# Interactive Planning of Competency-Driven University Teaching Staff Allocation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Featured Application

**The competency framework for matching student needs and lecturer competencies subject to disruptions caused by teacher absenteeism and curriculum changes. The model is adjusted to perform at the expected robustness level of resultant lecturer allocation. The main strength of the model, as well as the main contribution of this work, consists in that it can absorb the disruptions and produce robust teacher assignment schedules.**

## Abstract

## 1. Introduction

- In contrast to the common trend of dealing with personnel allocation and staff scheduling problems separately, we propose an integrated approach to the prototyping of robust competency-driven teacher allocations. This approach is based on the taxonomy of human resource allocation problems, which sets the context for the construction of realistic proactive strategies for the robust allocation of teachers.
- The solution is a declarative modelling-driven approach to the assessment of alternative variants of a competency framework robust to a selected set of anticipated types of personnel absence. The model searches for competency-driven teacher allocations robust to teacher absences under the Constraint Satisfaction Problem.
- Our approach shows the necessary qualities to replace typical operational research or computer simulation methods for workforce allocation and personnel scheduling with constraint programming-driven techniques. Its main advantage is that it accounts for the alternate usage of synthesis and analysis versions of APS, assuming the occurrence of disruptions in the course of planning solutions robust to teacher absenteeism.

## 2. Literature Review

- Assigning at most one task per agent (a one-to-one assignment).
- Assigning multiple agents to a task or assigning multiple tasks to the same agent (a one-to-many assignment).
- Multidimensional assignment problems assigning the members of three or more sets, such as matching jobs with workers and machines or assigning students and teachers to classes and time slots.

## 3. Competency-Driven Staff Allocation

#### 3.1. Problem Formulation

- Each course meeting can be conducted by only one competent teacher,
- The set of all course meetings for a specific course section (group) can be run by no more than three teachers,
- Lectures and seminars can only be delivered by professors and doctors,
- Each teacher should have a guaranteed teaching quota,
- All course meetings must be assigned to teachers,
- Others (following from various disruptions and/or individual needs of a university/organisational unit).

- (a)
- A task of a course can only be carried out by a competent teacher,
- (b)
- Each teacher must perform no less than 1 task and no more than 2 tasks,
- (c)
- All tasks of courses must be assigned to teachers.

${U}_{\omega}$ | is a family of $\omega $-number of simultaneous teacher absences: ${U}_{\omega}=\left\{{u}_{i}|{u}_{i}\subseteq \mathcal{P};\left|{u}_{i}\right|=\omega ;i=1\dots \left(\begin{array}{c}\left|\mathcal{P}\right|\\ \omega \end{array}\right)\right\}$, $\omega \in \left\{1,\mathrm{..},\left|\mathcal{P}\right|\right\};$ for example, for the case of an absence of two teachers in the example above (Table 3 and Table 4$),\text{}\mathrm{the}\text{}\mathrm{set}\text{}{U}_{2}=\left\{\left\{{P}_{1},{P}_{2}\right\},\left\{{P}_{1},{P}_{3}\right\},\left\{{P}_{2},{P}_{3}\right\}\right\}$ contains 3 absence scenarios. |

$L{P}_{\omega}^{}$ | is a subset of set ${U}_{\omega}$ ($L{P}_{\omega}^{}\subseteq {U}_{\omega}$) which contains absence scenarios for which competency framework, $G,$ guarantees permissible teacher allocation $X,$ to courses in the event of absences of teachers. In a special case, subset $L{P}_{\omega}^{}$ can be an empty set, i.e., $L{P}_{\omega}^{}=\varnothing $, which corresponds to a situation in which no suitable replacement enabling the execution of all the courses from set $\mathcal{Z}$ can be found for any of the 28 absence scenarios. |

- ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=0$ stands for lifelessness (robustlessness), i.e., there is no suitable replacement enabling the execution of all tasks from set $\mathcal{Z}$ for any of the possible cases of absence ($\omega $-number of simultaneous teacher absences).
- ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$ stands for full robustness, i.e., for each of the possible cases of absence ($\omega $-number of simultaneous teacher absences) there is at least one replacement guaranteeing the execution of all the tasks from set $\mathcal{Z}$.

#### 3.2. Model

- Tasks can only be executed by a competent teacher, i.e., ${\forall}_{k=1\dots m,i=1\dots n}{x}_{k,i}\le {q}_{i}\times {g}_{k,i}$.
- Teacher working time limits may not be exceeded, i.e., ${\forall}_{k=1\dots m,i=1\dots n}$ $\sum _{i=1}^{n}{x}_{k,i}^{}\times {l}_{i}\ge {s}_{k}^{}$ and $\sum _{i=1}^{n}{x}_{k,i}^{}\times {l}_{i}\le {z}_{k}^{}$.

$\mathcal{Z}$: | Set of courses: $\mathcal{Z}=\left\{{Z}_{1},\dots ,{Z}_{i},\dots ,{Z}_{n}\right\}$ |

$\mathcal{P}$: | Set of teachers: $\mathcal{P}=\left\{{P}_{1},\dots ,{P}_{k},\dots ,{P}_{m}\right\}$ |

${U}_{\omega}:$ | Family of scenarios parametrised by $\omega \u2014$the number of simultaneous teacher absences: ${U}_{\omega}=\left\{{u}_{i}|{u}_{i}\subseteq \mathcal{P};\left|{u}_{i}\right|=\omega ;i=1\dots \left(\begin{array}{c}\left|\mathcal{P}\right|\\ \omega \end{array}\right)\right\}$. |

$\Theta $: | A single scenario of absence $\omega $ of teachers, $\Theta \in {U}_{\omega}$ |

$L{P}_{\omega}^{\text{}}$: | Subset ${U}_{\omega}$ ($L{P}_{\omega}^{\text{}}\subseteq {U}_{\omega}$), which contains absence scenarios for which competency framework, $G,$ guarantees a permissible teacher allocation, $X,$ to courses in the event of absences of teachers. |

