INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 1240
OPTIMAL ALLOCATION OF RESOURCES IN FURNITURE PRODUCTION:
PROFIT MAXIMIZATION USING THE SIMPLEX METHOD
Arabova Farzanabonu Akmaljon kizi
Fergana state university
Practical mathematics direction
student
Email:
a07612720@gmail.com
Mamatova Zilolakhan Khabibullokhanovna
Fergana state university associate professor ,
pedagogy sciences according to philosophy Doctor of Philosophy (PhD)
E-mail:
mamatova.zilolakhon@gmail.com
ORCID ID
Abstract :
This in the article furniture working in the release optimal allocation of resources
issue Simplex method using Furniture company's six product type ( chair , table , sofa,
wardrobe , armchair , bed ) and six resource ( oil , cloth , metal , paint , labor ) power ,
energy ) based on structured linear programming model analysis The research is being
conducted purpose – limited resources under the circumstances weekly profit maximize for
the most good working release plan definition . Simplex of the method step calculations
resulting in an optimal solution as 10 sofas and 10 wardrobes working release offer is 1100
thousand
soum
maximum benefit brings . Article linear programming theoretical
Basics , Simplex of the method practical application and his/her working release processes
in optimization importance illuminates . Research results in real life economic problems
solution in doing mathematician models efficiency shows .
Abstract:
This article studies the issue of optimal resource allocation in furniture production
using the Simplex method. A linear programming model of a furniture company based on six
product types (chairs, tables, sofas, wardrobes, armchairs, beds) and six resources (wood,
fabric, metal, paint, labor, energy) is analyzed. The purpose of the study is to determine the
best production plan to maximize weekly profit under limited resources. As a result of step-
by-step calculations of the Simplex method, the optimal solution is proposed to produce 10
sofas and 10 wardrobes, which will bring a maximum profit of 1,100 thousand soums. The
article discusses the theoretical foundations of linear programming, the practical application
of the Simplex method, and its importance in optimizing production processes. The results of
the study demonstrate the effectiveness of mathematical models in solving real-life economic
problems.
Аннотация:
В статье рассматривается задача оптимального распределения ресурсов
при производстве мебели с использованием симплекс-метода. Проанализирована
модель линейного программирования мебельной компании на основе шести типов
продукции (стул, стол, диван, шкаф, кресло, кровать) и шести ресурсов (древесина,
ткань, металл, краска, рабочая сила, энергия). Целью исследования является
определение наилучшего плана производства для максимизации еженедельной
прибыли в условиях ограниченных ресурсов. В результате пошаговых расчетов
симплекс-метода оптимальным решением является изготовление 10 диванов и 10
шкафов, что принесет максимальную прибыль в размере 1 100 000 сумов. В статье
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
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page 1241
рассматриваются теоретические основы линейного программирования, практическое
применение симплекс-метода и его значение в оптимизации производственных
процессов. Результаты исследования демонстрируют эффективность математических
моделей при решении реальных экономических задач.
Kalit so‘zlar:
mebel ishlab chiqarish, resurs taqsimlash, optimal yechim, simpleks usul,
foydani maksimallashtirish, chiziqli dasturlash, ishlab chiqarish rejalashtirish, resurs
cheklovlari, yog‘och, mato, metall, bo‘yoq, ishchi kuchi, energiya, divan, shkaf, matematik
model, iqtisodiy samaradorlik, iteratsiya, pivot element.
Keywords:
furniture production, resource allocation, optimal solution, simplex method,
profit maximization, linear programming, production planning, resource constraints, wood,
fabric, metal, paint, labor, energy, sofa, wardrobe, mathematical model, economic efficiency,
iteration, pivot element.
Ключевые слова:
производство мебели, распределение ресурсов, оптимальное
решение, симплекс-метод, максимизация прибыли, линейное программирование,
планирование производства, ограничения ресурсов, древесина, ткань, металл, краска,
труд, энергия, диван, шкаф, математическая модель, экономическая эффективность,
итерация, опорный элемент.
Introduction .
Processes research and optimal management – decision acceptance to
do and systems to optimize scientific fields oriented .
1-Process research resources effective distribution for mathematician models , linear
programming , games theory and networks optimization such as from methods uses .
2-Optimal management systems the most good management strategies determination
with He is engaged in his work . main methods Pontryagin's Maximum principle and
Bellman's dynamic programming .
