**Yurii Dubas, Lviv Polytechnic National University, Ukraine**

**Nataliya Kunanets, Lviv Polytechnic National University, Ukraine**

*Context. **This work examines the possible influence of external factors on the student when choosing disciplines for the formation of an individual educational trajectory (IET). It has been established that the formation of IET depends not only on the student but also on other external factors. In particular, higher education institutions, teachers, and experts can have a significant influence on the student. *

*Objective. **In this work, we discussed the features of the educational process with a focus on the student and his individual capabilities and needs. After the analysis, it is clear that there is an urgent need for a tool that could while using information technology, provide the opportunity to include external factors in the IET formation process.*

*Methods. **To solve the described problem, we used the mathematical tool, known as Analytic Hierarchy Process (AHP). This method allows the ranking of multiple disciplines to determine which is the best suited according to the stated goal. The evaluation of disciplines for ranking is carried out according to the criteria defined by experts. These criteria are chosen based on the analysis of the characteristics of the disciplines when they are chosen by students. Hence, we can consider the opinion of experts as the influence of external factors.*

*Results. **We performed calculations with random input data to see if the implemented mathematical method works correctly with the system’s purpose. After the calculations, it became clear that the information system is able to rank the previously obtained disciplines according to the experts’ assessment. Thanks to this, it became possible to provide more precise disciplines to the user as recommendations. The disciplines obtained as a result of using the analytic hierarchy process can be recommended to the user for inclusion in an individual educational trajectory.*

*Conclusions. **During the research, we analyzed the possibility to include external factors in the process of the formation of individual trajectories. During research, It was proved that there is a huge necessity to include experts’ and teachers’ opinions based on criteria during the formation of IET. Experts’ or teachers’ influence on the student can be critical. In particular, the student’s choice of discipline may be completely different, apart from the opinion of experts. For confirmation, test calculations were carried out using the method of analytic hierarchy process with random data, which made it possible to determine the clear advantage of one alternative to others with respect to the goal. Thus, the developed information system can be considered one of the most variable and flexible due to the additional functions and capabilities described in this paper.*

**Keywords: **individual educational trajectory, information system, analytic hierarchy process.

# Introduction

Institutions of higher education, in accordance with modern requirements, are aimed at training competent specialists with a harmonious combination of individual and professionally important skills that contribute to self-development and self-realization in the chosen professional field. Considering this, it is appropriate to note that the student-centered orientation of the educational process is particularly relevant and valuable. This orientation includes taking into account the individual capabilities and needs of students, ensuring freedom of choice of components of educational programs, encouraging students to find and define individual goals in educational activities, and conscious and responsible selection of components for an individual educational trajectory (IET).

The concept of the educational trajectory can be understood as the personal direction of an education seeker in the educational space. Under different conditions of the organization of the educational process, different scenarios for determining this trajectory are possible, in particular:

- The traditional approach involves the definition by the professor of a single trajectory for achieving results by applicants for education.
- With a differentiated approach, the professor offers several possible trajectories for groups of applicants.
- With an individual approach, each applicant for education receives his own trajectory, which is implemented on the basis of what is more favorable for educational development in the opinion of the professor.

Any of the described approaches cannot be implemented without the participation of a professor, which does not allow for taking into account all the individual needs, interests, or abilities of the student. Thus, an educational environment is created with a low level of achievement of the goals set by the professor. This is due to the low level of motivation and the lack of personal meaning in the educational activities proposed for the student. One of the ways to solve the described problem is the organization of IET.

The formation and implementation of IET is a multifaceted process designed to ensure the development of the student’s independence and initiative, to give the opportunity for the fullest realization of his personal and cognitive potential, to help gain experience in choosing goals, future activities, an independent organization of activities, and self-evaluation.