$n:$ | Number of courses ($n\in \mathbb{N}$) |

${q}_{i}$: | Number of tasks from course ${Z}_{i}$ |

$m:$ | Number of teachers ($m\in \mathbb{N}$) |

$\omega :$ | Number of absent teachers, $\mathcal{P}$ ($\omega \in \mathbb{N}$), $\omega <m$ |

${l}_{i}$: | Duration of the task from course ${Z}_{i}$ (in hours) |

${s}_{k}^{}$: | Minimum working hours of teacher ${P}_{k}$$({s}_{k}\in \mathbb{N}$) |

${z}_{k}^{}$: | Maximum working hours of teacher ${P}_{k}$$\text{}({z}_{k}\in \mathbb{N}$) |

${}^{*}\mathcal{R}{}_{\mathcal{P}}^{\mathcal{Z}}$: | Predicted robustness of the competency framework (${}^{*}\mathcal{R}{}_{\mathcal{P}}^{\mathcal{Z}}\in \left[0,1\right]$) |

$G$: | Competency framework given by matrix $G={\left[{g}_{k,i}\right]}_{k=1\dots m;i=1\dots n},$ where: ${g}_{k,i}\in \left\{0,1\right\}$: ${g}_{k,i}^{}=\{\begin{array}{cc}1& {\mathrm{when}\text{}\mathrm{teacher}\text{}\mathrm{P}}_{\mathrm{k}}\text{}\mathrm{has}\text{}\mathrm{the}\text{}\mathrm{competencies}\text{}\mathrm{to}\text{}\mathrm{execute}\text{}\mathrm{task}\text{}\mathrm{from}\text{}\mathrm{course}{\mathrm{Z}}_{\mathrm{i}}\text{}\\ 0& \mathrm{in}\text{}\mathrm{remaining}\text{}\mathrm{cases}.\end{array}$Robustness of competency framework, $G,$ for the absence of $\omega $ teachers is described by function ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)$ (1). |

${G}^{\Theta}$: | Competency framework which takes into account absences of teachers defined in set $\Theta $: ${G}^{\Theta}={\left[{g}_{k,i}^{\Theta}\right]}_{k=1\dots m;i=1\dots n}$ where: ${g}_{k,i}^{\Theta}=\{\begin{array}{cc}1& \mathrm{when}\text{}k\notin {\Theta \text{}\mathrm{and}\text{}\mathrm{P}}_{\mathrm{k}}{\text{}\mathrm{have}\text{}\mathrm{the}\text{}\mathrm{competence}\text{}\mathrm{to}\text{}\mathrm{execute}\text{}\mathrm{task}\text{}\mathrm{of}\text{}\mathrm{Z}}_{\mathrm{i}}\text{}\\ 0& \mathrm{in}\text{}\mathrm{remaining}\text{}\mathrm{cases}.\end{array}$ |

$X:$ | Teacher allocation, $X={\left[{x}_{k,i}\right]}_{k=1\dots m;i=1\dots n}$, where: ${x}_{k,i}\in \left\{0,1,\dots ,{q}_{i}\right\}$ means the number of tasks from course ${Z}_{i}$ executed by teacher ${P}_{k}$. |

${X}^{\Theta}$: | Allocation in situations when teachers defined in set $\Theta $ are absent from work: ${X}^{\Theta}={\left[{x}_{k,i}^{\Theta}\right]}_{k=1\dots m;i=1\dots n}$, where: ${x}_{k,i}\in \left\{0,1,\dots ,{q}_{i}\right\}$ represents the number of tasks from course ${Z}_{i}$ executed by teacher ${P}_{k}$. |

${c}^{\Theta}$: | A variable that specifies whether there exists allocation ${X}^{\Theta}$ ensuring execution of tasks from courses, $\mathcal{Z}$. The value of variable ${c}^{\Theta}\in \left\{0,1\right\}$ depends on ancillary sub-variables: ${c}_{1,i}^{\Theta}$, ${c}_{2,k}^{\Theta}$, ${c}_{3,k}^{\Theta}$, which specify whether constraints (6)–(13) are satisfied. |

- The element ${g}_{k,i}^{\Theta}$ of matrix ${G}^{\Theta}$ that characterises the absence of teacher ${P}_{k}$ (${P}_{k}\in \Theta $) takes the value 0:$${g}_{k,i}^{\Theta}=\{\begin{array}{cc}{g}_{k,i}& \mathrm{when}\text{}{P}_{k}\notin \Theta \\ 0& \mathrm{when}\text{}{P}_{k}\in \Theta \end{array}$$
- Tasks are only executed by teachers who have the appropriate competence:$${x}_{k,i}^{\Theta}\le {q}_{i}\times {g}_{k,i}^{\Theta},\text{}\mathrm{for}\text{}k=1\dots m;i=1\dots n;\text{}\Theta \in {U}_{\omega}$$
- All tasks, ${q}_{i},$ from course ${Z}_{i}$ should be executed:$$({\sum}_{k=1}^{m}{x}_{k,i}^{\Theta}={q}_{i})\iff \left({c}_{1,i}^{\Theta}=1\right),\text{}\mathrm{for}\text{}i=1\dots n;\text{}\Theta \in {U}_{\omega}$$
- Workload of teacher ${P}_{k}$ is equal to or greater than the minimum number of working hours, ${s}_{k}^{}$:$$({\sum}_{i=1}^{n}{x}_{k,i}^{\Theta}\times {l}_{i}\ge {s}_{k}^{})\iff \left({c}_{2,k}^{\Theta}=1\right),\text{}\mathrm{for}\text{}{P}_{k}\in \mathcal{P}\backslash \Theta ;\text{}\Theta \in {U}_{\omega}$$
- Workload of teacher ${P}_{k}$ is not greater than the maximum number of working hours, ${z}_{k}^{}$:$$({\sum}_{i=1}^{n}{x}_{k,i}^{\Theta}\times {l}_{i}\le {z}_{k}^{})\iff \left({c}_{3,k}^{\Theta}=1\right),\text{}\mathrm{for}\text{}{P}_{k}\in \mathcal{P}\backslash \Theta ;\text{}\Theta \in {U}_{\omega}$$
- Robustness, ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right),$ is calculated as a ratio of the number of absence scenarios $\left|L{P}_{\omega}^{}\right|$ for which the competency framework is robust to the absence of $\omega $ teachers to all possible disruption scenarios ($\left|{U}_{\omega}\right|$):$${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=\frac{\left|L{P}_{\omega}^{}\right|}{\left|{U}_{\omega}\right|}\ge {}_{}{}^{*}R{}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)\text{},$$$$\left|L{P}_{\omega}^{}\right|={\sum}_{\Theta \in {U}_{\omega}}{c}^{\Theta}\text{},$$$${c}^{\Theta}={\prod}_{i=1}^{n}{c}_{1,i}^{\Theta}{\prod}_{k=1}^{m}{c}_{2,k}^{\Theta}{\prod}_{k=1}^{m}{c}_{3,k}^{\Theta}\text{}.$$