Literature analysis
Furniture working in the release optimal allocation of resources and profit maximize
issue modern economy and working to release management in the field important place
This
topic according to literature mainly linear programming , Simplex method and
resources effective management mathematician models around shaped . Linear programming
main theory Developed by George Dantzig in 1947 issued Simplex method with related is ,
this method resources distribution problems solution in doing wide applied . Danzig in their
work Simplex of the method algorithmic structure and his/her limited resources optimal
solution under the circumstances in finding efficiency in detail illuminated . This method
furniture working release such as in the fields one how many product types and resources
between balance determination for suitable tool as Local and international in literature
furniture working in the release from resources use optimization according to one row
research For example , the economy and working to release management according to general
in sources ( such as Hillier and Lieberman ) "Operations Research" by the authors ) Simplex
of the method various in the fields application examples with analysis This is done . in
sources furniture industry such as to resources related in the fields expenses minimize and
profit increase strategies general as a mathematical model cited .
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 1242
Research methodology
This of the research main purpose furniture working issuer the company limited
resources ( wood , cloth , metal , paint , labor ) power , energy ) within six kind of product
( chair , table , sofa , wardrobe , armchair , bed ) to release optimization and weekly profit is
to maximize . In the study mathematician to modeling based quantitative approach is used .
Linear programming Simplex method main tool as was chosen because it is restrictive
problems in solution effective and wide applicable algorithm is considered .
Research in the process following methods used . Mathematics
Modeling : Furniture
working release process linear programming model as expressed . Purposeful function ( profit)
maximize ) and resource restrictions equations and inequalities in the form of Simplex
method.Linear programming model solution for Simplex algorithm This was used . method
targeted the optimal value of the function determination for iterations through possible was
solutions of the territory edge points analysis does . Table in the form of calculation .
Simplex of the method steps table in format done increased , this while pivot element
selection , row and column operations clear and systematic accordingly to perform
opportunity gave . Analytical Approach : Results working of release economic efficiency
point of view from the point of view analysis was done , the optimal solution practical
importance was evaluated .
Analyses and results
Simplex method general if the borders equations and goal of functions equations
canonical to look has if not optimization linear issues solution for is used . In this case
equations system 's appearance as follows .
(
=
-
+
+
+
=
+
+
+
=
+
+
+
=
+
+
+
0
...
...
...
...
2
2
1
1
2
2
1
1
2
2
2
22
1
21
1
1
2
12
1
11
z
x
с
x
с
x
с
b
x
a
x
a
x
a
b
x
a
x
a
x
a
b
x
a
x
a
x
a
n
n
m
n
mn
m
m
n
n
n
n
1)
Simplex ( method ) in 2 steps is divided .
Stage 1 - Delimiter equations and goal functions canonical to look to bring
Stage 2 - Stage 1 as a result using simplex algorithm
harvest entered goal
function optimization .
Step 1 we build .
Artificial in stage 1 changes input way with , such as variables all to equations are
entered , equations to the system canonical appearance is given . Basis in character
variables was equations in the system and goal in functions uncommon variables and has a
coefficient of 1 was coefficients , from this exception . From this outside all to the system
artificial of variables from the sum consists of was additional equations is entered .
Then system of equations following to look has will be .
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 1243
=
-
+
+
+
=
-
+
+
+
=
+
+
+
+
=
+
+
+
+
=
+
+
+
+
+
+
+
+
+
+
+
0
...
0
...
...
...
...
2
1
2
2
1
1
2
2
1
1
2
2
2
2
22
1
21
1
1
1
2
12
1
11
W
x
x
x
z
x
с
x
с
x
с
b
x
x
a
x
a
x
a
b
x
x
a
x
a
x
a
b
x
x
a
x
a
x
a
m
n
n
n
n
n
m
m
n
n
mn
m
m
n
n
n
n
n
n
this on the ground :
xn
+1
, xn
+2
, … , x
n+m
- artificial variables ;
W = x
n+1
+ x
n+2
+ … + x
n+m
- their collection
All sizes non-negative to be need .
This for necessary in the case on the left side of the equation of variables gestures
change must be . x
n+1
, x
n+2
, … , x
n+m
variables last entered into the equation (W) for harvest
was system solution canonical to look has not . They disappearance
for - last to the
equation the first m equation will be added and the sum last from the equation is subtracted .
In this following equations system harvest It is .
=
-
+
+
+
=
+
+
+
+
=
+
+
+
+
=
+
+
+
+
+
+
+
0
...
...
...
...
