Mathematical Modeling of Company Activity

American Journal of Applied Science and Technology

38

https://theusajournals.com/index.php/ajast

VOLUME

Vol.05 Issue 03 2025

PAGE NO.

38-41

DOI

10.37547/ajast/Volume05Issue03-07

Mathematical Modeling of Company Activity

Aymatova Farida Khurazovna

Senior Lecturer of "Social and Exact Sciences", Tashkent State University of Economics, Uzbekistan

Shamsiyev Damin Najmiddinovich

associate professor at Tashkent State Technical University, Uzbekistan

Received:

13 January 2025;

Accepted:

15 February 2025;

Published:

15 March 2025

Abstract:

The article discusses the tasks of mathematical modeling of economic processes, particularly focusing on

the mathematical modeling of tourism company activities. Under given conditions, a mathematical model is
constructed, an optimal solution is found, and the results are analyzed.

Keywords:

Mathematical model, optimal solution, tourism, linear programming, objective function, constraints.

Introduction:

Currently, mathematical models are widely used in
such fields of science as physics, chemistry, biology, as
well as in technical and economic directions.
Mathematical models can be divided into analytical,
numerical and statistical types.
A mathematical model is a set of mathematical
formulas,

equations,

inequalities,

systems

of

equations, which allow to express the happening
events and processes with some accuracy.
Modeling of various interactions of management
organizations, producers, consumers of utility services,
as well as providing services to the residential stock is
complex, multi-criteria and dynamic, and it is
appropriate to use mathematical methods for solving
them.
Mathematical models make it possible to express
connections between various processes and realities in
the economy, to estimate various economic indicators
in advance, and to develop strategies for managing
economic objects. The relevance of modeling all
economic and management processes is based on the
obtained results, preliminary assessment of the
development of these processes, implementation of
effective management.
Today, tourism has become one of the leading sectors
of the world economy. In this regard, special attention
is paid to the organization of services in the field of
tourism in Uzbekistan based on international
standards, because our country has many world-

famous pilgrimage sites, cities and corners rich in
historical monuments.

Therefore, the issue of optimizing tourist plans for
tourist companies is urgent.

LITERATURE ANALYSIS
A lot of research has been done on solving economic
problems with the help of mathematical models.
Mathematical model and general methods of
application of mathematical methods in economics
have been developed (construction of functional
relationships, analysis and solving optimization
problems at various levels) [4].
In particular, researches aimed at optimization of
advertising business, tourist flow, etc. are also being
carried out. In general, to optimize the activities of
tourist organizations, mathematical models are built
and constraints and optimality criteria are formed [1].
These models differ according to the problem:
organization of family vacations, expansion of activities
with the help of advertising, modeling of the placement
of tourists' luggage [2]. Studying the laws of tourist
flows and their optimization is another important issue
[3].

RESEARCH METHODOLOGY

This article is focused on the issue of optimizing the
work of a tourist company, which requires
simultaneous variation of several group variables. A

American Journal of Applied Science and Technology

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American Journal of Applied Science and Technology (ISSN: 2771-2745)

mathematical model is built based on the results of the
previous season of the tourist company, and the issue
of creating an optimal plan for the next season is
solved. This problem is reduced to a linear

programming problem and is solved by the simplex
method. Of course, computer programs were used to
perform such calculations.

MAIN PART

n- the number of domestic tourist programs (n=1,2,...), let x

j

be the number of tourists in domestic tourist programs

conducted according to j- program (j=1,2,...n). Let a

ij

- be the coefficient of soum/tourist-sized expenses for i-service

in j- program, b

i

soum- the total expenses of a service, c- the cost of each tourist.