#### 3.3. Method

$\mathcal{V}\left(\omega \right)$ | $=\left\{{R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right),G,{G}^{\Theta},{X}^{\Theta}|\Theta \in {U}_{\omega}\right\}$—a set of decision variables which includes: robustness of competency framework ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)$, competency framework $G$, competency frameworks ${G}^{\Theta}$ for cases when the teachers from set $\Theta $ are absent, corresponding task allocations ${X}^{\Theta}$. |

$\mathcal{D}\left(\omega \right)$ | —a finite set of decision variable domains $\left\{{R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right),G,{G}^{\Theta},{X}^{\Theta}|\Theta \in {U}_{\omega}\right\}$ |

$\mathcal{C}\left(\omega \right)$ | —a set of constraints specifying the relationships between the competency framework and its robustness (constraints 1–10). |

## 4. University Staff Allocation Planning—Case of Course Cast

- May conduct: ${g}_{k,i}=1$,
- May conduct if it gains the missing competencies: ${g}_{k,i}\in \left\{0,1\right\}$,
- May not conduct and cannot get the missing competencies: ${g}_{k,i}=0$.

- Tasks from course ${Z}_{i}$ can only be executed by a competent teacher,
- Teacher working time limits (${s}_{k}^{}$ and ${z}_{k}^{}$) may not be exceeded.

- Method verification based on historical data: The verification includes an assessment of whether the use of the developed method would lead to the determination of $G$, a competency framework that would secure FECS (robustness ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$) against the effects inferred from the teacher absence in the academic year 2019/2020, e.g., the need to hire an additional teacher. In the case of successful assessment:
- Use of the developed method aimed at synthesis of competency frameworks robust to selected kinds of disruptions including:
- Absence of $\omega =1,\dots ,3$ teachers,
- Absence of teachers from the age group of pre-retirement and retirement employees.

#### 4.1. Method Verification Based on Historical Data

- Does there exist a minimal competency framework${G}_{OPT}$of FECS that guarantees robustness value${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$in the event of absence of teacher${P}_{18}$: Roach?

- Teacher ${P}_{29}$: Richardson can take over the substitution by conducting classes in:
- -
- Course ${Z}_{47}$ (50 h, 10 tasks)
- -
- Course ${Z}_{48}$ (75 h, 15 tasks)
- -
- Course ${Z}_{74}$ (80 h, 16 tasks)

- Teacher ${P}_{41}$: Thorpe can take over the substitution by conducting classes in:
- -
- Course ${Z}_{106}$ (15 h, 3 tasks)
- -
- Course ${Z}_{211}$ (20 h, 4 tasks)

- Teacher ${P}_{44}$: Lacroix can take over the substitution by conducting classes in:
- -
- Course ${Z}_{121}$ (30 h, 6 tasks)

- Teacher ${P}_{7}$: Crockett can take over the substitution by conducting classes in:
- -
- Course ${Z}_{125}$ (75 h, 15 tasks)

- Teacher ${P}_{22}:$ Meyer complements the competency from course ${Z}_{125}$,
- Teacher ${P}_{43}$: Whitehead complements the competency from course ${Z}_{125}$, etc.

#### 4.2. Synthesis of Competency Framework Robust to a Simultaneous Absence of $\omega $ Teachers

- Does there exist a (minimal) competency framework,${G}_{OPT}$, which guarantees robustness value${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$in in the event of absence of$\omega $teachers ($\omega =1,2,3$)?

- (a)
- ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(1\right)=$0.77,
- (b)
- ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(2\right)=$0.58,
- (c)
- ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(3\right)=$0.43.

- Teachers with what competencies should be employed to obtain$G$competency framework whose robustness level,${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1,$corresponds to situations when$\omega $teachers are absent ($\omega =1,2,3$)?

- (a)
- Conducting 21 courses:$\text{}{Z}_{4},{Z}_{7},{Z}_{23},{Z}_{24},{Z}_{25},{Z}_{45},{Z}_{93},{Z}_{103},{Z}_{114},{Z}_{131},{Z}_{132},\text{}{Z}_{134},{Z}_{135},{Z}_{157},{Z}_{158},$${Z}_{166},{Z}_{168},{Z}_{169}\mathrm{and}\text{}{Z}_{170}$. ($\omega =1$).
- (b)
- Conducting 71 courses: ${Z}_{4},{Z}_{5},{Z}_{6},{Z}_{7},{Z}_{9},{Z}_{10},{Z}_{19},{Z}_{21},{Z}_{22},{Z}_{23},{Z}_{24},{Z}_{25},{Z}_{26},{Z}_{27},{Z}_{28},$ ${Z}_{29},{Z}_{30},{Z}_{34},{Z}_{45},{Z}_{51},{Z}_{55},{Z}_{56},{Z}_{58},{Z}_{77},{Z}_{78},{Z}_{79},{Z}_{84},{Z}_{86},{Z}_{93},{Z}_{99},{Z}_{101},{Z}_{102},{Z}_{103},{Z}_{104},{Z}_{107},{Z}_{111},$ ${Z}_{114},{Z}_{115},{Z}_{117},{Z}_{120},{Z}_{130},{Z}_{131},{Z}_{132},{Z}_{133},{Z}_{134},{Z}_{135},{Z}_{136},{Z}_{137},{Z}_{149},{Z}_{153},{Z}_{156},{Z}_{157},{Z}_{158},{Z}_{159},{Z}_{160},$${Z}_{161},{Z}_{162},{Z}_{164},\text{}{Z}_{165},{Z}_{166},{Z}_{168},{Z}_{169},{Z}_{170},{Z}_{179},{Z}_{191},{Z}_{196},{Z}_{201},{Z}_{203},{Z}_{208}\mathrm{and}\text{}{Z}_{212}$ ($\omega =2$).
- (c)
- Conducting 129 courses ($\omega =3$): ${G}_{OPT}^{3},$ see in GitHub.

