Team composition and timetabling principles for multi-

Team composition and timetabling principles for multi-team membership in

education

Master thesis, MSc, Supply Chain Management

University of Groningen, Faculty of Economics and Business

January 27, 2020

Wouter Jans

Studentnumber: 3244733

e-mail:

w.h.jans@student.rug.nl

Supervisor/ university

Prof. Dr. J. Riezebos

Co-assessor/ university

Dr. O. Kilic

Supervisor/ field of study

C. Molenwater

ABSTRACT

CONTENT

2.2 Teams and Multi-Team Membership ... 6

2.3 Team composition ... 7

2.4 School timetabling problem ... 9

2.5 Integrated approach for composition and timetabling of teams ... 9

3 METHODS ... 11

3.1 Problem analysis ... 11

3.2 ILP model for composition and timetabling of teams ... 13

3.2.1 Input parameters of the model ... 13

3.2.2 Other parameters ... 16

3.3 Integer linear programming model ... 16

3.3.1 Decision variables ... 16

3.3.2 Objective function ... 16

3.3.3 Constraints ... 17

3.4 Constraints evaluation and ILP coding ... 21

4 RESULTS AND DISCUSSION ... 22

4.1 Outcome of the ILP model ... 22

4.2 Sensitivity Analysis ... 25

4.3 Discussion ... 26

5 CONCLUSION ... 27

REFERENCES ... 29

APPENDICES ... 33

Appendix A – Parameter matrices for the objective function ... 33

Appendix B – Parameter matrices for the constraints ... 36

Appendix C – Coding of the Integer Linear Programming model ... 39

1 INTRODUCTION

In the 20th _{century, team-based working is more of a standard than an exception in many}

organizations. Due to globalization and easy accessibility to information, organizations changed from hierarchically and functional structured organizations towards more collective and flexible team-based organizations (Lawler, Mohrman, & Ledford, 1995). Over the last couple of decades, there has been profound interest in team-based working (TBW) and scholars have given more attention to TBW. Research concludes that TBW has a positive effect on the performance and effectiveness of the organization (Delarue, Van Hootegem, Procter, & Burridge, 2008). However, TBW loses value when teams initiate it whilst having a lack of consideration for team dynamics and structured plans. (Hackman, 2002). Recent study has shown that formal planning activities are important and improve team effectiveness in the long term (Mathieu & Rapp, 2009).

A particular sector that is affected by the globalised developments is the educational sector. This sector experiences an increase in technological innovations, accessibility of information, cultural diversity and an ageing population (OECD, 2019). The educational institutions need to cope with these fast-changing developments in order to provide a sufficient and high-quality learning environment. The educational logistics needs to be improved in order to create a high-quality learning environment in educational institutes. However, educational institutes are less innovative and flexible compared to other organizations, as they are highly structured and systematically organized (Massimiliano, 2004; West & Markiewicz, 2008). Several initiatives are taken to eliminate protentional threats to the learning environment. For example, governments granted subsidies to assist on the ageing problem to attract people in the education sector. Furthermore, educational organization structures change more towards team-based structures. The educational sector believes that this structure can improve the effectiveness and quality of education (van Dartel & Koppens, 2019). Additionally, The Teaching and Learning International Survey 2013 (TALIS) shows that an increase in teacher collaboration increases job satisfaction and self-efficacy (European commision, 2013). Subsequently, more educational institutes start to explore various team-oriented working approaches like high performance teams and self-managing teams (Kluijtmans, Becker, Crijns, & Sewandono, 2005).

A rather new TBW approach is multi-team membership (MTM) (Mortensen, Woolley, & O’Leary, 2007). MTM refers to a team-based structure in which members are simultaneously acting in several teams and interdependently working towards shared goals. Because members work simultaneously in several teams, less members with certain expertise’s are required across all teams. Therefore, members can be positioned in numerous situations and act flexibly towards unpredictable changes (O’Leary, Woolley, & Mortenson, 2012). The retrieved literature focuses primarily on the correlation of the MTM working method and individual or team performances. In addition, study results also indicate contradictions concerning individual performance. On the one hand, individual performance increases while working in multiple teams, as it motivates individuals to perform better (Bertolotti, Mattarelli, Vignoli, & Macrì, 2015). On the other hand, time fragmentation among multiple teams could result in lower individual and team performances (Crawford, Reeves, Stewart, & Astrove, 2019). This can be anticipated by careful selection of team members and active coordination on scheduling (Chan, 2014; O’Leary, Mortensen, & Woolley, 2011). Thus, it can be concluded that team composition and schedule coordination are mediating factors and contribute to a successful correlation between MTM and performances.

and educational sectors are incomparable, as teams are organized based on a different value-added process. Although the difference is acknowledged, studies on team-oriented working in the educational sectors are lagging behind compared to other sectors (van Dartel & Koppens, 2019). Therefore, research is required on how team-oriented working can be applied in the educational sector and in which manner this affects the educational logistics.

This thesis seeks to enhance the knowledge about the application of MTM working approach combined with timetabling in the educational sector. In order to achieve this goal, the important principles of MTM will be explored in the literature and assessed on the applicability in the educational sector. The knowledge gained will be used to develop a model that could cope with the challenges in the sector. The model will address the team composition and timetabling as an integrated approach in order to create an effective and flexible organization. This will change the educational logistics and improve the quality of education. The associated research question for this research is stated as follows;

“How can team composition and planning improve MTM and support the educational sector in becoming an effective and flexible organization?”

This thesis project has the opportunity to conduct a case study at an educational institute which selected the MTM working approach as a possible solution to cope with the effects of globalization. The management of this particular institution believes that changing the institutional organization towards a more MTM-based structure could overcome the organizational issues. At the moment, the organization and workforce are structured based on the different levels of education and are therefore inflexible and ineffective. The workforce consists of regular teachers, subject-specific teachers and support staff. This research will provide the institution with an insight in team composition and a planning structure based on the MTM approach.

The remainder of this thesis is organised as follows; section 2 starts with a brief explanation of educational logistics, and will continue with reviewing the existing literature on teams and MTM, team composition and school timetabling. This section concludes with the presentation of the integrated model for composition and timetabling of teams. Section 3 presents the problem analysis and the integer linear programming model. This section also includes an explanation of the parameter matrices, decision variables, constraints and the objective function. The results of the formulated model are presented and discussed in section 4. Lastly, this thesis will end with a conclusion in section 5.

2 THEORETICAL BACKGROUND

2.1 Educational logistics

information flows that enable education institutions to provide qualitative and effective education.

The model of educational logistics highlights the importance of planning and timetabling in relation to the quality of education. The required flexibilization and personalized education influences the educational logistics. The current functional-orientated organizations have difficulties in creating a workable planning and timetable. Therefore, educational institutions have an interest in team-based working. In the following sections, the principles of team-based working are integrated in educational logistics.