If the privilege given to program j is k

j

, then the objective is a function

𝑷(𝒙) = 𝒄(𝒌

𝟏

𝒙

𝟏

+ 𝒌

𝟐

𝒙

𝟐

+ ⋯ + 𝒌

𝒏

𝒙

𝒏

) → 𝒎𝒂𝒙

(1)

and the constraints are expressed by the following n inequalities:

{

𝒂

𝟏𝟏

𝒙

𝟏

+ 𝒂

𝟏𝟐

𝒙

𝟐

+ ⋯ + 𝒂

𝟏𝒏

𝒙

𝒏

≤ 𝒃

𝟏

𝒂

𝟐𝟏

𝒙

𝟏

+ 𝒂

𝟐𝟐

𝒙

𝟐

+ ⋯ + 𝒂

𝟐𝒏

𝒙

𝒏

≤ 𝒃

𝟐

⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

𝒂

𝒎𝟏

𝒙

𝟏

+ 𝒂

𝒎𝟐

𝒙

𝟐

+ ⋯ + 𝒂

𝒎𝒏

𝒙

𝒏

≤ 𝒃

𝒎

(2)

𝒙

𝒋

≥ 𝟎, 𝒋 = 𝟏, 𝒏

̅̅̅̅̅

(3)

(1)-(3) is a well-known linear programming problem [4].
Using it, it is possible to determine the number of tourists x

j

that can be attracted to each program in the case

where the distribution of costs b

i

for services is known.

If a more complicated issue is the issue of optimal distribution of V expenses allocated for total tourist programs
with maximum attraction of tourists, then the above issue

𝒃

𝟏

+ 𝒃

𝟐

+ ⋯ + 𝒃

𝒏

≤ 𝑽

it will be necessary to fill with limitation.

Now we can solve this problem using the MS Excel program in the computer model method.
In the process of forming a mathematical model, it is difficult to determine the numerical values of coefficients

a

ij

of system (2).

If any i- condition of (2).

𝒂

𝒊𝟏

𝒙

𝟏

+ 𝒂

𝒊𝟐

𝒙

𝟐

+ ⋯ + 𝒂

𝒊𝒏

𝒙

𝒏

≤ 𝒃

𝒊

If we analyze , we see that

(𝒃

𝒊

)

determines the distribution of service costs among tourists of group

(𝒙

𝒋

)

through

coefficients a

ij

.

Evidently,

𝒂

𝒊𝒋

coefficient j

–

the number of tourists in the program

𝒍

𝒋

is related to the total costs

𝒃

𝒊

of type (i) service:

𝒂

𝒊𝒋

= 𝑸

𝒊𝒋

𝒃

𝒊

𝒔𝒐𝒖𝒎

𝒍

𝒋

𝒕𝒐𝒖𝒓𝒊𝒔𝒕

(4)

here,

𝑸

𝒊𝒋

- (i) type of service

𝒃

𝒊

is the part of the j-program corresponding to the total cost of

𝒃

𝒊

.

We will consider how the coefficient

𝒂

𝒊𝒋

is determined in a particular case in the following problem.

According to this year's plan, 3 tourist groups are planned:

1 group, university students

У

т

(𝒋 = 𝟏)

;

2 groups, factory workers

З

и

(𝒋 = 𝟐)

;

3 groups, retired parents

Н

о

(𝒋 = 𝟑)

.

The tourist company's plans for this year include four tourist destinations:

Samarkand;
Bukhara;
Khiva;
Shaxrisabz.

The results of the last tourism year were as follows:
Distribution of tourists by groups:

У

т

= 𝟓𝟎

,

З

и

=45,

Н

о

= 𝟑𝟎

.

Distribution of costs by services:
Samarkand

(𝒃

𝟏

) = 𝟐𝟓𝟎𝟎𝟎𝟎𝟎𝟎

, Bukhara

(𝒃

𝟐

) = 𝟐𝟎𝟎𝟎𝟎𝟎𝟎𝟎

,

Khiva

(𝒃

𝟑

) = 𝟑𝟎𝟎𝟎𝟎𝟎𝟎𝟎

, Shaxrisabz

(𝒃

𝟒

) = 𝟐𝟎𝟎𝟎𝟎𝟎𝟎𝟎

Based on last year's results, it was possible to fill in the following table 1.