#### 4.3. Synthesis of Competency Framework Robust to Absenteeism Caused by Pre-Retirement Aged Teachers

- Does there exist a (minimal) competency framework,${G}_{OPT},$of FECS that guarantees robustness value${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$in theevent of absence of$\omega $teachers from set$EM$($\omega =1,\dots ,9$)?

- Teachers with what competencies should be employed to obtain$G$competency framework whose robustness level${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$corresponds to the event of absence of$\omega $teachers from set$EM$($\omega =2,\dots ,9$)?

#### 4.4. Quantitative Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wang, L.; Shi, H.; Gan, L. Healthcare Facility Location-Allocation Optimization for China’s Developing Cities Utilizing a Multi-Objective Decision Support Approach. Sustainability
**2018**, 10, 4580. [Google Scholar] [CrossRef][Green Version] - Gola, A.; Kłosowski, G. Application of fuzzy logic and genetic algorithms in automated works transport organization. In Advances in Intelligent Systems and Computing; Springer: Cham, Switzerland, 2018; Volume 620, pp. 29–36. [Google Scholar]
- Moodley, R.; Chiclana, F.; Carter, J.; Caraffini, F. Using Data Mining in Educational Administration: A Case Study on Improving School Attendance. Appl. Sci.
**2020**, 10, 3116. [Google Scholar] [CrossRef] - Hmer, A.; Mouhoub, M. Teaching Assignment Problem Solver. In Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2010; Volume 6097, pp. 298–307. [Google Scholar]
- Faudzi, S.; Abdul-Rahman, S.; Abd Rahman, R. An Assignment Problem and Its Application in Education Domain: A Review and Potential Path. Adv. Oper. Res.
**2018**. [Google Scholar] [CrossRef] - Ongy, E. Optimizing Student Learning: A Faculty-Course Assignment Problem Using Linear Programming. J. Educat. Human Resour. Dev.
**2017**, 5, 1–14. [Google Scholar] - Whiddett, S.; Hollyforde, S. The Competencies Handbook; Institute of Personnel and Development: London, UK, 1999; 196p, ISBN 978-0-852-92735-9. [Google Scholar]
- Bombiak, E. Human resources risk as an aspect of human resources management in turbulent environments. In Shift! Major Challenges of Today’s Economy; Pînzaru, F., Zbuchea, A., Brătianu, C., Vătămănescu, E.M., Mitan, A., Eds.; Tritonic Publishing House: Bucharest, Romania, 2017; pp. 121–132. [Google Scholar]
- Ingels, J.; Maenhout, B. Optimised buffer allocation to construct stable personnel shift rosters. Omega
**2019**, 82, 102–117. [Google Scholar] [CrossRef] - Zamri, N.E.; Mansor, M.A.; Mohd Kasihmuddin, M.S.; Alway, A.; Mohd Jamaludin, S.Z.; Alzaeemi, S.A. Amazon Employees Resources Access Data Extraction via Clonal Selection Algorithm and Logic Mining Approach. Entropy
**2020**, 22, 596. [Google Scholar] [CrossRef] - Szwarc, E.; Bocewicz, G.; Banaszak, Z.; Wikarek, J. Competence allocation planning robust to unexpected staff absenteeism. Oper. Reliab.
**2019**, 21, 440–450. [Google Scholar] [CrossRef] - Szwarc, E.; Bocewicz, G.; Sitek, P.; Wikarek, J. Competence-oriented recruitment of a project team robust to disruptions. Lect. Notes Artif. Intell.
**2020**, 12034, 13–25. [Google Scholar] - Szwarc, E.; Wikarek, J. Proactive planning of project team members’ competences. Found. Manag.
**2020**, 12, 71–84. [Google Scholar] [CrossRef] - Osman, M.; Abo-Sinna, M.; Mousa, A. An effective genetic algorithm approach to multiobjective resource allocation problems (MORAPs). Appl. Math. Comput.
**2005**, 163, 755–768. [Google Scholar] [CrossRef] - Yin, P.; Wang, J. A particle swarm optimization approach to the nonlinear resource allocation problem. Appl. Math. Comput.
**2006**, 183, 232–242. [Google Scholar] [CrossRef] - Lin, C.; Gen, M. Multi-criteria human resource allocation for solving multistage combinatorial optimization problems using multiobjective hybrid genetic algorithm. Expert Syst. Appl.
**2008**, 34, 2480–2490. [Google Scholar] [CrossRef] - Burkova, I.; Titarenko, B.; Hasnaoui, A.; Titarenko, R. Resource allocation problem in project management. E3S Web Conf.
**2019**, 97, 01003. [Google Scholar] [CrossRef][Green Version] - Nesterov, Y.; Shikhman, V. Dual subgradient method with averaging for optimal resource allocation. Eur. J. Oper. Res.
**2018**, 3, 907–916. [Google Scholar] [CrossRef][Green Version] - Ivanova, A.; Pasechnyuk, D.; Dvurechensky, P.; Gasnikov, A.; Vorontsova, E. Numerical methods for the resource allocation problem in networks. arXiv
**2020**, arXiv:1909.13321. [Google Scholar] - Kłosowski, G.; Gola, A.; Świć, A. Application of Fuzzy Logic Controller for Machine Load Balancing in Discrete Manufacturing System. Lect. Notes Comput. Sci.
**2015**, 9375, 256–263. [Google Scholar] - Janardhanan, M.N.; Li, Z.; Bocewicz, G.; Banaszak, Z.; Nielsen, P. Metaheuristic algorithms for balancing robotic assembly lines with sequence-dependent robot setup times. Appl. Math. Model.
**2019**, 65, 256–270. [Google Scholar] [CrossRef][Green Version] - Bouajaja, S.; Dridi, N. A survey on human resource allocation problem and its applications. Oper. Res. Int. J.
**2016**, 17, 339–369. [Google Scholar] [CrossRef] - Gunawan, A.; Ng, K.M.; Poh, K.L. A Mathematical Programming Model for A Timetabling Problem. World Congress in Computer Science. In Proceedings of the International Conference on Scientific Computing, Las Vegas, NV, USA, 26–29 June 2006; pp. 42–47. [Google Scholar]
- Wang, Y.Z. An application of genetic algorithm methods for teacher assignment problems. Expert Syst. Appl.
**2002**, 22, 295–302. [Google Scholar] [CrossRef] - Zibran, M. A Multi-Phase Approach to University Course Timetabling. Ph.D. Thesis, Faculty of Arts and Science, University of Lethbridge, Alberta, CA, USA, 2007. [Google Scholar]
- Tadic, I.; Marasovic, B. Analyze of human resource allocation in higher education applying integer linear programming. In Economic and Social Development: Book of Proceedings; Beros, M.B., Recker, N., Kozina, M., Eds.; Varazdin Development and Entrepreneurship Agency: Varazdin, Croatia; Faculty of Management University of Warsaw: Warsaw, Poland; University North: Koprivnica, Croatia, 2018; pp. 266–276. [Google Scholar]
- Ferreira, P. Application of scheduling techniques to Teacher-Class Assignments. Master’s Thesis, Instituto Superior Técnico, Lisboa, Portugal, 2015. [Google Scholar]
- Hultberg, T.H.; Cardoso, D.M. The teacher assignment problem: A special case of the fixed charge transportation problem. Eur. J. Oper. Res.