2.2 Teams and Multi-Team Membership

Over the last two decades, the structure of work has been changed from a functional to a more team-based structure. A variety of reasons have been given for this and many of them have been caused by the globalization of the world. The increase in competition, adaptability, the demanding nature of jobs in terms of skills diversity and expertise, have forced organizations to be more flexible (Kozlowski & Ilgen, 2006). The definition of teams best suited for this research is: “a set of two or more people that interact dynamically, interdependently and adaptively towards a common and valued goal/objective/mission” (Salas, Dickinson, Converse, & Tannenbaum, 1992, p. 4). In other words, multiple employees work simultaneously together and independently on task-related activities and need to adapt to the constant changing environment in order to achieve their goals. Four different types of teams are addressed in the literature and are recognized by organizations: work teams, parallel teams, project teams, and management teams (Cohen & Bailey, 1997). The four teams could be described as production teams for goods and services (work teams), problem-solving teams (parallel teams), time-limited and non-repetitive teams (project teams) and coordinating and strategic teams (management teams). The teams within this thesis are identified as work teams that are well defined, structured and often self-managed (Cohen & Bailey, 1997).

Despite the well-defined and structured characteristics of work teams, they are still complex and dynamic systems that need to be carefully designed and managed in order to successfully perform. Critical to the performance or effectiveness of teams is the design and organization of taskwork and teamwork. Taskwork consists of work-related activities that are assigned to members within the team and need to be executed in order to perform (Wildman et al., 2012), whereas teamwork refers to the manner in which team members work with each other and also involves shared behaviours (i.e. what members do), attitudes (what team members feel or believe) and cognitions (what team members think or know) (Salas, Shuffler, Bedwell, & Lazzara, 2015). The basics for designing teams are associated with taskwork, the activities that need to be performed in order to achieve a common goal. Whenever a team is specialized or an expert concerning a certain topic, it is defined as an expert team. Expert teams are a group of people that possess unique and expert-level knowledge, skills and experiences related to the task performance (Salas et al., 2006). Expert teams have the ability to quickly understand and solve problems because of their expertise and take on a flexible and adaptive approach towards (un)certain situations (Chi, Feltovich, & Glaser, 1981; Hatano & Inagaki, 1984). Focusing on expertise of people while designing team-based structure will create more synergy in teams and improve teamwork (Salas, Burke, & Stagl, 2004).

knowledge companies with a flat and matrixed organizational structure (O’Leary et al., 2012). Besides the benefits of organising taskwork in a more effective way, several researchers have highlighted that time fragmentation of members across multiple teams influences teamwork within teams. Several studies suggest that the effectiveness of an individual improves by using more efficient work practices, better utilization of time, increased access to external resources (e.g. other teams within the organization) and an equally balanced workload (Cummings & Haas, 2012; De Vries, Walter, Van Der Vegt, & Essens, 2014; O’Leary et al., 2011). However, MTM could potentially harm the effectiveness because of the time fragmentation across multiple teams, reduced team cohesion, less within-team routines and mental agreements in teams and the potential for lags and delays (De Vries et al., 2014; O’Leary et al., 2011; Pluut, Flestea, & Curseu, 2014; Zika-Viktorsson et al., 2006). Beyond the doubts about the benefits and detriments of MTM, the literature has a common understanding that the negative effects could be reduced by carefully considering the composition of teams and active coordination of planning. This study will now further explore the composition of work teams and planning structures in order to make MTM successful.

2.3 Team composition

Team composition is the process of selecting individuals for working in teams based on the individuals’ attribute and characteristics. The right composition has a substantial influence on teamwork and performance (Hackman & Wageman, 2005). The relationship of team composition characteristics and team effectiveness is a much studied subject, and resulted in multiple models and frameworks. Research focuses mainly on how different types of teams (i.e. project, work, parallel or management teams) could be composed, considering task design, team composition and organizational context (Cohen & Bailey, 1997). Task design and team composition are frequently considered together and are interrelated. Task design addresses how work activities are structured and coordinated within teams and determines tasks, responsibilities and authority (Stewart & Barrick, 2000), whereas team composition refers to the process of finding the right people for the task.

Teams are composed by assessing characteristics of individuals that will suit the purpose of the team. The literature defines two approaches to determine characteristics of individuals. First, Cohen & Bailey (1997) determined three variables that should be considered: (1) someone’s individual knowledge, skills and attitudes (KSAs), (2) the diversity of KSA and other characteristics amongst individuals in the teams and (3) the size of the teams. In literature, the three variables are known as the KSAOs. The second approach is selecting individuals based on their personality. Well-known and accepted models are amongst others the five-factor model (FFM), also known as the Big five or OCEAN personality traits (McCrae & Costa, 2008) and Myers-Briggs Type Indicator (MBTI) (Briggs Myers, McCaulley, Quenk, Hammer, & Mitchell, 2009). Both models predict and explain the behaviour of individuals and use quantitative measures to indicate the relative comparability of personality types, for example within teams. In the educational sector, the combination of the two approaches is used to determine someone’s capabilities in teaching students. Qualities of teachers are commonly categorized as knowledge, skills, attitudes, and personality (Timmering, Snoek, & Dietze, 2009).

more moments of contact with the students and focus on creating a safe learning environment for the students. Whereas, cross-grade teams will be more focused on transmitting knowledge on the specific subjects to students across education levels. Creating a safe learning environment depends on their teaching skills, which can be divided into verbal competences, pedagogic competences, dyadic competences and organizational competences (Büttner, Pijl, Bijstra, & Van Den Bosch, 2018; Timmering et al., 2009). The proficiency level on specific subject knowledge depends on someone’s interest, educational background and years of experiences (Salas et al., 2006). The requirements in terms of proficiency of knowledge and skills depends on the organisation structure.

The required personalities for each team can be determined using Myers-Briggs Type Indicator method. This method is more practical than for example the Five-Factor Model, and is more suitable for network configurations (Neil & Petty, 2019). The MBTI method has proven to be valid and reliable in educational and business environments. Composing teams by considering MBTI is found to improve team outcome (McCaulley, 1974). The MBTI self-assessment instrument categorized persons along four dimensions: attitude (extraversion or introversion), perceiving (sensing or intuition), judging (thinking or feeling) and lifestyle preference (perceiving or judging). The test predicts a preference for each dimension, resulting is 16 different personality types (Briggs Myers & McCauley, 1985). The individuals could be tested according the required KSAOs and personalities for each team.

Based on the team requirements and characteristics of individuals, the teams can be composed. Several allocation methods can be found in literature, Mathieu, Tannenbaum, Donsbach, & Alliger (2014) for example, conducted an extensive review on group composition methods. A number of methods can be used to find the best configuration of characteristics within a team, namely through using the team-generic KSAOs (Stevens & Campion, 1994), finding the right person for the position in the team (Cooke et al., 2003), balancing the KSAs within a team (Bell, Villado, Lukasik, Belau, & Briggs, 2011), or assessing the relative contribution of an individual to performance (Humphrey, Morgeson, & Mannor, 2009). The existing models concentrate on either the individual focus and undermining the team focus, or the other way around, so they ignore or oversimplify individual contribution compared to the overall performance (Wolfson & Mathieu, 2017). Additionally, the models focus mainly on temporal (project) teams with a traditional team setup, where team composition is stable over time, members are only assigned to one team, work towards a common goal and with well-defined tasks and roles. This is, however, not the case in the MTM structure, where teams are more fluid (individuals move on and off teams) and often overlap (multi membership), so members have several goals to achieve (Tannenbaum, Mathieu, Salas, & Cohen, 2012). This lead to implications in practice, managers struggling with the composition of teams and tasks in a dynamic and multi-member team setting (Mortensen & Haas, 2018).

