APPLICATION OF METHODS OF MATHEMATICAL STATISTICS IN MARKETING AND PERSONNEL MANAGEMENT

ПРИМЕНЕНИЕ МЕТОДОВ МАТЕМАТИЧЕСКОЙ СТАТИСТИКИ В МАРКЕТИНГЕ И УПРАВЛЕНИИ ПЕРСОНАЛОМ

Багаутдинова Нигина Ильдаровна

студент, кафедра Цифровые системы и модели, Казанский государственный энергетический университет,

РФ, г. Казань

Айтуганова Жанна Илевна

научный руководитель, канд. пед. наук, доц., кафедра иностранных языков, Казанский государственный энергетический университет,

РФ, г. Казань

АННОТАЦИЯ

В данной статье проводится анализ методов математического моделирования в области маркетинга и управления. В качестве объекта исследования рассмотрим задачу, основанную на профессиональных предпочтениях. Для решения данной задачи будет применен венгерский алгоритм, а также построена матрица предпочтений.

ABSTRACT

This article provides an analysis of methods of mathematical modeling in the field of management. The object of study is a problem based on professional preferences. To solve this problem, the Hungarian algorithm will be applied, and a preference matrix will be constructed.

Ключевые слова: управление персоналом, методы математического моделирования, профессиональные предпочтения.

Keywords: personnel management, methods of mathematical modeling, professional preferences.

In the context of fierce competition and the variability of the labor market, effective human resource management is critically important. An optimal distribution of employees across vacancies, considering their skills, interests, and career goals, increases motivation, reduces turnover, and enhances the overall productivity of the company.

As management tasks become more complex, intuitive methods are giving way to mathematical modeling and algorithms. These provide accuracy, objectivity, and automation in decision-making, allowing for the identification of optimal solutions while accounting for various constraints.

The Hungarian algorithm serves as a promising solution—a classic tool for assignment problems that allows for finding the optimal match between employees and positions, taking into account their professional preferences.

The relevance of this article is determined by the need for a systematic approach to personnel management in order to effectively implement HR strategies and enhance the overall effectiveness of organizations. Special attention is given to the development of preference matrices as a starting point for the algorithm.

The aim of this article is to develop a model for the optimal distribution of employees across positions based on their professional preferences, using a preference matrix and the Hungarian algorithm.

Within the framework of mathematical modeling, the task is reduced to determining the optimal distribution of candidates across vacancies that ensures maximum team efficiency and achievement of objectives.

The main idea is to find a distribution of candidates across positions that maximizes the objective function. Thus, the problem is reduced to finding an extremum while satisfying certain conditions and constraints.

The mathematical model of the problem consists of directly finding the optimal value of the objective function:

with the following constraints:

⁡⁡ , if the i-th candidate is not assigned to the j-th position, and xij =1, otherwise. The elements (c{ij}) of matrix C may represent the time spent by employees on tasks, thus solving a minimization problem.

For optimal candidate distribution across company vacancies, the Hungarian algorithm is frequently employed to maximize overall efficiency.

Example: There are five candidates (X₁, X₂, X₃, X₄, X₅) and five positions (Y₁, Y₂, Y₃, Y₄, Y₅). The compatibility ratings reflecting the professional preferences of the candidates for each position are compiled based on testing using the Holland system.

How the Holland Test Works:

1. The candidate fills out a questionnaire indicating preferences and interests in various areas.
2. The data is analyzed, forming the candidate's interest profile.
3. Based on this profile, the most suitable professions are identified.

The Holland test aims to uncover professional interests and inclinations. Examples of questions include:

• What do you prefer: working with people, things, ideas, or technologies?
• What is more important to you: helping others or creating something new?
• Do you feel more comfortable working in a team or independently?
• Analysis or practice?

Education, art, engineering, or management?

A scale from 1 to 10 is used to evaluate the degree of interest in each area.

Based on the questionnaire data and testing, the following matrix of ratings C was obtained, where rows represent candidates and columns represent positions according to their profiles:

Indicators of professional suitability of candidates for each role, where a higher value corresponds to a better fit.

The Hungarian algorithm is typically used for minimization. Therefore, to maximize the sum of ratings, the matrix needs to be transformed. To do this:

Determine the maximum value in the matrix: (maxC=10).

Multiply all elements by -1 to convert the problem into a minimization problem.

Add 11 (one more than the maximum value) to each element to make all elements positive.

The transformation looks like this:

Cij′​=11−Cij

The updated matrix looks like this:

This step allows for the use of the classic Hungarian algorithm.

The next stage is to achieve the maximum number of zeros in the matrix to simplify the search for optimal assignments. For each row, we find the minimum element: Row 1: minimum – 3; Row 2: minimum - 2; Row 3: minimum - 1; Row 4: minimum - 3; Row 5: minimum - 1.

We subtract these values from all elements of the corresponding rows.

Updated matrix:

Now each row has at least one zero element.

Next, we seek the minimum values across the columns: Column 1: minimum – 0; Column 2: minimum – 0; Column 3: minimum - 0; Column 4: minimum - 0; Column 5: minimum – 0.

We subtract these values from the corresponding columns: updated matrix:

Now the matrix contains numerous zeros, and we can begin searching for optimal assignments. We look for a set of zeros such that exactly one zero element is selected from each row and column. We will draw lines through rows and columns to cover all zeros. If the number of lines equals the size of the matrix, the solution is found. For this, we use the independent zero searching method of the Hopcroft-Karp algorithm or a greedy choice:

Table 1.

Results for the independent search method of zero of the Hopcroft-Karp algorithm.

Note that in this case, since the zero elements in the rows and columns do not intersect, this is a valid assignment.

Based on the selected zero elements, we return to the original matrix C (before transformation) to determine the final ratings:

X₁ → Y₄ (value 8)

X₂ → Y₂ (value 9)

X₃ → Y₃ (value 10)

X₄ → Y₁ (value 8)

X₅ → Y₃ (value 10)

Thus, candidate X₁ will be assigned to position Y₄, candidate X₂ to position Y₂, candidate X₃ to position Y₃, candidate X₄ to position Y₁, and candidate X₅ to position Y₃.

The total combined rating of the entire team will be: 8 + 9 + 10 + 8 + 7 = 42.

Matrix methods and the Hungarian algorithm are indeed powerful tools for solving optimization problems. They help optimize resource distribution, improve decisionmaking processes, enhance work efficiency, and reduce costs.

Regular use of such methods contributes to economic development and enables companies to remain competitive in the market.

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