1 table. Classification of tourists into

(𝑙

𝑗

)

groups

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American Journal of Applied Science and Technology (ISSN: 2771-2745)

𝑖

City

𝑙

𝑗

(tourist)

𝑗
= 1

𝑗
= 2

𝑗
= 3

Expenses

(сўм)

𝑏

𝑖

∑ 𝑙

𝑖𝑗

3

𝑗=1

(tourist)

1 Samarkand

17

12

8

25000000

37

2 Bukhara

11

7

7

20000000

25

3 Khiva

13

16

9

30000000

38

4 Shaxrisabz

9

10

6

20000000

25

We assume that the costs of i-type services are equally distributed among all customers in i-line, then the total
costs of i-type services of j-programme customers is equal to

Ж

𝑖𝑗

=

𝑏

𝑖

∑

𝑙

𝑖𝑗

3

𝑗=1

𝑙

𝑖𝑗

(5)

where

𝑙

𝑖𝑗

is the number of tourists in the j- program who used the i- type of service.

So, according to what has been said

𝑎

𝑖𝑗

=

Ж

𝑖𝑗

𝑙

𝑗

=

𝑏

𝑖

𝑙

𝑖𝑗

𝑙

𝑗

∑

𝑙

𝑖𝑗

3

𝑗=1

(6)

where

𝑙

𝑗

−

is the number of tourists in the j program,

∑

𝑙

𝑖𝑗

3

𝑗=1

−

is the number of tourists in row i,

𝑙

𝑖𝑗

is the number

of tourists at the intersection of row i and column j.

According to formulas (4) and (6).

𝑄

𝑖𝑗

=

𝑙

𝑖𝑗

∑

𝑙

𝑖𝑗

3

𝑗=1

The calculated values of coefficients

𝑎

𝑖𝑗

according to the formula (6) are presented in Table 2.

Table 2

𝑗

𝑖

1

2

3

1

𝑎

11

= 230000

𝑎

12

= 180000

𝑎

13

= 180000

2

𝑎

21

= 176000

𝑎

22

= 124500

𝑎

23

= 124500

3

𝑎

31

= 205000

𝑎

32

= 280000

𝑎

33

= 237000

4

𝑎

41

= 144000

𝑎

42

= 178000

𝑎

43

= 160000

putting the found values of the coefficients

𝑎

𝑖𝑗

into the mathematical model under the conditions

𝑘

1

= 𝑘

2

= 1

and

𝑘

3

= 0,93

(1)-(3)

{

230𝑥

1

+ 180𝑥

2

+ 180𝑥

3

≤ 25000

176𝑥

1

+ 124,5𝑥

2

+ 124,5𝑥

3

≤ 20000

205𝑥

1

+ 280𝑥

2

+ 237𝑥

3

≤ 30000

144𝑥

1

+ 178𝑥

2

+ 160𝑥

3

≤ 20000

(7)

in limitations

𝑃(𝑥) = 𝑐(𝑥

1

+ 𝑥

2

+ 0,93𝑥

3

) → 𝑚𝑎𝑥

(8)

we find the optimal value of

𝑥

1

, 𝑥

2

, 𝑥

3

by checking the objective function to the maximum.

RESULTS AND CONCLUSIONS
Simplex calculations in MS Excel gave the results

𝒙

𝟏

= 𝟓𝟏, 𝒙

𝟐

= 𝟒𝟗, 𝒙

𝟑

= 𝟐𝟓

. So, if the values of

𝒙

𝟏

, 𝒙

𝟐

, 𝒙

𝟑

are the

same as above, the net income will be the highest if tourists are attracted. If the cost of each trip is the same c in
all directions, the net revenue

𝑷(𝒙) = 𝒄(𝒌

𝟏

𝒙

𝟏

+ 𝒌

𝟐

𝒙

𝟐

+ 𝒌

𝟑

𝒙

𝟑

) = 𝒄(𝟓𝟏 + 𝟒𝟗 + 𝟎, 𝟗𝟑 ∙ 𝟐𝟓) = 𝟏𝟐𝟑, 𝟐𝟓𝒄

we will be able to calculate through

In our case, the values of the coefficients

𝒂

𝒊𝒋

are appropriate for the last tourist season, and the results obtained in

the next year, when the conditions have not changed, give the exact solution. Otherwise, it can be considered as a
target with some degree of accuracy for the next year, and according to the obtained results, it will be necessary to

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American Journal of Applied Science and Technology (ISSN: 2771-2745)

adjust the values of the coefficients in the above manner.

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