2.4 School timetabling problem

Timetabling has been an extensively studied research area due to rapid changes in practice and theory over the past 40 years. Timetabling is the process of planning and scheduling and is applicable in a variety of settings, including the educational setting. In the educational sector, timetabling often refers to school timetables, university timetables and exam timetables. Wren (1995) provides the following general definition of timetabling: “The allocation of given resources to specific objects being placed in space time, in such way as to satisfy as nearly as possible a set of desirable objectives, subjected to constraints.” Nowadays, school timetabling research is mainly focused on trying to find solutions that cope with, on the one hand, the decrease in available resources (i.e. teachers, rooms, equipment), and the increasing number of students that demand specialized education on the other hand. As a result of budget cuts by the government and a scarcity of teachers in society, educational institutes are forced to use their resources more effectively and efficiently (Oude Vrielink, Jansen, Hans, & van Hillegersberg, 2019). Furthermore, changing from a conventional education towards a more personalized and student-centered approach, asks for more flexibility from various facets in educational logistics (Jonassen & Land, 2012; Oude Vrielink et al., 2019).

The construction of school timetabling models is carried out in many different ways and relies on the manner in which academics and educational institutions define the problem requirements and the hard (mandatory) and soft (desirable) constraints (Pillay, 2014). It is extremely difficult to construct and apply a general model due to the variety of constraints, diversity of the problem and specific requirements of the education institution (Jat & Yang, 2009). Solving the problem would mean allocating a set of classes, teachers and rooms to timeslots to satisfy the defined hard and soft constraints (Post et al., 2012). Traditional optimization techniques are used to solve the problems. The most popular solving methods are: evolutionary algorithms, tabu search, integer programming and many hybrid forms of the optimization techniques. Pillay (2014) conducted a systematic research into the different techniques and concluded that all techniques solved the problem and came up with a feasible and good qualitative solution. Nevertheless, criticism have been given that the majority of the studies come up with an intelligent research tool that could not be used in a practical application at the educational institutions. When developing new timetables, the user should be able to make manual changes to the constraints and the weighting of the constraints. Unlike others, integer or linear programming is one of the techniques that could be easily adjusted because of the mathematical representation (Ribic, Turcinhozic, & Muratovic-Ribic, 2015).

The main advantage of integer linear programming (ILP) is the distinction between hard constraints defined by an equation and soft constraints defined through the objective function (Ribic et al., 2015). Additionally, ILP provides the possibility to weigh certain constraints or part of the objective function to differentiate in importance (Schaerf, 1999). The hard and soft constraints could be categorized into seven categories: problem requirements constraints, no clashes constraints, resource utilization constraints, workload constraints, timeslot distribution constraints, preference constraints and lesson constraints (Pillay, 2014; Ribic et al., 2015). The seven categories consist of several constraints that could be used while constructing the school timetabling model. Which constraint could actually be used, depends on the specific problem and requirements of the educational institute.

2.5 Integrated approach for composition and timetabling of teams

functional organization into a team-based organization with specific within-grade and cross-grade level teams. Working in teams enables the institution to become more effective and flexible towards (un)certain changes in the environment. Multi-membership gives an organization the opportunity to provide specialized and quality education to all their students. Cross-grade level teams, hereafter referred to as expert teams, are responsible for transmitting specific knowledge and provide personalized education when needed. Whereas within-grade level teams, hereafter referred as level teams, provide less-specialized education and are responsible for creating a safe learning environment by having a close relationship with the students and parents.

This approach creates an effective and flexible organization in a number of different ways. Flexibility is created by shifting the responsibility of providing education from individuals towards the team, making the organization less vulnerable in case of temporary staffing problems and decreasing the student-teacher relationship dependency. The introduction of expert teams creates the opportunity to educate students on their own level and expert areas. Providing personalized education is more effective and will benefit the learning curve of the student. Additionally, expert teams could easily decide to switch students between different levels and specific expert subjects. The same flexibility is created in the level teams; the teams can decide which subjects will be taught and when.

In terms of timetabling, flexibility is created by scheduling teams instead of an individual to a particular timeslot. This allows teams to postpone the decision about which staff member and course will be educated. Regardless of the decision of the team, the students will know in advance which expert or level team is providing education. The planning of individuals and timetabling of teams is separated, making this approach more flexible and effective.

Figure 2.1 shows the main components of the integrated approach for composition and timetabling of teams in a MTM environment. The individuals presented on the left side of the figure represent the staff members of the educational institute. They are allocated to an expert or level team based on level of knowledge, skills, competences and personality. The required level of knowledge and skills is determined by the taskwork which differs for each team. Teamwork is considered by the personality traits of MBTI. The MTM structure is created by allocating individual staff members to both expert teams and to level teams. Staff members divide their working hours over multiple teams that have different goals. This structure results in that the capacity for providing education depends on the availability of individuals and their allocation to multiple teams. The timetables for all education levels and year can be constructed knowing the team composition and their capacity for providing education. The allocation of team to timeslot, education level and year is presented on the right side of the figure. Teams are allocated to particular timeslots considering the school timetabling principles. The “person figure” in the timetable represents that particular expert team and the shaded squares are associated with the different level teams.

3 METHODS

3.1 Problem analysis

This thesis has the opportunity to apply the formulated integrated approach at a special needs educational institute in the northern part of the Netherlands. This particular educational institute is facing challenges with finding and contracting a sufficient number of well-educated employees. Developments in globalization, such as ageing of society, results in a staffing problem for educational institutes in general. The staffing problem in the special needs education (SNE) is also closely related with the biased salary system. The working environment within a SNE is more complex and dynamic compared to primary education, however, teachers and specialists earn the same salary as teachers in a primary education institute. The new generation of teachers therefore prefers to work for primary institutions instead of SNE institutions. The national government is currently working on a solution for the salary issue within SNE.

The staffing problem results in a number of related problems. The most important problem is the influence on the quality of education. Studies have shown that a shortage in personnel and a high staff turnover influence the quality of education. Some barriers for delivering high quality education include a higher workload, constant changes, conflicting priorities, timetabling and poor staff morale. Poor quality of education influences the development of skills and competences of students and will impact the employability negatively, which directly affects the identity of the educational institute (Dicker, Garcia, Kelly, & Mulrooney, 2018). This specific SNE institution acknowledged this problem and is actively searching for solutions to overcome the barriers for delivering high quality education.

Educational logistics and a qualified workforce are the most important factors for providing high quality education for children with SNE. Close collaboration between the workforce, consisting of teachers, teacher’s assistants and specialists (e.g. psychologists, therapists or social workers), should be carefully organized. The management of the institution and multidisciplinary workforce should create a stable, secure and predictable learning environment for children with SNE (Bellour, Bartolo, & Kyriaopoulou, 2017).

The educational institute is part of regional expertise cooperation in the northern part of the Netherlands that provides primary and secondary special needs education to up to 2300 students. This institution is indicated as SNE for students that have EBD (cluster 4), meaning that the students have learning and behavioural difficulties as a result from physical, intellectual and communications disorders (FPIES, 2018). Education is provided to around 185 students from the age of 9 to 18 years old. The organization and logistics are arranged customary to the Dutch secondary education levels of pre-vocational education (VMBO) and higher continued education (HAVO). The education levels are specified into four levels: practical education (VMBO-Arbeid), basic profession-orientated learning path (VMBO-BB), theoretical learning path (VMBO-TL) and higher continued education (HAVO). Students in secondary education usually attend an education level for four years straight.

