or j max => -2).

13. Find and assign Tt to b.

- For step 12. (a, b, c, d), if Tt with the right level q for course i and sufficient Ut is available, then assign Tt to b and set w = 1.

- Else, search for alternative i in schedule and return to step 12. - If not possible, set w = 2 for initial i.

14. Do h + 1 and return to step 8. until all blocks in the second half of the schedule have been

assigned.

4.2.3. Validation of the batching policy

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5. SIMULATION STUDY

This section will describe the design of the simulation study and the determination of the warm-up period, the number of replications and the run-length. An existing simulation model with a simple rule of thumb, which is part of this model, is used for the simulation study in this research. The developed model shows the academic processes of secondary schools at an operational level and is a non-terminating simulation model. This means that the model does not have a natural end point (Robinson, 2004). This is the case for an educational system, because the pupils continue with the lecture material the next day where they stopped the previous day. Furthermore, the pupils from all different schoolyears make use of the same resources in the school (e.g., classrooms and teachers). The main overarching purpose of this model is to study the effect that the personalized learning system would have on the operational performance measures, which will be explained later. However, this research uses the model for a more detailed purpose. Namely, to validate the newly developed flexible batching tool and to analyze its effect on the performance measures.

The model is based on real life data from a case study of one of the schools that is participating in the ‘Zo.Leer.Ik!’project. They provided their data so a realistic simulation model could be built. The scope of the model considers two academic programs: HAVO and VWO (from which respectively three and four years of pupils are part of the model). Throughout the academic programs pupils have to complete 36 or 48 chapters for 10 different classes. In order to do so, pupils must cover the required material of each chapter by means of lectures and /or self- study. After pupils covered the material of each chapter, there will be a test in which it is decided if the pupils continue to the next level. Furthermore, each schoolyear a new batch of pupils is added to the first schoolyear in the system. However, a real life school does not start with an empty school each year. This is because each real life school already contains several pupils, which worked their way through one or more schoolyears. Therefore, pupils from the second, third and fourth schoolyears are generated at the beginning of a simulation run. A uniform distribution is used to generate these pupils. A more detailed description of the model is written by Vis, Lopez Alvarez and Kokx (2016), in which the different assumptions, parameters and variables are discussed.

5.1. Design of the simulation study

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adjusted in the experiments. An overview and definition of the experiments is displayed in Table 7 in section 5.4. The first group of experiments aims to compare the two different control strategies. Experiments 1 till 3 are included in this group. The second group of experiments is used to adjust the baseline settings of the new batching policy in order to perform a sensitivity analysis and to test the robustness of the policy. The minimum batch size for a lecture or self-study and the teachers’ maximum utilization are examples of factors that are adjusted in these experiments. With these results it will be able to analyze what relations exist between the parameters, variables and performance measures and in what way they can be influenced. Experiments 3 till 43 are part of this second group.

Table 4: design of the simulation study

Factor Settings

1. Control strategy Simple, New

2. Number of educational levels (k) 2

3. % HAVO pupils (Gk1) 75

4. % VWO pupils (Gk2) 25

5. Number of modules HAVO per course (M1) 36

6. Number of modules VWO per course (M2) 48

7. Number of effective hours needed to complete a module (Hij)

20 8. Self-study contribution/Lecture contribution 1/2 9. Number of new pupils per year (N) 200 10. Number of teachers level 1 (Tqi1) 11

11. Number of teachers level 2 (Tqi2) 35

12. Availability pupils (%) (Al) 95 13. Availability teachers (%) (At) 95 14. Probability to pass a test (%) (pi) 95

15. Minimum batch size for a lecture (Cbw1) 5, 10, 15, 20, 25, 30 16. Minimum batch size for self-study (Cbw2) 5, 10, 15, 20, 25, 30