# Review of the literature

According to the National Strategy for the Development of Education in Ukraine for 2012-2021, the main direction of the development of the educational space is the implementation of the (student-centered) concept of individualized education [1]. The implementation of individualized training is provided by the definition of IET and the step-by-step passage of the route [2], which allows it to be implemented. Pedagogical aspects of the creation and implementation of IET (route) at various stages of education, in the study of individual disciplines, and the organization of education in educational institutions of higher education are considered in the works of a large number of researchers (E. Alexandrova, I. Kankovsky, S. Kliminska, G. Klimova, etc.) [3,4,5]

For a full-scale analysis of individual education, it is necessary to consider the developments in higher education institutions abroad. In particular, let’s consider the experience of universities in the UK, where there is a close concept of “Personal Development Planning” (“individual development planning”). Norman Jackson defines this concept as a structured and accompanied process, implemented directly by the student. It consists in the fact that the student analyzes his learning process, progress and results achieved, and also draws up a plan for his individual, educational and professional development [6].

A large number of studies have described that students’ decisions on how they construct their educational trajectories are deeply embedded and subject to their context. This may refer to their environment, including their peers, family context, context of growing up and neighborhood. In addition, not only may individually experiences matter, but also the organizational structures of the educational institution in which students are embedded (meso-level), or further, the regional or national structure of the educational system (macro-level) [7].

Several studies encompassed some meso-theoretical concepts, i.e., how the organizational context of the educational institutions influences how students proceed through individual education. For example, Meggiolaro et al. [8] argued that larger educational institutions provide more and better services to students. Thus, this is beneficial for students’ persistence. They also refer to the well-known finding that more selective programs and institutions also have higher persistence rates.

Bringing the concept of individualization into the research focus, we should also mention the existing problem of higher education institutions, which, although considered to be open social systems, are actually operationally closed. Although they are included in the global socio-economic reality, their survival depends on constant re-adaptation to interaction with their environment. In particular, it creates a model of an “educational supermarket” where a one-sided understanding of individualization spreads, i.e., when selected decision-makers take relevant decisions and then make students follow them only to later satisfy the job market.

To address this problem, E. de Corte suggests focusing more on competencies that empower students. Since learners are not passive recipients of information, constructive competencies will stimulate their effortful involvement in the processes of knowledge and skills acquisition; self-regulation competencies will facilitate monitoring of an ongoing learning process by providing one’s own feedback and by keeping oneself concentrated and motivated; collaborative competencies will guide learners from a “solo” activity to a “distributed” one [9]. In agreement with this approach, G. Prozumentova suggests implementing sustainable managerial mechanisms, capable of eliminating anthropological and humanitarian deficits and facilitating “a personal action in a person’s education” [10].

Considering the reviewed studies, it can be concluded that although the process of forming the IET is implemented by the student, he, first of all, is only the subject of the choice of differentiated education offered by the educational institution. Thus, the student, as it were, “offers” to the educational institution his individual educational route within the framework of a certain educational and professional program, taking into account his educational needs, cognitive profile, and other individual characteristics. The main work on the formation of the IET, after taking into account the analysis of all the personal qualities of the student, takes place with the participation of the educational institution, teachers, and experts.

The considered studies are necessary to analyze both the general concept and the influence of other external factors on the formation of IET. This is about the role not only of the students themselves in the process of formation but also the influence of professors, experts, and even educational institutions.

*The purpose of the work is* to use a mathematical tool that could provide the inclusion of expert opinions for the formation of IET.

# Research results

To solve the problem of IET formation, a model of the information system (IS) in the form of a recommendation system was developed. The created IS provides the user with the opportunity to get a recommended list of disciplines for their further inclusion in IET. The process of forming recommendations consists of two steps:

- Passing a career guidance test by the user, which allows determining his professional guidance.
- Determination of the semantic proximity between professional guidance and potential discipline in the educational program.

The created system is capable of providing basic recommendations, and its developers strive to ensure high accuracy of the results and try to ensure that the recommendations provided by the system are not very general, and correlate with specific existing educational and professional programs, taking into account the personal qualities of the user.