The workforce of this institution consists of around 50 people who all have different background and disciplines. The staffing problem concerns the employees that are interacting with the students. This group is divided into employees that provide education (i.e. mentor-teachers and expert-mentor-teachers) and supporting employees (i.e. behavioural specialists, traineeship coordinators and internal coaches). In the current situation, mentors are assigned to a particular education level and education year. The mentors are present in the classroom for the majority of the day and act as focal point for students and parents. The expert teachers provide technical education (i.e. Language, Math, Physics/Chemistry, Biology) for all education levels and education years. Some expert teachers provide education in a designated classroom due to materials needed that are present in a particular classroom (i.e. hospitality, technical, land scape, nursing). The supportive group of employees supports the teachers when facing emotional and behavioural problems with specific students and facilitate personalized education.

Considering the current situation and the developments in the sector, a distinction in teams can be made based on expertise and educational level. Expert teams are responsible for providing expertise education (i.e. language and math), as well as personalized expertise education to students in need. Furthermore, they will accumulate and disseminate knowledge to other staff members. The level teams are associated with the Dutch educational levels. The less expertise courses are provided by the level teams and they are also responsible for the relationship with the students and parents. The multidisciplinary teams are composed of staff members that provide education and support employees.

The integrated approach for composition and timetabling of teams is developed as a weighted linear assignment model with four individual assignments. ILP is a simple technique that determines the optimal use of resources to minimize or maximize a certain parameter. The model includes a number of individuals, a number of expert- and level teams and a number of timeslots. This model determines optimal use of resources (individuals and timeslots) to maximize four ambitions. The first ambition is the individual-expert team assignment that should maximize the available content knowledge in each expert team. The second ambition is the individual-level team assignment that should maximize the level of competences and skills within in each level team. The third and fourth ambition is the expert team-timeslot and level team-timeslot assignment, which should maximize the time preference value. The ILP model is subjected to a number of constraints, these constraints are developed based on the team composition and school timetabling literature. The weighted ILP is coded in Mosel-language and solved in the optimizing software Xpress IVE.

3.2 ILP model for composition and timetabling of teams

The integrated team composition and timetable model contains three terms. The first term allocates individuals to expert teams based on their proficiency on content knowledge. While doing this, the objective is to assign a sufficient number of individuals, who prefer a certain expert team, in a way that they cover the required number of hours the expert teams needs to teach. The model considers a set of individuals 𝑖 ∈ 𝐼 where 𝑖 = 1, … ,37 which are teachers and teaching assistants who are full-time or part-time available for providing education. Individuals have certain characteristic (i.e. availability, competences, personality profile) that are used throughout the model. Expert teams are defined by the educational institutions themselves, in this case a set of expert teams 𝑗 ∈ 𝐽 and each team represents a expertise where in this case j=1: language, j=2: math, j=3: behaviour and mental resilience and j=4: work field.

The second term of the model contains the allocation of individuals to level teams. Individuals will in most cases be allocated to both expert- and level teams. Level teams are responsible for teaching the courses that are related to the education level of the students and maintaining a relationship with the student and their environment. The main objective is to create a quality learning environment for the students. The individuals assigned to the level teams should possess competences and skills for creating this learning environment. This model considers a set of education levels 𝑘 ∈ 𝐾 and is defined along the Dutch secondary education system where k=1: Airbeid/BB, k=2: BB/KB, k=3: KB/TL and k=4: TL/HAVO. The set of education levels K also represents the four level teams in the model.

The third term of the model is the actual scheduling of expert and level teams in such a way that several weekly timetables are created for all educational levels and years of education. The main objective is to schedule the teams on their preferred timeslots and ensure the number of scheduled timeslots complies with the Dutch educational law. In the Netherlands, students usually spend four years in secondary education before going to college or university. Therefore, the set of educational years is defined as 𝑙 ∈ 𝐿 where 𝑙 = 1, … ,4. In the Netherlands, education is provided five days a week, from Monday until Friday. The number of teaching hours per day differs between education institutions. The model considers nine teaching hours per day, resulting in 45 teaching timeslots per week. The set of timeslots is defined as 𝑡 ∈ 𝑇 where t=1: Monday 1st _{hour, t=2: Monday 2}nd _{hour and t=45: Friday 9}th _hour.

3.2.1 Input parameters of the model

determine the outcome of the model while allocating individuals to different teams and allocating the teams to a particular timeslots, educational levels and years.

Table 3.1

Input parameters for the objective function

Symbol Description Value

𝐷₅₆ Knowledge level of individual i in expertise j 𝐷₅₆ ∈ {0,2,04, … ,1} 𝐹₅ Teaching competence level of individual i 𝐹₅ ∈ {0,2,0,4, … ,1}

𝐺6= Time preference for expert team j in timeslot t 𝐺6= ∈ [0,1; 0,3; 0,5; 0,7; 0,9] 𝐻_D= Time preference for level team k in timeslot t 𝐻_D= ∈ [0,1; 0,3; 0,5; 0,7; 0,9]

The first parameter matrix is the level of knowledge of an individual for a certain expertise. Individuals have been asked to indicate their level of knowledge for the four expertise’s using a 5-likert scale. Where 𝐷56 = 0,2 is having the lowest level and 𝐷56 = 1 the highest level for a particular expertise. The parameter matrix 𝐷₅₆ is shown in table 3.2.

Table 3.2

Parameter matrix D(i,j)

Individuals Expert team 1 2 3 4 1 0,2 0,8 0,8 0,4 2 1 0,4 0,8 0,2 3 0,2 0,4 0,8 1 … … … … … 37 0,6 1 0,2 0,8

Note. The entire parameter matrix is added in the Appendix table A-1

The second parameter matrix is the average competence level of individuals. The competences are associated with someone’s qualities for teaching (EBD) students, better known as teacher qualities (Büttner et al., 2018). The teacher should be able to satisfy the three phycological needs (i.e. competences, autonomy and relatedness) of a student which represent the necessary conditions for students’ optimal learning development. The following competences are considered: verbal competences, engaging in learner-centric and teacher-student relationship, teaching strategies and interaction styles. In order to identify individual competences, the observation tool, modified Norwegian Teacher self-efficacy test of Büttner et al. (2018) can be used. The measurement tools have proven to be adequate and provide a graded level for each individual competence. The average score of the five competences has been used to define the competence level of an individual, where 𝐹₅ = 0,2 is the lowest and 𝐹₅ = 1 is the highest level of competence, resulting in the parameter matrix 𝐹₅, which is shown in table 3.3.

Table 3.3

Parameter matrix F(i)

Individuals Teaching competences

1 0,8

3 0,8

… …

37 0,4

Note. The entire parameter matrix is added in the Appendix, table A-1.

Criticism have been raised that the majority the timetables models have not the possibility to make manual changes by the users (Pillay, 2014). The soft constraint for timeslot preference gives the user the possibility to influence the outcome of the model. The third parameter matrix is time preference of an expert team for a particular timeslot and weekday, table 3.4 presents the time preference matrix for expert team 1. The education institute prefers to allocate expert courses to the fourth, fifth and sixth timeslot of the day, the time preference value is therefore the highest compared to the other timeslots. Furthermore, the education institute prefers to start every day with education provided by the level team. The time preference value for the expert teams is therefore low and the preference value for level teams is high in the first two timeslots (see table 3.5). The time preference matrices for the other teams are added in Appendix A, tables A-3 until A-7.