**1997**, 121, 463–473. [Google Scholar] [CrossRef] - Thongsanit, K. Solving the Course—Classroom Assignment Problem for a University. Silpakorn U. Sci. Tech. J.
**2014**, 8, 46–52. [Google Scholar] - Domenech, B.; Lusa, A. A MILP model for the teacher assignment problem considering teacher’s preferenes. Eur. J. Oper. Res.
**2016**, 249, 1153–1160. [Google Scholar] [CrossRef] - Bettinelli, A.; Cacchiani, V.; Roberti, R.; Toth, P. An overview of curriculum-based course timetabling. TOP
**2015**, 23, 313–349. [Google Scholar] [CrossRef] - Hertz, A.; Robert, V. Constructing a course schedule by solving a series of assignment type problems. Eur. J. Oper. Res.
**1998**, 108, 585–603. [Google Scholar] [CrossRef] - Abdullah, S.; Ahmadi, S.; Burke, E.K.; Dror, M.; McCollum, B. A tabu-based large neighbourhood search methodology for the capacitated examination timetabling problem. J. Oper. Res. Soc.
**2007**, 58, 1494–1502. [Google Scholar] [CrossRef][Green Version] - Alvarez-Valdes, R.; Crespo, E.; Tamarit, J.M. Design and implementation of a course scheduling system using tabu search. Eur. J. Oper. Res.
**2002**, 137, 512–523. [Google Scholar] [CrossRef] - Daskalaki, S.; Birbas, T.; Housos, E. An integer programming formulation for a case study in university timetabling. Eur. J. Oper. Res.
**2004**, 153, 117–135. [Google Scholar] [CrossRef] - Lukas, S.; Aribowo, A.; Muchri, M. Solving timetable problem by genetic algorithm and heuristic search case study: Universitas pelita harapan timetable. Real World Appl. Genet. Algorithms
**2013**, 378, 303–317. [Google Scholar] - De Carolis, D.M. Competencies and imitability in the pharmaceutical industry: An analysis of their relationship with firm performance. J. Manag.
**2003**, 29, 27–50. [Google Scholar] [CrossRef] - Boucher, X.; Bonjour, E.; Grabot, B. Formalization and use of competencies for industrial performance optimization: A survey. Comput. Ind.
**2007**, 58, 98–117. [Google Scholar] [CrossRef][Green Version] - Manavizadeh, N.; Hosseini, N.S.; Rabbani, M.; Jolai, F. A simulated annealing algorithm for a mixed model assembly U-line balancing type-I problem considering human efficiency and Just-In-Time approach. Comput. Ind. Eng.
**2013**, 64, 669–685. [Google Scholar] [CrossRef] - Schniederjans, M.; Kim, G. A goal programming model to optimize departmental preference in course assignments. Comput. Oper. Res.
**1987**, 14, 87–96. [Google Scholar] [CrossRef] - Nembhard, D.A.; Bentefouet, F. Selection, grouping, and assignment policies with learning-by-doing and knowledge transfer. Comput. Ind. Eng.
**2015**, 79, 175–187. [Google Scholar] [CrossRef] - Moreira, J.J.; Reis, L.P. Multi-Agent System for Teaching Service Distribution with Coalition Formation. Adv. Intell. Syst. Comput.
**2013**, 206, 599–609. [Google Scholar] - Boyatzis, R. Competencies in the 21st century. J. Manag. Dev.
**2007**, 27, 5–12. [Google Scholar] [CrossRef] - Spencer, L.; Spencer, S. Competence at Work: Model for Superior Performance; John Wiley & Sons: New York, NY, USA, 1993; 384p, ISBN 978-0-471-54809-6. [Google Scholar]
- Dubois, D. Competency-Based HR Management; Black Well Publishing: Hoboken, NJ, USA, 1998. [Google Scholar]
- Rothwell, W.; Bernthal, P.; Colteryahn, K.; Davis, P.; Naughton, J.; Wellins, R. ASTD Competency Study: Mapping the Future; ASTD Press: Alexandria, VA, USA, 2004. [Google Scholar]
- Chouhan, V.S.; Srivastava, S. Understanding Competencies and Competency Modeling—A Literature Survey. IOSR J. Bus. Manag.
**2014**, 16, 14–22. [Google Scholar] [CrossRef] - Antosz, K. Maintenance—Identification and analysis of the competency gap. Eksploatacja i Niezawodnosc
**2018**, 20, 484–494. [Google Scholar] [CrossRef] - Xian-Ying, M. Application of assignment model in PE human resources allocation. Energy Proced.
**2012**, 16, 1720–1723. [Google Scholar] [CrossRef][Green Version] - Gunawan, A.; Ng, K.M. Solving the Teacher Assignment Problem by Two Metaheuristics. Int. J. Inf. Manag. Sci.
**2011**, 22, 73–86. [Google Scholar] - Pentico, D.W. Assignment problems: A golden anniversary survey. Eur. J. Oper. Res.
**2007**, 176, 774–793. [Google Scholar] [CrossRef] - Dück, V.; Ionescu, L.; Kliewer, N.; Suhl, L. Increasing stability of crew and aircraft schedules. Transp. Res. Part C Emerg. Technol.
**2012**, 20, 47–61. [Google Scholar] [CrossRef] - Ehrgott, M.; Ryan, D.M. Constructing robust crew schedules with bi-criteria optimization. J. Multi Criteria Decis. Anal.
**2002**, 11, 139–150. [Google Scholar] [CrossRef] - Hazir, O.; Haouari, M.; Erel, E. Robust scheduling and robustness measures for the discrete time/cost trade-off problem. Eur. J. Oper. Res.
**2010**, 207, 633–643. [Google Scholar] [CrossRef] - Ingels, J.; Maenhout, B. The impact of reserve duties on the robustness of a personnel shift roster: An empirical investigation. Comput. Oper. Res.
**2015**, 61, 153–169. [Google Scholar] [CrossRef] - Topaloglu, S.; Selim, H. Nurse scheduling using fuzzy modelling approach. Fuzzy Sets Syst.
**2010**, 161, 1543–1563. [Google Scholar] [CrossRef] - Potthoff, D.; Huisman, D.; Desaulniers, G. Column generation with dynamic duty selection for railway crew rescheduling. Transp. Sci.
**2010**, 44, 493–505. [Google Scholar] [CrossRef][Green Version] - Nawaz, N. Human Resource Information Systems—A review. Int. J. Manag. IT Eng.
**2013**, 3, 74–98. [Google Scholar] - Nawaz, N.; Gomes, A.M. Human Resource Information System: A Review of Previous Studies. J. Manag. Res.
**2017**, 9, 92–120. [Google Scholar] - Draganidis, F.; Mentzas, G. Competency based management: A review of systems and approaches. Inf. Manag. Comput. Secur.
**2006**, 14, 51–64. [Google Scholar] [CrossRef] - Cieśla, B.; Gunia, G. Development of integrated management information systems in the context of Industry 4.0. Appl. Comput. Sci.
**2019**, 15, 37–48. [Google Scholar] - Wikarek, J.; Sitek, P. A data-driven approach to constraint optimization. Adv. Intell. Syst. Comput.
**2020**, 920, 135–144. [Google Scholar] - Ardjmand, E.; Ghalehkhondabi, I.; Weckman, G.R.; Young, W., II. Application of decision support systems in scheduling/planning of manufacturing/service systems: A critical review. Int. J. Manag. Decis. Mak.
**2016**, 15, 248–276. [Google Scholar] [CrossRef] - Sitek, P.; Wikarek, J. A multi-level approach to ubiquitous modeling and solving constraints in combinatorial optimization problems in production and distribution. Appl. Intell.
**2018**, 48, 1344–1367. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Model of Constraint Optimisation Problem (COP)-based synthesis of a robust competency framework.

