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volume 4, issue 3, 2025
745
PLANT TO OBTAIN MAXIMUM BENEFITS FROM PRODUCTION
Rozaliyev Sherzodjon Avazjonovich
Fergana state university information technologies department manager ,
pedagogy sciences according to philosophy Doctor of Philosophy (PhD)
E-mail:
ORCID ID
Abduhalilova Owner Abdurasul daughter
Fergana State University Practical mathematics 3rd year student ,
group 22-08 student
E-
mail:
sohibaabduhalilova159@gmail.com
Abstract:
Simplex method – linear programming issues effective solution for used strong is an
algorithm . Simplex schedule using iterative calculations done increased , optimal solution This
is found in method resources distribution , production release planning and logistics in the fields
wide is used . This in my article confectionery factory from resources effective use and
maximum benefit to take issue linear programming and simplex method using analysis as I'm
leaving .
Key words:
Simplex method , linear programming , optimal plan , goal function , constraint
conditions , pivot element , simplex table , production release optimization , resources
distribution , maximum profit , mathematics modeling , linear equations , organization
efficiency , economic optimization , product working production , confectionery factory , costs
reduction , mathematics programming , executable iterations , business planning .
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Modern economic under the circumstances enterprise and factories main purpose – limited
resources under the circumstances maximum to income is to achieve . release process effective
organization to grow and profit optimization today's of the day current from issues is one .
Especially , one how many product working removable in factories resources right
distribution , production release size designation and the most high economic benefit to take for
clear to the calculations rely on important importance profession will reach .
Such issues in solution mathematician modeling methods , in particular , linear programming and
his/her effective solution method was simplex method wide is used . Simplex method through
at the factory working release possible was
products number of them expenses and causing
benefit in consideration taken and optimally worked release plan to compose possible will be .
This in the article simplex method based on product working release plan to compose and
maximum benefit to take opportunities study
in sight Research
during working release
resources ( raw materials , labor) power , time and hk ) restrictions under how from products
how much working release need is determined .
Literature analysis
The firm's product working in the release maximum benefit to take plan according to literature
analysis working release processes optimization , resources effective distribution and profit
maximum to the level to deliver according to various methods to determine help gives . L.
Kantorovich's " Mathematical programming and economic analysis " (1959 ) working release
optimal plan in processes to compose and resources distribution methods statement G. Dantzig "
Linear programming and his/her in the book " Applications " (1963) simplex method and make it
real release to the conditions application
issues covered . R. Dorfman, P. Samuelson and R.
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volume 4, issue 3, 2025
746
Solow " Linear programming and economic analysis " (1958 ) optimal planning , constraints and
goal function based on decision acceptance to do discussion IG Bashmakov's "Optimal
production " release systems " (2005 ) modern working release processes to optimize related
theoretical and practical approaches showing
Also , GN Nemchinov 's " Linear economic
models " (1972) book economic in systems linear programming and analysis methods to be used
dedicated .
Research methodology
This research firm product working in the release maximum benefit to take plan to compose
according to linear programming methods to apply Research
methodology empirical and
theoretical analysis own
inside Research
during literature analysis optimal performance
through release plan formation according to there is scientific sources is studied . Various
mathematician modeling methods , including simplex
method , graphic method and dual
method using working release processes optimization opportunities analysis Comparative
analysis through various economic models compared and their confectionery products working
release to the process compatibility is determined . In this working release resources limited ,
product types benefit level and demand conditions into account is obtained . Experimental
analysis and theoretical basically of the optimal plan formulated to practice implementation to be
completed to study aimed at to be , to work release size increase and expenses reduce according
to recommendations working Qualitative
analysis methodological aspects , work release
process conditions and the results quality in terms of to evaluate is based on . Research
methodology working release plan thorough planning , resources effective distribution and
maximum benefit to take for scientific approaches to determine These methods are aimed at
using working release process further improvement and economic efficiency increase possible .
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 .
=
-
+
+
+
=
-
+
+
+
=
+
+
+
+
=
+
+
+
+
=
+
+
+
+
+
+
+
+
+
+
+
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
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volume 4, issue 3, 2025
747
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
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 , then W 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 .
One factory 4 types product making cookies (
�
1
)
, pasta
(�
2
)
, lag'mon ( )
�
3
)
and bread
(
�
4
) This
products working release 4 types for resource flour (R1), sugar (R2 ), salt (R3)
and eggs (R4) are required . Each product for expendable resource amount table in the form of
given . The factory purpose – products working from issuing removable general profit maximum
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volume 4, issue 3, 2025
748
to do
�
1
�
2
�
3
�
4
max
Z
�
1
�
2
�
3
�
4
�
1
�
2
�
3
�
4
�
1
�
2
�
3
�
4
�
1
�
2
�
3
�
4
�
1
�
2
�
3
�
4
�
1
�
2
�
3
�
4
�
5
�
6
�
7
�
8
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volume 4, issue 3, 2025
749
�
1
�
5
�
3
�
4
�
2
�
6
�
7
�
8
max
Z
=700/3.
�
1
�
5
�
3
�
6
�
2
�
4
�
7
�
8
max
Z
and according to the calculation book it is equal to 1700/7.
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volume 4, issue 3, 2025
750
�
7
�
5
�
3
�
6
�
2
�
4
�
1
�
8
1
x
=2
2
x
=30
3
x
=0
4
x
=6
max
Z
=244
Conclusion:
So it can be seen that we should have a production volume of 2 kg of cookies, 30
kg of pasta, 0 kg of lagmanish, and 6 kg of bread. Then we can get maximum profit from the
production of the product.
Answer: the maximum profit is 244.
References
1.Hamdi A. Taha – " Operation Research : Introduction "
2. Frederick S. Hiller and Gerald J. Lieberman – " Operation to research " introduction "
3. TM Sobirov , MA Musaeva – " Linear programming "basics "
4. Mokhtar S. Bazoaa – " Linear " programming and network streams "
5. Jorge Nosedal and Stephen Wright – " Digital optimization "