Table 3.4

Parameter matrix G(j,t) – Time preference table for expert team 1

Hours

Workday

Monday Tuesday Wednesday Thursday Friday

1 0,1 0,1 0,1 0,1 0,1 2 0,1 0,1 0,1 0,1 0,1 3 0,7 0,7 0,7 0,7 0,1 4 0,9 0,9 0,9 0,7 0,7 5 0,9 0,9 0,9 0,7 0,7 6 0,9 0,9 0,9 0,7 0,7 7 0,7 0,7 0,7 0,1 0,7 8 0,5 0,5 0,5 0,1 0,5 9 0,5 0,5 0,5 0,1 0,5

Note. The preference table for the other expert teams are added in Appendix A.

Table 3.5

Parameter matrix H(k,t) – Time preference table for level team 2.

Hours

Workday

Monday Tuesday Wednesday Thursday Friday

1 0,9 0,9 0,9 0,9 0,3 2 0,9 0,9 0,9 0,9 0,3 3 0,7 0,7 0,7 0,7 0,3 4 0,7 0,7 0,7 0,7 0,3 5 0,5 0,5 0,5 0,5 0,3 6 0,7 0,7 0,1 0,1 0,3 7 0,7 0,1 0,1 0,1 0,3 8 0,5 0,1 0,1 0,1 0,3 9 0,5 0,1 0,1 0,1 0,3

3.2.2 Other parameters

Besides the input parameter matrices mentioned in the previous section, some additional input parameters are defined for the constraints. The parameter matrices are explained in de constraint section and can be found in the Appendix B.

Table 3.6

Input parameters for the constraints.

Symbol Description Value

𝑎5= Availability of individual i in timeslot t 𝑎5=∈ [0,1]

𝑅_6DGH

Number of timeslots t required for expert team j, level k and year l 𝑅6DGH ∈ {0,1,2, … , 𝑛}

𝑅DGG Number of timeslots t required for level team k and year l 𝑅DGG ∈ {0,1,2, … , 𝑛}

𝑆6H Level of knowledge required for expert team j 𝑆6H ∈ {0,2,0,4, … ,1}

𝑆DG Level of teaching skills required for level team k 𝑆DG ∈ {0,2,0,4, … ,1}

𝐸𝐼5 Personality dimension focus of attention EI of individual i 𝐸𝐼5∈ [0,1]

𝑆𝑁5 Personality dimension information intake SN of individual i 𝑆𝑁5 ∈ [0,1]

𝑇𝐹5 Personality dimension decision making TF of individual i 𝑇𝐹5∈ [0,1]

𝐽𝑃5 Personality dimension way of working JP of individual i 𝐽𝑃5∈ [0,1]

3.3 Integer linear programming model

The ILP model will be presented in this section. The model contains four families of decision variables with a total of 3896 variables. Additionally, the objective function is defined and this section explains the twenty-four families of constraints.

3.3.1 Decision variables

Four decision variables are defined that will indicate whether an individual will be assigned to a particular expert team (𝑋56) or level team (𝑌5D). Both decision variables will initiate a soft constraint in the objective function. The other two decision variables indicate the allocation of an expert team (𝐸𝑇6=DG) and level team (𝐿𝑇=DG) to a particular timeslot, education level and year. The four decision variables can only be binary variables.

3.3.2 Objective function

The objective function is defined as a maximalization problem that contains three terms. A weight factor (𝑊𝐸, 𝑊𝐿 𝑎𝑛𝑑 𝑊𝑇) between 0 and 1 could be added to each term to Let us explain the following decision variables:

𝑋₅₆ = 1 If individual 𝑖 ∈ 𝐼 is assigned to expert team 𝑗 ∈ 𝐽; 𝑋₅₆ = 0, otherwise 𝑌5D = 1 If individual 𝑖 ∈ 𝐼 is assigned to level team 𝑘 ∈ 𝐾; 𝑌5D = 0, otherwise 𝐸𝑇_6=DG = 1 If expert team 𝑗 ∈ 𝐽 is assigned to timeslot 𝑡 ∈ 𝑇 and provide classes to _{education level 𝑘 ∈ 𝐾 and education year 𝑙 ∈ 𝐿 ; 𝐸𝑇}

6=DG = 0, otherwise 𝐿𝑇=DG = 1 If level team 𝑘 ∈ 𝐾 is assigned to timeslot 𝑡 ∈ 𝑇 and provide classes to _{education year 𝑙 ∈ 𝐿 ; 𝐿𝑇}

differentiate in importance. The first term multiplies the individual knowledge level of an expertise (𝐷56) by the decision variable (𝑋56), which can either take the value 0 or 1. The function should maximize the available knowledge in all the expert teams. The second term relates to the level of teaching competences (teacher quality) of individuals in each level team. The term multiplies the level of competences of an individual (𝐹₅) by the decision variable (𝑌_5D) that can take a 0 or 1. The third and last term is associated with the constructing of a weekly timetable. The previous composed teams will be allocated to unique timeslots, educational level and year. Particular expert teams could be allocated to unique timeslots by the decision variable 𝐸𝑇_6=DG (value 0 or 1). The value will be multiplied by the preference value of that particular expert team (𝐺₆₌). The second part of term three is based on the same assumption, when a particular level team is allocated indicated by decision variable 𝐿𝑇_=DG it will be multiplied by the preference value of that particular level team (𝐻D=).

𝑀𝑎𝑥: 𝑊𝐸 W W 𝑋₅₆𝐷₅₆ 6∈X 5∈Y + 𝑊𝐿 W W 𝑌_5D 𝐹₅ D∈[ 5∈Y + 𝑊𝑇 W W(W W 𝐸𝑇_6=DG𝐺₆₌ D∈[ 6∈X + W 𝐿𝑇_=DG𝐻_D= D∈[ ) G∈\ =∈] 3.3.3 Constraints

Constraint – Individuals possess the right amount of knowledge for an expert team Individuals can only be allocated to an expert team if they possess a certain level of knowledge. Parameter matrix 𝐷₅₆ indicates the level of knowledge of each individual. Professionals from the institute have been asked to indicate the required level and importance for each expert team, resulting in the parameter matrix 𝑆₆H _{(see table B-1). Individuals may be} assigned to expert team when possessing a minimal level of knowledge indicated by 𝑆₆H_{, but it} could also mean that individual will not be assigned to expert team. In the case when the knowledge level for an expertise is lower than required 𝑆₆H_{, it is not allowed to assign that} individual to the expert team.

𝑋56 = ^0 𝑜𝑟 1 ₀

𝑖𝑓 𝐷₅₆ ≥ 𝑆₆H 𝑖𝑓 𝐷56 < 𝑆6H

The following linear restrictions, using Big M with M=1, will make sure that the above-mentioned constraint is valid.

𝑋₅₆ ≤ 1 + 𝐷₅₆− 𝑆₆H _{∀ 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 (1)}

𝑌_5D = ^0 𝑜𝑟 1 0

𝑖𝑓 𝐹₅ ≥ 𝑆_DG 𝑖𝑓 𝐹5 < 𝑆DG

The following linear restrictions, using Big M with M=1, will make sure that the above-mentioned constraint is valid.