$\mathit{G}$ | Competencies Enabling the Execution of Courses, ${\mathit{Z}}_{\mathit{i}}$ | |||
---|---|---|---|---|

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ||

Teachers | ${P}_{1}$ | 1 | 1 | 0 |

${P}_{2}$ | 0 | 0 | 1 | |

${P}_{3}$ | 1 | 1 | 0 |

$\mathit{X}$ | Number of Tasks of the Course | |||
---|---|---|---|---|

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ||

Teachers | ${P}_{1}$ | 1 | 0 | 0 |

${P}_{2}$ | 0 | 0 | 2 | |

${P}_{3}$ | 0 | 1 | 0 |

$\mathit{G}$ | Competencies Enabling the Execution of the Courses, ${\mathit{Z}}_{\mathit{i}}$ | |||
---|---|---|---|---|

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ||

Teachers | ${P}_{1}$ | 1 | 1 | 1 |

${P}_{2}$ | 0 | 0 | 1 | |

${P}_{3}$ | 1 | 1 | 1 |

$\mathit{X}$ | Number of Tasks of the Course | |||
---|---|---|---|---|

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ||

Teachers | ${P}_{1}$ | 1 | 0 | 1 |

$-$ | - | - | - | |

${P}_{3}$ | 0 | 1 | 1 |

${\mathit{Z}}_{\mathit{i}}$ | ${\mathit{q}}_{\mathit{i}}$ | ${\mathit{l}}_{\mathit{i}}$ |
---|---|---|

${Z}_{1}$: History of technics 1 | 16 | 5 |

${Z}_{2}$: History of technics 2 | 5 | 5 |

${Z}_{3}$: Inventics | 12 | 5 |

${Z}_{4}$: Economics | 9 | 5 |

${Z}_{5}$: Foundations of mathematical analysis | 20 | 5 |

… | … | … |

${Z}_{74}$: Programming in. NET environment | 21 | 5 |

… | … | … |

${Z}_{213}$: Distributed information processing systems | 6 | 5 |

${Z}_{214}$: Artificial intelligence methods | 6 | 5 |

**Table 6.**Competency framework, $G,$ of FECS teaching staff (source: https://github.com/erykszw/TAP).

$\mathit{G}$ | ${\mathit{Z}}_{1}:$ History of Technics 1 | ${\mathit{Z}}_{2}:$ History of Technics 2 | ${\mathit{Z}}_{3}:$ Inventics | ${\mathit{Z}}_{4}:$ Economics | ${\mathit{Z}}_{5}:$ Foundations of Mathematical Analysis | ${\mathit{Z}}_{6}:$Mathematical Analysis and Linear Algebra | … | … | ${\mathit{Z}}_{74}:$ Programming in .NET Environment | … | … | ${\mathit{Z}}_{212}:$ Selected Branches of Physics | ${\mathit{Z}}_{213}:$ Distributed Information Processing Systems | ${\mathit{Z}}_{214}:$ Artificial Intelligence Methods |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${P}_{1}:\text{}$Mills | 1 | 1 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 1 | 0 | 0 |