17. Number of classrooms (Cc) 38

18. Maximum teacher utilization (%) total (Utmax) 45, 67.5, 100

19. Maximum teacher utilization (%) lecture (Utw1max) 22.50, 45, 33.75, 50, No limit

20. Maximum teacher utilization (%) self-study (Utw2max) 22.50, 33.75, 45, 50, No limit

21. Hours per day (h) 7

22. Pupils learning speed per course (Sli) 0.8/0.85/0.90/0.95/1

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Some of the factors in Table 4 are related. Examples are factors 10-11 and factors 18-20. Namely, factors 18-20 describe the teachers’ utilization for the supervision of an hour with self-study and for giving a class. Factor 18 is the total teachers’ utilization and 19 and 20 are specified for the self-study supervision and giving a lecture. The baseline settings of this total teachers’ utilization is set at 67.5%, from which 45% is for giving a lecture and 22.50% for supervising self-study. This is adapted from the expert sessions with experts from the educational sector and these values are based on the 45% of giving lectures which are prescribed by the agreements for secondary schools in the Netherlands (VO-Raad, 2014). In the 22.50% of supervising self-study, it is assumed that the teachers are able to prepare other lectures or to do office work. However, it might result in interesting findings if this utilization is adjusted. Therefore, multiple experiments will be executed with different utilization rates (e.g., 50%/50%, 22.50/45%, 33.75%/33.75%), as can be seen in Table 4. For instance, it might be the case that the adjusted teachers’ utilization cause alterations in the waiting time for pupils to start a module for a specific course. But this also might differ per course. Furthermore, factors 12-14 relate to the availability of the pupils and teachers and the probability to pass a test. These are all set at 95%, because there might be, like in real life, some ill pupils or teachers. And it might be possible for a pupil to not pass a test the first time. The next factors that are important for the experiments are factors 15 and 16. These factors display the minimum batch size that is needed to start a lecture or to supervise self-study. The base setting is set at five pupils, but in the experiments the effect on the performance measures of other minimum amounts are tested. Factors 17 and 21 display the amount of classrooms and the amount of hours per day. These are set at 38 classrooms and 7 hours per day respectively. This is based on the input data from a specific school in the Netherlands, which has 38 classrooms and 8 blocks of 45 minutes in a day (excl. breaks). Finally, the learning speed (factor 22) might differ per pupil and per course. The values of this factor are therefore randomly selected for each pupil per course, with the values that are displayed at Table 4.

5.1.2. Schedule

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For each level (HAVO and VWO) specific schedules are defined. For each year (1 till 4 for VWO and 1 till 3 for HAVO) and level, different schedules are used. Furthermore, a schedule is divided in grey and white blocks. The grey blocks represent the fixed hours and the white blocks represent the courses that are offered in case there are not enough teachers available for the fixed course in the grey blocks. Subsequently, the white blocks make sure that the courses that are not available often in the grey blocks, are sufficiently offered to the pupils in the total schedule. This means that the courses that are offered less in the fixed hours of the schedule, are compensated in the white blocks. An example is the course IT. This course is only offered twice in the fixed hours of the original schedule. The reason for this might be that the course requires less time for the pupils to finish a module or that this course requires less total modules than other courses. However, at this moment the real reason for this is not known yet and needs to be examined further in more detail. Therefore, in this research the assumption is made that all courses need to be offered equally. The simulation model first looks at the courses in the grey block, but if it is the case that a lot of pupils are not able to receive a lecture, the two white blocks in the same hour are checked. When the teachers of these courses are not available as well, the pupils will do self-study for the grey block in the specific hour. An example of this schedule (for year 2 of VWO) is displayed in Figure 3.

Figure 3: example of an extensive schedule for pupils of year 2 VWO

To indicate that the schedule really is just a tool for the simulation model to work and to test whether the newly developed batching policy also works with different schedules, a simpler schedule is used. This schedule offers fewer possibilities to the pupils and is therefore more fixed than the schedule in Figure 3. An example of this schedule is displayed in Figure 4. The simple rule of thumb with these two schedules is incorporated in the experiments, to be able to benchmark the results with the results of the new batching policy.

Y2/VWO Monday Tuesday Wednesday Thursday Friday

Dutch PE PE Crafts Math

Economy Biology IT Economy IT

IT Economy Biology IT Economy

Math PE PE Music Math

Economy Biology History French Music

IT History Biology History Economy

English Dutch Music Music French

History Music IT English IT

Music English Biology IT Music

IT Math French Biology Economy

Music IT History Music Geography

Geograph Music Music Economy Biology

Music English Dutch Latin Economy

Economy Geography Geography Music Geography

Geograph Economy IT History Music

History French Dutch Drawing Geography

IT Economy IT Economy IT

Economy IT Economy IT Biology

Biology Economy English History Biology

IT French IT French IT

French IT Math IT French

Biology English Crafts History Geography

Dutch Music French Math Dutch

English French English French IT

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Figure 4: example of a simple schedule for pupils of year 2 VWO