The absence of external influence on the process of formation of the IET leads to a model for choosing disciplines that do not take into account the recommendations of professors. Most studies indicate that recommendations from professors or experts are not mandatory or important, as they limit the individuality of the student. However, potential recommendations from an expert may force a student to change their views on educational competencies. This is due to the fact that, for example, the experience of an expert in a particular field can provide the student with the necessary information to form the final vision of his own professional activity in the future. The experience of the teacher or professor may be less authoritative, but this depends more on the relationship between them and the students and the direct interest of the student in related professional activities if the professor is involved in such.

Considering this, there is a decision-making problem when choosing the best discipline for a selective block of an educational and professional program, which cannot be solved without the support of experts. Accordingly, to take into account the recommendations of experts in solving the problem of decision-making, a mathematical tool was used – the analytic hierarchy process.

The analytic hierarchy process (AHP) is a mathematical tool for a systematic approach to solving complex decision-making problems. The main application of the method is to support decision-making through the hierarchical composition of the problem and the rating of alternative solutions [11]. It provides a comprehensive and rational framework for structuring a decision problem, presenting and quantifying its elements, linking these elements to common goals, and evaluating alternative solutions [12].

As a concept for improving the IS, the analytic hierarchy process will be used to rank academic disciplines at the user’s request. The AHP application algorithm consists of five steps:

- Building a hierarchical structure of the task.
- Defining the priorities of the elements in the hierarchical structure and creating a matrix of pairwise comparisons.
- Calculating the weight coefficients based on pairwise comparisons.
- Testing the comparison matrix for consistency.
- Performing hierarchical synthesis.

Let’s test an example of using AHP with the following input data. Say, the user used the system, and the largest weight coefficients were assigned to the following disciplines:

- Computer linguistics.
- Problem-oriented programming.
- IT project planning.

These disciplines belong to the selective components of the educational and professional program “System Analysis” for the first level (bachelor) of higher education as for the 2022 academic year. The ranking of disciplines, respectively, will be carried out based on the following criteria:

- Personal hobbies and interests.
- Potential salary.
- Professional prospects.
- Ability to work in related fields.

These criteria are selected based on the analysis of the main characteristics of the disciplines when they are chosen by students.

Thus, the first step in the application of AHP is the construction of a hierarchical structure of the task, which will include the goal, criteria, alternatives, and other factors influencing the choice. Each element of the hierarchy represents different aspects of the problem. A tree of hierarchies with criteria and alternatives can serve as a practical implementation of the hierarchical structure. The tree of hierarchies with previously described alternatives and criteria is shown in Fig. 1.

The top level of the hierarchy represents the purpose for which the AHP is applied, and the second and third levels represent the criteria and disciplines, respectively.

The next step is to determine the priorities, which represent the relative importance or superiority of the elements in the constructed hierarchy tree. Disciplines will prioritize each decision criterion, while criteria will be prioritized based on their importance to achieving the goal. The priorities will then be combined across the hierarchy to determine the overall priority for each discipline. A discipline with a high priority will be the best alternative, and the ratio of discipline priorities will indicate their relative strength in relation to the goal.

Priorities will be derived by performing a series of calculations: pairwise comparisons involving all elements of the hierarchy. Elements at each level will be compared two by two, given their contribution to the elements above them. The results of these comparisons will be entered into a matrix, which is processed mathematically to obtain the priorities of all elements at the level.

In the matrix of pairwise comparisons, it is required to set the relative weight of each discipline. To do this, we use the relationship scale developed by Saaty. It is shown in Table 1. This scale allows the expert to set a certain number, which determines the correspondence between the degree of advantage of one factor over another.