𝑌_5D ≤ 1 + 𝐹₅− 𝑆_DG _{∀ 𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 (2)}

Constraint – Sufficient number of individuals assigned to expert or level teams

The following two constraints connects the processes of allocation of individuals to teams and the allocation of teams to timeslots. The model should ensure that the individuals assigned to a particular team should together have sufficient capacity to teach the courses. Therefore, the sum of all individuals assigned to a particular team times the available timeslots of the individuals should be equal or bigger than the required timeslots for providing education. The availability matrix (𝑎₅₌) and the matrix for the required timeslots for each expert team, educational level and educational year (𝑅_6DGH _{) are added in Appendix B, table B-2 and B-3.}

W W 𝑋₅₆𝑎₅₌ =∈] ≥ W W 𝑅_6DGH G∈\ D∈[ 5∈Y ∀ 𝑗 ∈ 𝐽 (3)

The following constraint ensures that sufficient individuals will be assigned to the level teams. The required timeslots matrix for each level team and education year (𝑅_DGG _{) is added in} Appendix B. Additionally, in consultation with the educational institute, the decision was made to create overcapacity in each level team in order to cope with the absenteeism of individuals. The level teams are the focal point to the students, therefore extra capacity is created in the level teams instead of the expert teams. In the Netherlands, the percentage of absenteeism due to illness in the SNE is 7 percent (DUO, 2019). Information about other reasons of absenteeism (i.e. maternity leave, parental leave or other special leave) is not available, therefore the percentage for other reasons of absenteeism is set at 13 percent. The percentage of absenteeism should be determined by each individual educational institute and can be used as input for this model. W W 𝑌_5D𝑎₅₌ =∈] ≥ 1.2 W W 𝑅_DGG G∈\ D∈[ 5∈Y ∀ 𝑘 ∈ 𝐾 (4)

Constraint –Team size for expert teams and level teams

W 𝑋₅₆ ≥ 4 5∈Y ∀ 𝑗 ∈ 𝐽 (5) W 𝑌_5D ≥ 4 5∈Y ∀ 𝑘 ∈ 𝐾 (6) W 𝑋56 ≤ 10 5∈Y ∀ 𝑗 ∈ 𝐽 (7) W 𝑌_5D ≤ 7 5∈Y ∀ 𝑘 ∈ 𝐾 (8)

Constraint – Restriction for the number of individuals assigned to an expert or level team Individuals can only be assigned to one expert team and one level team (constraint 9 and 10). All individuals should be allocated to at least one expert or one level team (constraint 11) W 𝑋₅₆ 6∈X ≤ 1 _{∀ 𝑖 ∈ 𝐼 (9)} W 𝑌_5D ≤ 1 D∈[ ∀ 𝑖 ∈ 𝐼 (10) W Wh𝑋₅₆ + 𝑌_5Di ≥ 1 D∈[ 6∈X ∀ 𝑖 ∈ 𝐼 (11)

Constraint – Leadership personality in a level team

Literature states that in each team an individual with a leadership personality should be added (Salas et al., 2004). Following the MBTI test, the personality profile that includes ETJ is indicated as a leader. The MBTI personality test will show whether an individual is Extrovert (E) or introvert (I) and gives 1 if someone is extrovert and 0 if someone is introvert. This is the same for the other personality dimension: T=1, F=0, J=1 and P=0. Thus, four personality dimensions results in a personality profile that consists of four binary numbers. The personality profile of all individuals is added in table B-5 in Appendix B. This constraint only uses three of the four personality dimensions.

W 𝑌5D(𝐸𝐼5 ∗ 𝑇𝐹5∗ 𝐽𝑃5 ) 5∈Y

≥ 1 _{∀ 𝑘 ∈ 𝐾 (12)}

Constraint – Heterogeneity in level teams

et al., 2017). Furthermore, diverse teams are more flexible and members can more easily switch between different teams (O’Leary et al., 2011). The more diverse a team is in terms of the dimension of perceiving, judging and attitude, the higher the congeniality. Team composition of personality should aim for heterogeneity and not for homogeneity. A healthy balance or heterogeneity is created by allowing a certain range of -1 and +1 for each level team. The three personality dimensions will be checked separately. The sum of all people assigned to a level team times the personality value of that individual should be bigger or equal (for the -1 range) and smaller or equal (for the +1 range) than the average number of individuals with that personality that could be assigned to all teams.

W 𝑆𝑁₅𝑌_5D 5∈Y ≥ k∑ 𝑆𝑁5∈Y 5 |𝐾| − 1n ∀ 𝑘 ∈ 𝐾 (13) W 𝑆𝑁₅𝑌_5D 5∈Y ≤ k∑ 𝑆𝑁5∈Y 5 |𝐾| + 1n ∀ 𝑘 ∈ 𝐾 (14) W 𝑇𝐹₅𝑌_5D 5∈Y ≥ k∑5∈Y𝑇𝐹5 |𝐾| − 1n ∀ 𝑘 ∈ 𝐾 (15) W 𝑇𝐹₅𝑌_5D 5∈Y ≤ k∑5∈Y𝑇𝐹5 |𝐾| + 1n ∀ 𝑘 ∈ 𝐾 (16) W 𝐸𝐼5𝑌5D 5∈Y ≥ k∑5∈Y_|𝐾|𝐸𝐼5 − 1n ∀ 𝑘 ∈ 𝐾 (17) W 𝐸𝐼5𝑌5D 5∈Y ≤ k∑5∈Y𝐸𝐼5 |𝐾| + 1n ∀ 𝑘 ∈ 𝐾 (18)

Constraint – Schedule teams only when individuals are available

Constraints 19 and 20 ensures the number of times a specific expert- or level team is allocated, is not more than the number of individuals in that specific team. This is applicable for each timeslot in all education levels and in all years. Furthermore, this can only be done if the individual is available in that allocated timeslot. For example, expert team 1 consist of four individuals which are all available in timeslot ten. In all education levels and years, the model can only allocate this team four times to timeslot ten. Otherwise, there will not be an individual available in that timeslot.

W W 𝐸𝑇6=DG ≤ W 𝑋56𝑎5= 5∈Y G∈\ D∈[ ∀ 𝑡 ∈ 𝑇, 𝑗 ∈ 𝐽 (19) W W 𝐿𝑇_=DG ≤ W 𝑌_5D𝑎₅₌ 5∈Y G∈\ D∈[ ∀ 𝑡 ∈ 𝑇, 𝑘 ∈ 𝐾 (20)

Constraint – All courses need to be scheduled

Constraint – Only one expert or level team per timeslot in educational level and year This constraint ensures that only one team, either expert team or level team, is assigned to a timeslot within an educational level and educational year.

𝐸𝑇_6=DG + 𝐿𝑇_=DG ≤ 1 ∀ 𝑗 ∈ 𝐽 𝑡 ∈ 𝑇, 𝑘 ∈ 𝐾, 𝑙 ∈ 𝐿 (23)

Constraint – Expert team sequences in timetable

The education institution argued that expert team j=1 (Language) and j=2 (Math) is not allowed to provide education in the succeeding timeslot in the same education level and education year. The institution state that EBD students are not able to concentrate for two succeeding timeslots while educating this expertise.