5.1.2. Simple rule of thumb

To be able to run the model it needed a policy to batch the different pupils in the model. A simple rule of thumb was chosen to be able to analyze the first results. Moreover, it will serve as a benchmark for the new batching policy. This policy first ranked the pupils on which module they need to work on. If this ranking is finished, it selects the largest group and just pushes as much pupils as possible in the available spots in a classroom. After that, the policy tries to allocate a teacher with the right skills and utilization to the group of pupils in the classroom. When a teacher is assigned to the group the lecture will start. If there is no appropriate teacher or classroom available, the policy searches for available teachers from the (if available) alternative courses in the schedule for that specific block. If for these courses no appropriate teacher is available either, the pupils are allocated to do self-study for the initial course in the schedule. This cycle is executed for the first half of the blocks in the schedule and is repeated until these blocks are finished. Subsequently, for the other half of the blocks in the schedule, this cycle is executed reversibly. This means that the policy first searches for the smallest groups and tries to assign them to teachers and classrooms. This prevents that the smaller groups are assigned to do self-study every block of the schedule.

5.2. Warm-up period

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system. The schools want to offer the pupils sufficient choices in courses and lectures, to let them continue with their learning material when they want to. The results of this run and the Welch’s method are displayed in Figure 5.

Figure 5: Welch’s method: plot of moving average of WT of pupils (window = 5)

It can be noticed in Figure 5 that the line of the waiting times stabilizes after 600 new pupils have entered the system. Each year 200 new pupils enter the system (adapted from Table 4). This means that after three schoolyears the system is in a steady state, from which follows that the warm-up period is set at three years. In the simulation model three schoolyears are equal to 5460 hours. That is because the model needs to run the yet to be determined run-length plus 5460 hours of the warm-up period. However, as explained in section 4, the existing simulation model uses a uniform distribution to generate the pupils in year 2, 3 and 4 as this mimics the situation of a real life school. Because the warm-up period of 5460 hours causes long computation times and the available time is limited, it is chosen to get around those long computation times and make use of this possibility to uniformly generate the pupils. Though, is does has to be determined whether these generated pupils effect the results in a bad way. That is the reason the distribution is tested with 2 alternative options with different lower and upper values for the uniform distribution. This is done to check whether the results were dissimilar for the different options and values. The values used for these alternative options are displayed in Table 5.

Table 5: initial and alternative values uniform distribution for generated pupils

Initial values (modules) Alternative values (modules)

Lower Upper Lower Upper Lower Upper

Year 2 8 16 10 20 12 24

Year 3 16 36 18 34 24 36

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The results of the runs with different values for the uniform distribution did not show large differences among the variables (e.g., waiting times of pupils (in days), cycle times of pupils (in days)) at first sight (Table 6).

Table 6: results multiple options uniform distribution for generated pupils

That means that, presumably, a warm-up period is not necessary. Hence, it was chosen not to use an empty system at the start of the experiments, but to generate the pupils in years two, three and four according to the uniform distribution. An additional check is performed by executing a run with a run-length of 10 times the initial warm-up period (Robinson, 2004) without the uniformly generated pupils. This is to verify whether the results from this run differ from the results of the runs with the uniformly generated pupils. Because the initial warm-up period is set at 600 pupils, which equals 5460 hours (three schoolyears), a run with 10 times the warm-up period is executed. These results are shown in Figure 6.

Figure 6: plot of moving average of WT of pupils (window = 5) with 54600 hours

Figure 6 shows that the results of this run are similar to the results in Figure 5, which means that the waiting time per module stabilizes from the point of 600 new pupils. Furthermore, the average waiting- and cycle times of pupils do also correspond to the results in Table 6. Namely, the average waiting time of HAVO and VWO pupils was 2.21284 days and 2.30758 respectively days. The average cycle time of HAVO and VWO pupils was 24.36198 days and

0 1 2 3 4 5 6 W ai ti ng ti me (i n da ys )

Number of pupils in the system

Moving average of the WT of pupils

Uniform distribution settings AVG WT

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results without the group of generated pupils are very similar. Based on this, it is decided that in the simulation study the pupils of years two, three and four will be generated by means of the uniform distribution as described before.