${P}_{2}:\text{}$Garner | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{3}:\text{}$Ray | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{4}:\text{}$MacPherson | 0 | 0 | 0 | 0 | 0 | 0 | … | … | {0,1} | … | … | 0 | 0 | 0 |

${P}_{5}:\text{}$Burnham | 0 | 0 | 0 | 0 | 0 | 0 | … | … | {0,1} | … | … | 0 | 0 | 0 |

${P}_{6}:\text{}$Davis | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | {0,1} |

${P}_{7}:\text{}$Crockett | {0,1} | {0,1} | {0,1} | 0 | 0 | 0 | … | … | {0,1} | … | … | 0 | 1 | {0,1} |

${P}_{8}:\text{}$Hudson | {0,1} | {0,1} | {0,1} | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

… | … | … | … | … | … | … | … | … | … | … | … | … | … | … |

${P}_{18}:$Roach | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 1 | … | … | 0 | 0 | {0,1} |

… | … | … | … | … | … | … | … | … | … | … | … | … | … | … |

${P}_{47}:\text{}$Fox | 0 | 0 | 0 | 1 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{48}:\text{}$Porterfield | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{49}:\text{}$Johnson | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

Teacher | ${\mathit{s}}_{\mathit{k}}^{}$ | ${\mathit{z}}_{\mathit{k}}^{}$ | Teacher | ${\mathit{s}}_{\mathit{k}}^{}$ | ${\mathit{z}}_{\mathit{k}}^{}$ | Teacher | ${\mathit{s}}_{\mathit{k}}^{}$ | ${\mathit{z}}_{\mathit{k}}^{}$ |
---|---|---|---|---|---|---|---|---|

${P}_{1}:$Mills | 180 | 360 | ${P}_{18}:$Roach | 180 | 360 | ${P}_{35}:$Morrow | 180 | 360 |

${P}_{2}:$Garner | 360 | 600 | ${P}_{19}:$Schneider | 240 | 480 | ${P}_{36}:$ Fitch | 240 | 480 |

${P}_{3}:$Ray | 180 | 360 | ${P}_{20}:$. Reyes | 240 | 480 | ${P}_{37}:$ Clay | 240 | 480 |

${P}_{4}:$MacPherson | 180 | 360 | ${P}_{21}:$Barnes | 120 | 240 | ${P}_{38}:$Manning | 240 | 480 |

${P}_{5}:$Burnham | 360 | 600 | ${P}_{22}:$Meyer | 360 | 600 | ${P}_{39}:$Ramsey | 180 | 360 |

${P}_{6}:$Davis | 120 | 240 | ${P}_{23}:$ Sharpe | 120 | 240 | ${P}_{40}:$Hansen | 240 | 480 |

${P}_{7}:$Crockett | 128 | 360 | ${P}_{24}:$ Sinclair | 160 | 360 | ${P}_{41}:$Thorpe | 180 | 360 |

${P}_{8}:$Hudson | 240 | 480 | ${P}_{25}:$Mahoney | 180 | 360 | ${P}_{42}:$Rice | 340 | 600 |

${P}_{9}:$Whittaker | 240 | 480 | ${P}_{26}:$Kirkland | 240 | 480 | ${P}_{43}:$Whitehead | 240 | 480 |

${P}_{10}:$ Middleton | 360 | 600 | ${P}_{27}:$Slaughter | 240 | 500 | ${P}_{44}:$Lacroix | 20 | 120 |

${P}_{11}:$ Sloan | 330 | 600 | ${P}_{28}:$Gardner | 360 | 600 | ${P}_{45}:$Nichols | 50 | 120 |

${P}_{12}:$Flynn | 360 | 600 | ${P}_{29}:$Richardson | 190 | 480 | ${P}_{46}:$Cooley | 20 | 120 |

${P}_{13}:$Pope | 240 | 480 | ${P}_{30}:$Byrne | 240 | 480 | ${P}_{47}:$Fox | 30 | 120 |

${P}_{14}:$Buckley | 240 | 480 | ${P}_{31}:$Curran | 240 | 480 | ${P}_{48}:$Porterfield | 150 | 300 |

${P}_{15}:$Johnston | 360 | 600 | ${P}_{32}:$Owens | 180 | 400 | ${P}_{49}:$ Johnson | 50 | 100 |

${P}_{16}:$Bullock | 180 | 360 | ${P}_{33}:$Hoover | 345 | 600 | |||

${P}_{17}:$Dowling | 360 | 600 | ${P}_{34}:$Reynolds | 240 | 480 |

**Table 8.**Teacher allocation, $X,$ of the teaching staff employed by FECS (source: https://github.com/erykszw/TAP).

$\mathit{X}$ | ${\mathit{Z}}_{1}:$ History of Technics 1 | ${\mathit{Z}}_{2}:$ History of Technics 1 | ${\mathit{Z}}_{3}:$ Inventics | ${\mathit{Z}}_{4}:$ Economics | ${\mathit{Z}}_{5}:$ Foundations of Mathematical Analysis | ${\mathit{Z}}_{6}:$Mathematical Analysis and Linear Algebra | … | … | ${\mathit{Z}}_{74}:$ Programming in .NET Environment | … | … | ${\mathit{Z}}_{212}:$ Selected Branches of Physics | ${\mathit{Z}}_{213}:$ Distributed Information Processing Systems | ${\mathit{Z}}_{214}:$ Artificial Intelligence Methods |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${P}_{1}:\text{}$Mills | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 15 | 0 | 0 |

${P}_{2}:\text{}$Garner | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{3}:\text{}$Ray | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{4}:\text{}$MacPherson | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{5}:\text{}$Burnham | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{6}:\text{}$Davis | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{7}:\text{}$Crockett | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

… | … | … | … | … | … | … | … | … | … | … | … | … | … | … |

${P}_{18}:$Roach | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 80 | … | … | 0 | 0 | 0 |

… | … | … | … | … | … | … | … | … | … | … | … | … | … | … |

${P}_{47}:\text{}$Fox | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{48}:\text{}$Porterfield | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

${P}_{49}:\text{}$Johnson | 0 | 0 | 0 | 0 | 0 | 0 | … | … | 0 | … | … | 0 | 0 | 0 |

**Table 9.**Analysis of robustness, ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right),$ of FECS competency framework (Table 6).