2
2
1
1
2
2
1
1
2
2
2
2
22
1
21
1
1
1
2
12
1
11
z
x
с
x
с
x
с
b
x
x
a
x
a
x
a
b
x
x
a
x
a
x
a
b
x
x
a
x
a
x
a
n
n
m
m
n
n
mn
m
m
n
n
n
n
n
n
=
=
=
=
-
=
-
-
+
+
-
+
-
m
i
i
n
m
i
mn
m
i
i
m
i
i
b
W
x
a
x
a
x
a
1
1
2
1
2
1
1
1
...
=
=
m
i
ij
i
a
d
1
and
=
=
m
i
i
b
W
1
0
designation we enter .
In that case Simplex Step 1 of the method beginning for last equations system :
=
-
+
+
+
=
+
+
+
+
=
+
+
+
+
=
+
+
+
+
+
+
+
0
...
...
...
...
2
2
1
1
2
2
1
1
2
2
2
2
22
1
21
1
1
1
2
12
1
11
z
x
с
x
с
x
с
b
x
x
a
x
a
x
a
b
x
x
a
x
a
x
a
b
x
x
a
x
a
x
a
n
n
m
m
n
n
mn
m
m
n
n
n
n
n
n
d
1
x
1
+ d
2
x
2
+ … + d
n
x
n
– W = - W
0
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ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
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page 1244
Simple of the method first in the phase usual simplex algorithm to z using suitable W
function This minimization is need as follows :
1) d
j
-2 values is found if all sizes negative If W is minimize possible not , if W>0 ,
the path placed solution possibility no .
If the sizes some d
j
<0 if so , of the unknown d
s
=min( d
j
)d
s
<0 condition according
to to the base incoming S - index is selected .
2) Then from the base b
r
/ a
rs
=min(b
i
/ a
is
)a
is
>0 condition according to from the
base of the unknown IV to be released index is found .
3) 2nd system all equations is changed . In this d
j
and W
0
those of change additional
functions service except for : r all columns for d
j
= d
j
-d
s
a
rj
/ a
rs
, r column for d
r *
=-d
s
/
a
rs
W
0
=W
0
+ds b
r
/ a
rs
Then 13 points all sizes non-negative unless until repeated .
4) W is defined , if W=0 , then it is clear that all artificial variables 0 g a equals . Then
equations (2) from the system last equation and all artificial variables lost (2) system again
is written . Harvest made system canonical to look has If W<0 , the solution is no .
Stage 2 obtained in Stage 1 system 's algorithm using from optimization consists of .
A piece of furniture working issuer company 6 types kind of product working
produces : Chair (A), Table (B), Sofa (C), Wardrobe (D), Armchair (E) and Bed (F). Each
product working release 6 types for Resources required : oil (m³), fabric (m²), metal ( kg),
paint ( l), labor power ( hours ) and energy ( kWh ). Each of the product resource
requirements and benefit following in the table quoted :
Product Wood
(m³)
Fabric
(m²)
Metal
( kg )
Paint
(l)
Worker
power
( hours )
Energy
( kWh )
Profit
(
thousand)
soum )
Chair
(A)
1
0
0.5
0.2
2
1
20
Table
(B)
2
0
1
0.5
3
2
30
Divan
(C)
3
4
0
1
5
3
60
Shkaf
(D)
4
0
2
0.8
4
2
50
Kreslo
(E)
2
2
0.5
0.4
3
1.5
35
Karvat
(F)
5
3
1.5
1.2
6
4
80
In the company per week following resources there is :
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ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
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page 1245
Oil tank : 70 m³
Fabric: 40 m²
Metal : 20 kg
Paint : 15 liters
Worker Power : 80 hours
Energy : 50 kWh
Question : Company weekly the benefit maximize for every one from the product how much
working release need ?