𝐸𝑇_6=DG + 𝐸𝑇_6=opDG ≤ 1 ∀ 𝑡 ∈ 𝑇 − 1, 𝑘 ∈ 𝐾, 𝑙 ∈ 𝐿, 𝑗 = 1, 𝑗 = 2 (24)

3.4 Constraints evaluation and ILP coding

The existing literature provide a number of requirements for team compositions and school timetabling problems. These requirements are included in the formulated ILP model by means of soft and hard constraints. Table 3.7 provides an overview of which constraints represents the specific problem requirement. The team composition problem can be seen a nearly complete problem, the formulated ILP problem includes the requirements that are used for creating effective teams (i.e. KSAO’s and personality traits). On the other hand, the school timetabling problem cannot be seen as a full school timetabling problem. Ribic et al. (2015) presents the most complete ILP model which consist of 37 constraints divided over seven categories. The formulated ILP model consists of only five unique constraints.

The formulated ILP model is coded according the Mosel language in the optimization software Xpress IVE. The in the previous section mentioned input parameter matrices are initialized from an Excel workbook. The entire programming code and explanation are added in Appendix C.

Table 3.7

Constraints evaluation of the ILP model

Problem requirements Constraints included in the ILP

model

Team composition

problem

1. Expertise knowledge (1) and SC in objective function 2. Skill competences (2) and SC in objective function

3. Team size (5), (6), (7) and (8)

4. Leadership personality (12)

5. Diversity within teams (13), (14), (15), (16), (17) and (18) 6. Problem specific (3), (4), (9), (10) and (11)

School timetabling

problem

1. Problem requirements (21) and (22)

2. No clashes (23)

3. Resource utilization (19) and (20)

4. Workload

5. Timeslot distribution

6. Preference SC in objective function

7. Lessons (24)

Note. SC = Soft constraint

4 RESULTS AND DISCUSSION

4.1 Outcome of the ILP model

The developed ILP model have been coded and solved in Xpress IVE with a 2,3Ghz Intel Core i5 processor. The computing time was in the seconds. The required level of knowledge and competences for the team composition of expert and level teams have been provided by the management of the educational institute. Unfortunately, the information about the level of competences and personality profiles of the individuals were not available. The input parameters for the level of competences are therefore randomly chosen. The frequency of personality profiles in the US have been used to give all individuals an MBTI personality profile (Neil & Petty, 2019). The input parameters and required levels used for computing these results can be found in Appendix A and B.

The ILP model is developed a weighted maximisation problem with four soft constraints in the objective function and is subjected to twenty-four hard constraints. The weights for this outcome are agreed upon WE=0.3, WL=0.5 and WT=0.7. The composition of level teams with sufficient capacity and constructing a weekly timetable is more important than the composition of expert teams. The hard constraints should be satisfied in order to come up with team composition and timetable for the teams.

At first the initial developed model did not come with a solution due to the heterogeneity constraints (constraints 13 until 16). These constraints ensure that a diversity of personalities is allocated to the different level teams. The allowance range for the constraints had to be increased in order to get a solution. Constraints 13 and 14 had to be changed from -1 and +1 to -2 and +2. The range for constraints 15 and 16 had to be increased to -3 and +3. This increase resulted in a solution for the team composition and timetables.

The model allocated the 37 individuals to one of the expert teams and also allocated 28 of them to one of the level teams. The allocation solution of individuals is presented in table 4.1. The weighted maximised available knowledge in each expert team is 10,74 and the total level of competences in each level team has a weighted value of 8,8. The model allocated 10 individuals to expert teams 1, 2 and 4. Expert team 3 consist of 7 individuals. In this solution 7 individuals are assigned to each level team. Additionally, the model constructed 16 different timetables by allocating the teams to the timeslots, education levels and years. The timetable for education level 2 (BB/KB) year 1 is presented in table 4.2, the rest of the timetables can be found in Appendix D. The weighted maximum time preference value for the 16 timetables is together 344.12. The computed maximum value for this solution is 363,66.

Table. 4.1

Allocation results of individuals to expert teams and level teams.

Individuals

Expert team allocation Level team allocation

Team number Value D(i,j) Team number Value F(i)

1 3 0,8 3 0,8 2 1 1 3 0,6 3 4 1 2 0,8 4 2 1 4 0,8 5 4 1 2 0,6 6 4 0,8 1 0,8 7 4 1 4 0,8 8 1 1 3 0,8 9 2 1 4 0,6 10 1 1 1 0,8 11 4 1 3 0,4 12 2 1 4 0,4 13 1 1 - - 14 2 1 1 0,8 15 1 1 1 0,6 16 3 0,8 2 0,8 17 3 1 3 0,6 18 1 1 - - 19 2 0,8 - - 20 1 1 - - 21 4 1 3 0,6 22 1 1 1 0,6 23 1 1 2 0,6 24 3 0,8 2 0,6 25 1 1 2 0,6 26 4 1 - - 27 2 1 4 0,6 28 4 1 - - 29 3 1 - - 30 2 1 3 0,4 31 4 1 - - 32 3 0,8 - - 33 2 1 1 0,6 34 3 0,8 4 0,6 35 2 1 4 0,4 36 4 1 1 0,6 37 2 1 3 0,4

knowledge for that team (constraint 1). Furthermore, the model ensures that the assigned individuals have sufficient capacity for providing the expertise courses to all education levels and years (constraint 3). Lastly, all individuals are assigned to only one expert team (constraint 9). Thus, it can be concluded that all team composition constraints for expert teams are satisfied.

The second part of the objective function is the allocation of individuals to level teams in a way that it maximizes the teaching competences in each level team and satisfy all hard constraints. The model only allocates individuals to a level team if they possess the minimal required level of teaching competences for that team (constraint 2). All level teams consist of minimal 4 and maximum 7 individuals (constraints 4 and 6). The composed level teams have sufficient capacity and 20 percent overcapacity for providing education to all education years (constraint 12). In addition, to ensure effective teamwork, multiple constraints are added to create diversity among personalities and make sure that all level teams have a leader. In all level teams is an individual allocated with the leadership personality (constraint 12) and the adjusted constraints for creating diverse level teams in terms of personalities also worked (constraints 13 until 18). The composed solution for the level teams satisfies all constraints.

Thirdly, the composed expert- and level teams are allocated to certain timeslots, education levels and years. The objective is to allocate the teams to unique timeslots in such a way that the preference value for all allocations is maximized. The timetable solution will be evaluated by considering the solution for education level 2 year 1 (Table 4.2). Starting with the level team, the preference value for the first timeslots on the day is higher for level team than for the expert teams. Level teams are therefore allocated to the first three timeslots on Monday, Tuesday and Wednesday (grey shaded). This is not the case on Thursday morning, this is explained by the fact that expert team 4 (yellow shaded) have a higher preference value compared to rest of the teams on Thursday and Friday. Furthermore, the preference value for expert teams 1, 2 and 3 is the highest on the fourth, fifth and sixth timeslot on all workdays. Expert team 1 (red shaded) is therefore allocated to fourth, fifth and sixth timeslot on Monday, expert team 2 (blue shaded) is allocated to the sixth slot on Wednesday and expert team 3 (green shaded) to the fourth timeslot on Wednesday. The preference value in the objective function have proven to be working.