5.3. Number of replications and run-length

In stochastic simulation studies it is important to use multiple replications, because randomness exists during a simulation run (Robinson, 2004). This means that every run produces different results and therefore the required number of replications needs to be determined. In order to determine this number of replications Robinson (2004) described three different methods. The first one is a rule of thumb from Law and McComas (1990) that recommend that at least three to five replications should be performed. However, it is not unusual for a model to need more than three to five replications to obtain a reliable view of the output data. Therefore, the second method is a graphical method, which displays the cumulative mean of the output data from multiple replications (Robinson, 2004). Moreover, the third method is the confidence interval method. This is a statistical means for showing how accurately the mean average of a value is being estimated (Robinson, 2004). The use of these methods is elaborated in Appendix I. However, in non-terminating simulation studies multiple replications are only needed when the outcomes of the different experiments are compared to each other by means of a t-test (Robinson, 2004). In this research it is chosen not to compare the results of the different experiments by means of a t-test, due to time limitations. Nevertheless, in order to obtain as realistic results as possible, multiple replications are used for the different experiments. As can be seen in in Appendix I, the required number of replications is determined at 20 replications. Therefore, this number of replications is used for the simulation study.

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5.4. Experiments

As was mentioned before in section 5.1., two groups of experiments are defined for the simulation study. The second group is subdivided into five subgroups. In these experiments the factors of Table 4 are slightly modified per experiment to gain insights in the impact of these changes on the results. Table 7 shows an overview of the experiments with the different values for the factors. Furthermore, the software package that was used to conduct the experiments is Arena™, version 14.7 (Rockwell Automation Technologies, Inc.)

Table 7: definition of experiments

Group Experiments Control strategy Max total teacher utilization (%) (Utwmax) Max lecture teacher utilization (%) (Utw1max) Max self-study teacher utilization (%) (Utw2max) Minimum batch size lecture (Cbw1) Minimum batch size self-study (Cbw2) 1. Exp. 1 Simple (extensive schedule) 67.50 45 22.50 5 5

Exp. 2 Simple (simple schedule)

67.50 45 22.50 5 5

Exp. 3 New (extensive schedule) 67.50 45 22.50 5 5 s1 s2 s3 2. s4 Exp. 4 New 67.50 45 22.50 10 10 Exp. 5 New 67.50 45 22.50 15 15 Exp. 6 New 67.50 45 22.50 20 20 Exp. 7 New 67.50 45 22.50 25 25 Exp. 8 New 67.50 45 22.50 30 30 Exp. 9 New 67.50 45 22.50 30 5 Exp. 10 New 67.50 33.75 33.75 5 5 Exp. 11 New 67.50 33.75 33.75 10 10 Exp. 12 New 67.50 33.75 33.75 15 15 Exp. 13 New 67.50 33.75 33.75 20 20 Exp. 14 New 67.50 33.75 33.75 25 25 Exp. 15 New 67.50 33.75 33.75 30 30 Exp. 16 New 67.50 33.75 33.75 30 5 Exp. 17 New 67.50 22.50 45 5 5 Exp. 18 New 67.50 22.50 45 10 10 Exp. 19 New 67.50 22.50 45 15 15 Exp. 20 New 67.50 22.50 45 20 20 Exp. 21 New 67.50 22.50 45 25 25 Exp. 22 New 67.50 22.50 45 30 30 Exp. 23 New 67.50 22.50 45 30 5 Exp. 24 New 100 50 50 5 5 Exp. 25 New 100 50 50 10 10 Exp. 26 New 100 50 50 15 15 Exp. 27 New 100 50 50 20 20 Exp. 28 New 100 50 50 25 25 Exp. 29 New 100 50 50 30 30 Exp. 30 New 100 50 50 30 5

Exp. 31 New 45 No limit No limit 5 5

S5 Exp. 34 New 45 No limit No limit 20 20

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Table 8: important performance measures Subject Performance measures

Pupils 1. Average time per module per course

2. Distribution among the modules

3. Average waiting time for a module per course

Teachers 4. Actual utilization rate (%) of teaching hours (NB: for level 1 without the lectures for VWO years 5-6 and HAVO years 4-5)

5. Actual utilization rate (%) of self-study hours

6. Actual utilization rate (%) of preparation hours

Classrooms 7. Actual utilization rate (%) for lectures

8. Actual utilization rate (%) for self-study hours