Table 1

**Pair of pairwise comparisons**

Relative importance (points) | Definition | Explanation |

1 | of equal importance | both elements make the same contributions |

3 | one element is a little more important than the other | experience allows us to put one element a little higher than another |

5 | essential advantage | experience allows us to establish an unconditional advantage over each other |

7 | significant advantage | one element is so important than the other that it is practically significant |

9 | the absolute superiority of one over the other | the obviousness of the advantage is confirmed by the majority of factors |

2,4,6,8 | intermediate estimates between adjacent statements | compromise decision |

inverse values of the numbers above | in case, when comparing one element with another, one of the above numbers (1-9) is obtained, then when comparing the second with the first, we will receive the inverse value |

The validity of this scale has been proven practically in comparison with many other scales. When using this scale, the expert, comparing two objects in the context of achieving the goal located at the highest level of the hierarchy, must match this comparison with a number in the range from 1 to 9 or the opposite of it. In cases where it is difficult to distinguish how many intermediate gradations from absolute to weak advantage or this is not required in a particular task, a scale with a smaller number of gradations can be used. The maximum scale has two ratings: 1 – the objects are equivalent; 2 – the advantage of one object over another. [11] Thus, comparing alternatives with the criterion “Personal hobbies and interests” is shown in Table 2.

Table 2

**Alternatives compared with respect to criteria “Personal hobbies and interests”**

Alternative | Alternative weight | Alternative | Alternative weight |

Computer linguistics | 7 | Problem-oriented programming | 1 |

Computer linguistics | 4 | IT project planning | 1 |

Problem-oriented programming | 9 | IT project planning | 1 |

Next, we build a matrix of pairwise comparisons, which is filled in according to the rules that are shown in Fig 2.

Our matrix based on the given input data and the criterion “Personal hobbies and interests” will look like this:

For this matrix, it is necessary to perform mathematical calculations to determine its numerical characteristics: eigenvector, largest value, consistency index, and ratio sequence index.

First, we calculate the value of the eigenvector. The algorithm for finding the eigenvector consists of three steps:

- For each row of the matrix of pairwise comparisons, find the geometric mean of its elements.
- Find the sum of all geometric means.
- Divide each geometric mean by their sum (normalization).

Accordingly, given the input data, we obtain the following eigenvector:

After finding the eigenvector, it is necessary to evaluate the consistency of the judgments of experts on the data from which the matrix of pairwise comparisons was built. The process of evaluating judgments consists in finding three quantities:

- Maximum eigenvalue.
- Consistency index.
- Ratio sequence index.

The maximum eigenvalue can be found using the following formula:

(1) |

To calculate the consistency index, we use the following equality:

(2) |

The following formula allows for determining the ratio sequence index:

(3) |

is a random indicator for (equal to the number of alternatives we use as input data), the value of which is the same for all subsequent calculations of alternative weights. Next, we perform all the necessary calculations:

Similarly, calculations are made for all other criteria. For the “Potential salary” criterion comparison of alternatives can be seen in Table 3.

Table 3

**Alternatives compared with respect to criteria “Potential salary”**

Alternative | Alternative weight | Alternative | Alternative weight |

Computer linguistics | 1 | Problem-oriented programming | 5 |

Computer linguistics | 3 | IT project planning | 1 |

Problem-oriented programming | 8 | IT project planning | 1 |

Matrix of pairwise comparisons:

Eigenvector:

For the “Professional prospects” criterion comparison of alternatives can be seen in Table 4.

Table 4

**Alternatives compared with respect to criteria “Professional prospects”**

Alternative | Alternative weight | Alternative | Alternative weight |

Computer linguistics | 4 | Problem-oriented programming | 1 |

Computer linguistics | 1 | IT project planning | 6 |

Problem-oriented programming | 1 | IT project planning | 7 |

Matrix of pairwise comparisons:

Eigenvector:

For the criterion “Ability to work in related fields”, a comparison of alternatives can be seen in Table 5.

Table 5

**Alternatives compared with respect to criteria “Ability to work in related fields”**

Alternative | Alternative weight | Alternative | Alternative weight |

Computer linguistics | 3 | Problem-oriented programming | 1 |

Computer linguistics | 4 | IT project planning | 1 |

Problem-oriented programming | 6 | IT project planning | 1 |

Matrix of pairwise comparisons:

Eigenvector:

Now that we evaluated the alternatives with respect to their strength in meeting the criteria, we would need to evaluate the criteria with respect to their importance in reaching the goal. A comparison can be seen in Table 6.