Table 4.2

Timetable result– Education level 2 and year 1

Hours

Workday

Monday Tuesday Wednesday Thursday Friday

1 1 10 19 28 37 2 2 11 20 29 38 3 3 12 21 30 39 4 4 13 22 31 40 5 5 14 23 32 41 6 6 15 24 33 42 7 7 16 25 34 43 8 8 17 26 35 44 9 9 18 27 36 45

Note. The numbers in the right corner corresponds with the timeslot number.

timeslot, thus each team have enough individuals available. The required number for each expert- and level team are scheduled (constraint 21 and 22) and only one team is allocated to a timeslot in each education level and year (constraint 23). Constraint 24 could not be added by the researcher, therefore expert team 1 and 2 are scheduled in succeeding timeslots.

4.2 Sensitivity Analysis

Sensitivity analysis is performed to check the robustness of the solutions and to check whether the model is working properly. The formulated problem in the previous section is tested by changing one variable at the time. In figure 4.1a changes the maximum number of individuals in an expert team by considering 6, 8, 10, 12 and 14. In figure 4.1b illustrates the results of changing the maximum number of individuals in level teams by considering 6, 8, 10, 12 and 14. Figure 4.1c considers changes in total number of available timeslots of all individuals.

In figure 4.1a and 4.1b, it is surprising to see that any change in team composition results in a decrease or increase of the objective function and that this change doesn’t affect the preference value (Term 3). The increase of the objective function can be explained by allowing more individuals to be allocated to expert or level teams, gives the model the opportunity to allocate the individuals to teams where they possess the highest knowledge or level of competences. However, the change in team composition should also affect the allocation of teams to timeslots. This observation implies that the composed teams does not function as an input for the timetabling process.

Furthermore, the parameter set of the formulated problem has in total of 1400 unique timeslots available in one week. By decreasing the available timeslots, the connection between team composition and timetabling can be tested. The analysis shows that decreasing the available unique timeslots doesn’t affect the objective value (figure 4.1c). Decreasing the available number to 600 results in no solution which is related to the fact that in total 636 slots need to be scheduled for the teams (constraint 21 and 22). As long as the available number of timeslots is higher than the minimal requested, a change in the available timeslots doesn’t affect the composition of expert- and level teams.

The sensitivity analysis shows that the interaction between the allocation of individuals to teams and the allocation of teams to timeslots is missing in the developed model. Constraint (a) Different expert team sizes (b) Different level team sizes (c) Different available timeslots Figure 4.1 – Sensitivity analysis for the ILP for composition and timetabling of teams. The objective value (shown in blue) is the maximum value of the objective function and TERM 3 (shown in orange) is the time preference value of the expert- and level teams together.

18 and 19 supposed to create this interaction in the formulated ILP model. Thus, it can be concluded that the model is not working properly on these constraints. The robustness of the solutions is therefore questionable.

4.3 Discussion

The purpose of this thesis was to enhance the knowledge of the MTM working approach by considering team composition and planning principles. This thesis enhances the literature by proposing an organizational structure for educational logistics that incorporates MTM. Additionally, the insights in MTM and proposed organizational structure functioned as an input for the development of an integer linear programming model. The model simultaneously addresses team composition and timetabling while considering the aspects that makes MTM effective. Although this model is not fully functioning and is only tested for one set of parameters, several general insights can be found that can support the education sector in becoming a more effective and flexible organization.

Firstly, individuals working in multiple teams at the same time can harm the performance of individuals and teams (Pluut et al., 2014). While designing MTM structures and composing teams, it is therefore important to simultaneously consider taskwork (activities) and teamwork (how people work together). The proposed organizational structure gives an insight on how the taskwork an teamwork can be considered by designing work teams in the education. Structuring the organization alongside the required specializations for providing good quality education results in expert teams, these teams can take on a more flexible and adaptive approach towards uncertain situations (i.e. staffing problems) (Chi et al., 1981; Salas et al., 2006). This thesis gives an insight in how educational institutes can design their activities along expertise’s, by dividing the organization in expert teams (specialized in certain educational topics) and in level teams (specialized in providing a good learning environment). Structuring the organization ensure qualitative education. Additionally, more intense collaboration between teachers with the same expertise increases job satisfaction (Vangrieken et al., 2015). Nevertheless, this thesis approached the topic from an organizational perspective and examined the approach less on the pedagogical perspective of education.

The second contribution is that this thesis gives an insight in how individuals in the education can be assessed in terms of their knowledge, skills, attitudes and personalities. Timmering et al. (2009) identified the important teacher qualities and argued that individuals should possess certain teaching skills and attitudes towards special need students. The teacher can be assessed on their competences with the proposed measurements tools of Büttner et al. (2018) and used as one of the inputs for the team composition of educational work teams. However, the assessment tools should be seen as a development tool and helps the teachers by achieving their goals within the teams (Büttner et al., 2018). The other input for the team composition is the MBTI test. In this way the quantitative result of the test by Büttner et al. (2018) can be combined with the binary results of the MBTI tests. This thesis showed that combination of competences and personality profiles can be used for the composition of teams. The results show that this approach is successful in generating teams with sufficient teaching competences (taskwork) and teamwork.

shows that this feature is not working properly in de coded version of the model. In addition, it is important to include the ability to make manual changes in the process of constructing timetables (Ribic et al., 2015). This function is incorporated and has proven to be effective in the developed ILP model.

A limitation of this thesis is that the current version of the ILP model did not come up with a feasible solution. In order to judge whether this approach and allocation model could provide a feasible solution depends on the sensitivity analysis of the improved model. Furthermore, the required information for the input parameters for the model were not available and couldn’t be obtained during this research period. Some input parameters were therefore based on literature and randomized. Next to that, only a few of the ILP school timetabling constraints mentioned in literature are considered in this model. The required information and additional school timetabling constraints should be used in order to construct a model that reflects a practical situation. Lastly, the formulated approach for MTM and timetabling can only be used at educational institutes where an obvious distinction can be made between expert teams and level teams. This distinction is less common in regular or mainstream education institutes.

As for directions for future research, it might be interesting to add the iterative process with a randomized starting point. The iterative process will help with finding the optimal solution. This addition and the incorporation of extra school timetabling constraints, increases the variables and therefore computing time in creating an optimal solution. Therefore, it might be interesting to look whether the problem could be solved using a metaheuristic approach, like tabu research or simulated annealing. Metaheuristics optimization methods are more suitable for complicated problems and offer the possibility to make a trade-off between solution quality and computing time (Glover & Sörensen, 2015).

5 CONCLUSION

Globalised developments and a scarcity in resources have forced organization to use their resources more effectively and they are obliged to create a more flexible organization. MTM is a team-oriented working approach that offers organizations this flexibility. However, incomplete implementation of MTM is likely to have a negative effect on individual and team performance. This thesis extends the knowledge on MTM by addressing how team composition and planning can improve the MTM by proposing an integrated approach for the composition and planning of teams. Furthermore, this research shows how this approach can support the educational sector in becoming a more effective and flexible organisation.

While designing a multi-team organization structure, it is important to simultaneously consider the conditions for taskwork that improve individual performance and the conditions for teamwork that are beneficial to team performance. Structuring the educational organization by defining expert- and level teams ensures the quality of education and creates an environment in which a multidisciplinary team can work efficiently. This structure also provides a good basis for creating practical timetabling solution. For the composition of teams, it is important to allocate individuals to teams based on their knowledge, skills, attitudes and personality. Considering taskwork and teamwork simultaneously will support organisations in adapting to changes in the environment and will ultimately diminish the negative effects of MTM on performance.