Number of Absent Teachers, $\mathit{\omega}$ | Robustness Level, ${\mathit{R}}_{\mathcal{P}}^{\mathcal{Z}}\left(\mathit{\omega}\right)$ | Calculation Time (s) |
---|---|---|

1 | 0.35 | 0.9 |

2 | 0.1 | 1.1 |

3 | 0.03 | 1.4 |

4 | 0.01 | 2.2 |

5 | 0 | 4.5 |

6 | 0 | 6.8 |

7 | 0 | 9.2 |

**Table 10.**Synthesis of competency framework, ${G}_{OPT},$ following robustness level condition ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)\ge {}^{*}\mathcal{R}{}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)$ assumed on $\omega $ teacher absenteeism.

Number of Absent Teachers, $\mathit{\omega}$ | Expected Robustness Level, ${}^{*}\mathit{R}{}_{\mathcal{P}}^{\mathcal{Z}}\left(\mathit{\omega}\right)$ | Obtained Robustness Level, ${\mathit{R}}_{\mathcal{P}}^{\mathcal{Z}}\left(\mathit{\omega}\right)$ | Number of Changes Introduced to the Competency Framework, $\mathit{G}$ | Calculation Time (s) |
---|---|---|---|---|

1 | 0.2 | 0.23 | 9 | 13.1 |

0.4 | 0.49 | 62 | 13.5 | |

0.6 | 0.77 | 138 | 14.2 | |

0.8 | ^{1)}⨯ | ⨯ | 14.5 | |

1 | ⨯ | ⨯ | 14.9 | |

2 | 0.2 | 0.29 | 121 | 47.7 |

0.4 | 0.58 | 415 | 49.4 | |

0.6 | ⨯ | ⨯ | 51.6 | |

0.8 | ⨯ | ⨯ | 52.8 | |

1 | ⨯ | ⨯ | 54.4 | |

3 | 0.2 | 0.27 | 170 | 1068 |

0.4 | 0.43 | 660 | 1185 | |

0.6 | ⨯ | ⨯ | >1200 | |

0.8 | ⨯ | ⨯ | >1200 | |

1 | ⨯ | ⨯ | >1200 | |

4 | 0.2 | 0.31 | 752 | >1200 |

0.4 | ⨯ | ⨯ | >1200 | |

0.6 | ⨯ | ⨯ | >1200 | |

0.8 | ⨯ | ⨯ | >1200 | |

1 | ⨯ | ⨯ | >1200 |

^{1)}

**⨯**—no acceptable solution, i.e., there is no competency framework, G, which guarantees an expected value of robustness: ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)\ge {}^{*}\mathcal{R}{}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)$.

**Table 11.**Synthesis of competency framework, ${G}_{OPT},$ under robustness level condition ${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$ assumed on $\omega $ teacher absenteeism while taking into account the employment of additional staff.

Number of Absent Teachers, $\mathit{\omega}$ | Obtained Robustness Level, ${\mathit{R}}_{\mathcal{P}}^{\mathcal{Z}}\left(\mathit{\omega}\right)$ | Number of Competencies in the Team of Newly Employed Teachers | Calculation Time (s) |
---|---|---|---|

1 | 1 | 21 | 14.4 |

2 | 1 | 71 | 51.5 |

3 | 1 | 129 | 1131 |

4 | 1 | 155 | >1200 |

5 | 1 | 184 | >1200 |

6 | 1 | 197 | >1200 |

7 | 1 | 204 | >1200 |

**Table 12.**Synthesis of competency framework, ${G}_{OPT},$ guaranteeing full robustness (${R}_{\mathcal{P}}^{\mathcal{Z}}\left(\omega \right)=1$) for different numbers of teachers and courses.

Number of Teachers, $\mathit{m}$ | Number of Courses, $\mathit{n}$ | Number of Absent Teachers, $\mathit{\omega}$ | Calculation Time (s) |
---|---|---|---|

50 | 300 | 1 | 25 |

50 | 300 | 2 | 53 |

50 | 300 | 3 | 1005 |

100 | 400 | 1 | 134 |

100 | 400 | 2 | 345 |

100 | 400 | 3 | 6200 |

150 | 500 | 1 | 234 |

150 | 500 | 2 | 865 |

150 | 500 | 3 | 11,240 |

200 | 600 | 1 | 1540 |

200 | 600 | 2 | 5980 |

200 | 600 | 3 | >18,000 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Szwarc, E.; Wikarek, J.; Gola, A.; Bocewicz, G.; Banaszak, Z.
Interactive Planning of Competency-Driven University Teaching Staff Allocation. *Appl. Sci.* **2020**, *10*, 4894.
https://doi.org/10.3390/app10144894

**AMA Style**

Szwarc E, Wikarek J, Gola A, Bocewicz G, Banaszak Z.
Interactive Planning of Competency-Driven University Teaching Staff Allocation. *Applied Sciences*. 2020; 10(14):4894.
https://doi.org/10.3390/app10144894

**Chicago/Turabian Style**

Szwarc, Eryk, Jaroslaw Wikarek, Arkadiusz Gola, Grzegorz Bocewicz, and Zbigniew Banaszak.
2020. "Interactive Planning of Competency-Driven University Teaching Staff Allocation" *Applied Sciences* 10, no. 14: 4894.
https://doi.org/10.3390/app10144894