Mathematician shape :
Variables :
x
1
– Number of chairs (A)
x
2
– Number of tables (B)
x
3
– Number of sofas (C)
x
4
– Number of cabinets (D)
x
5
– Chair (E) number
x
6
– Number of beds (F)
Purposeful function ( maximize ):
Z = 20x
1
+ 30x
2
+ 60x
3
+ 50x
4
+ 35x
5
+ 80x
6
Checks :
1. Fat hungry :
x
1
+ 2x
2
+ 3x
3
+ 4x
4
+ 2x
5
+ 5x
6
≤ 70
Fabric:
4x
3
+ 2x
5
+ 3x
6
≤ 40)
2. Metal :
0.5x
1
+ x
2
+ 2x
4
+ 0.5x
5
+ 1.5x
6
≤ 20
3. Paint
: 0.2x
1
+ 0.5x
2
+ x
3
+ 0.8x
4
+ 0.4x
5
+ 1.2x
6
≤ 15
4. Worker power :
2x
1
+ 3x
2
+ 5x
3
+ 4x
4
+ 3x
5
+ 6x
6
≤ 80
5. Energy :
x
1
+ 2x
2
+ 3x
3
+ 2x
4
+ 1.5x
5
+ 4x
6
≤ 50
6. Negative not happened condition :
x
1
, x
2
, x
3
, x
4
, x
5
, x
6
≥ 0
Support
�
�
�
�
�
�
�
�
�
�
�
�
RHS
�
�
1
2
-11/3
4
-4/3
0
40/3
�
�
0
0
4/3
0
2/3
1
40/3
�
�
0.5
1
-2
2
-0.5
0
5
�
��
0.2
0.5
-0.6
0.8
-0.2
0
5
�
��
2
3
-3
4
-1
0
10
←
�
��
1
2
-7/3
2.5
-7/6
0
50/3
�
-20
-30
100/3
-50
35/3
0
933.33
↑
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Journal:
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page 1246
Free numbers support column to the elements let's be and the most the youngest we
will get
Pivot row :
x
11
(row 5), pivot element: 4.
Now and simplex table to compose we will get
Score doer column and lines place will be replaced .
↑
Pivot column choice module according to best
big negative value selectively we will
get that is only negative the value is -25/6 for yourself we will get
Free numbers support column to the elements let's be and the most the youngest we
will get
Pivot element = 4/3 (row 2, column 3)
Score doer column and lines place will be replaced .
Support
�
�
�
�
�
�
�
�
�
�
�
�
RHS
�
�
-1
-1
0
0
0
-0.5
10
�
�
0
0
1
0
0.5
0.75
10
�
�
-0.5
-0.5
0
0
0.25
0.375
5
�
��
-0.2
-0.1
0
0
0
0
3
�
��
0.5
0.75
0
1
0.125
0.5625
10
�
��
-0.25
0.125
0
0
-0.3125
0.34375
15
Support
�
�
�
�
�
�
�
�
�
�
�
�
RHS
�
�
-1
-1
-2/3
0
-1/3
0
10/3
�
�
0
0
4/3
0
2/3
1
40/3
←
�
�
-0.5
-0.5
-0.5
0
0
0
0
�
��
-0.2
-0.1
0
0
0
0
3
�
��
0.5
0.75
-0.75
1
-0.25
0
2.5
�
��
-0.25
0.125
-11/24
0
-13/24
0
125/12
�
5
7.5
-25/6
0
-5/6
0
1058.33
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American Academic publishers, volume 05, issue 04,2025
Journal:
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page 1247
�
5
7.5
0
0
3.3333
3.125
1100
Z line negative values there is no more , so for This is the optimal solution .
x
3
= 10 x
3
= 10 (Sofa)
x
4
= 10 x
4
= 10
(Shkaf)
x
1
, x
2
, x
5
, x
6
= 0
Maximal foyda : Z=60×10+50×10=600+500=1100 Z = 60 \times 10 + 50 \times 10 = 600
+ 500 = 1100 Z = 60 × 10 + 50 × 10 = 600 + 500 = 1100 ming so' m .
Furniture working in the release Optimal resource allocation : Simplex method using
profit maximize topic within furniture working issuer the company limited from resources
( wood – 80 m³, fabric – 40 m², metal – 25 kg, paint – 15 l, labor power – 90 hours , energy –
50 kWh ) efficient use and profit maximize issue studied . Simplex method using calculated
optimal solution this showed that the company 10 sofas (
x
3
=10 ) and 10 wardrobes per week
( x
4
= ��
) must be produced, the remaining products (chairs, tables , armchairs , beds ) must
be produced This strategy will bring weekly profit up to 1100 thousand soums
x
1
= x
2
= x
5
= x
6
= �
( Z =60×10+50×10=1100 ) . Results resources effective in
distribution linear programming importance and Simplex of the method practical
application confirms this . approach working release processes optimization and economic
efficiency increase for important tool as service does .
Literature:
1. Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press.
2. Hillier, F. S., & Lieberman, G. J. (2010). Introduction to Operations Research. McGraw-
Hill Education.
3. Taha, H. A. (2017). Operations Research: An Introduction. Pearson Education.
4. Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury
Press.
5. Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury
Press.