Table 6

**Criteria compared with respect to reaching the Goal**

Criteria | Criteria weight | Criteria | Criteria weight |

Personal hobbies and interests | 3 | Potential salary | 1 |

Personal hobbies and interests | 4 | Professional prospects | 1 |

Personal hobbies and interests | 6 | Ability to work in related fields | 1 |

Potential salary | 1 | Professional prospects | 3 |

Potential salary | 8 | Ability to work in related fields | 1 |

Ability to work in related fields | 1 | Professional prospects | 5 |

The corresponding matrix of pairwise comparisons for the criteria will have the following form:

Eigenvector:

The last step in the execution of the AHP is to carry out a hierarchical synthesis. The eigenvector of criteria weight coefficients, which is necessary for hierarchical synthesis, was preliminarily calculated. Next, we multiply the matrix of weights of alternatives by the resulting vector of criteria weights. Calculations can be seen in Fig. 3.

Figure 3 – Calculating vectors.

To conclude the algorithm, we have to sum up all the weights of the alternatives with respect to the criteria. This is required to find the largest value. Calculation results can be seen in Table 7.

Table 7

**Overall priorities for all of the alternatives**

Alternatives | Personal hobbies and interests | Potential salary | Professional prospects | Ability to work in related fields | Goal |

Problem-oriented programming | 0,357 | 0,028 | 0,058 | 0,032 | 0,476 |

Computer linguistics | 0,127 | 0,068 | 0,023 | 0,019 | 0,239 |

IT project planning | 0,035 | 0,007 | 0,236 | 0,005 | 0,285 |

Totals: | 0,519 | 0,103 | 0,317 | 0,056 | 1 |

According to the table, the alternative (discipline) “Problem-oriented programming” with a weight of 0.476 has the highest priority in relation to the goal.

Thus, due to the use of the hierarchy analysis method, the process of obtaining more accurate recommendations during the formation of the IET is provided. The experiment showed that in a situation where the recommendation of several disciplines passes, the external decision of the experts can greatly affect the result. The process of determining the best discipline depends largely on the given criteria and the assigned weight of alternatives, however, this was the purpose of demonstrating the importance of external influence (decisions of experts) in the formation of the IET.

It is clear that all the recommendations received are not mandatory for their inclusion in the IET. The user can deny additional recommendations of experts when choosing disciplines and choose those that he likes best. In this case, the hierarchy analysis method will not be used and the system will be limited to determining professional inclination and finding semantically similar educational and professional programs and disciplines in them.

# Conclusions

This article describes the actual scientific problem of the formation of IET. The focus of the research was carried out on the individualization of the educational process and its dependence on the actions of the higher educational institution. Professors and experts are presented as external factors influencing the formation of IET.

To solve the problem, a previously developed tool for providing assistance in the formation of the IET was implemented. The peculiarity of system behavior was also analyzed when trying to solve the problem of the influence of external factors on the provision of recommendations in the formation of the IET.

Thus, in order to solve the new problem posed, an improvement of the considered tool was proposed. Accordingly, the improved model is aimed at the possibility of taking into account the opinions of experts in the decision-making process when choosing disciplines for IET. For this, the method of analysis of hierarchies was used, as it allows to optimally solve the problem of decision making.

The forthcoming work on the developed model will be aimed at its practical implementation. It is planned to use web technologies to ensure the relevance, availability, and smooth operation of the system.

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3. E. de Corte, “Improving Higher Education Students’ Learning Proficiency by Fostering their Self-regulation Skills”, European Review. 2016. Vol. 24 (2), pp. 264-276.

4. G. N. Prozumentova, “Educational Innovations: The phenomenon of ‘personal presence’ and management potential”, Tomsk: Tomsk State University, 2016.

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