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 1190
OPTIMAL PRODUCTION PLAN FOR A CONFECTIONERY FACTORY USING
THE SIMPLEX METHOD
Mamatova Zilolakhan Khabibullokhanovna
Fergana state university associate professor ,
pedagogy sciences according to philosophy Doctor of Philosophy (PhD)
Orchid : 0009-0009-9247-3510
E-mail:
Gafforova Zakhrabanu Ikhtiyorjon kizi
Fergana State University Practical mathematics 3rd year student , group 22-08 student
E-mail:
zaxrobonugofforova@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 .
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
Confectionery optimal factory operation release 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. 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
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 1191
Nemchinov 's " Linear economic models " (1972) book economic in systems linear
programming and analysis methods to use dedicated .
Research methodology
This research confectionery optimal factory operation release 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 - Optimization of the objective function obtained as a result of stage 1 using
the simplex algorithm .
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 . In addition, the system will not allow
all 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 1192
=
-
+
+
+
=
-
+
+
+
=
+
+
+
+
=
+
+
+
+
=
+
+
+
+
+
+
+
+
+
+
+
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
here:
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 .
To do this, if necessary, add the left -hand 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 .
This results in the following system of equations.
=
-
+
+
+
=
+
+
+
+
=
+
+
+
+
=
+
+
+
+
+
+
+
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, the final system of equations for the start of the 1st stage of the Simplex
method 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
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 1193
d
1
x
1
+ d
2
x
2
+ … + d
n
x
n
– W = - W
0
the simplex method , the function W corresponding to z is minimized using the usual
simplex algorithm . The purpose of this minimization is 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 The index of the unknown IV to be extracted is found.
3) 2nd system all equations is changed . In this d
j
and W
0
those of change additional
functions service It turns out : for all columns except r , d
j
=d
j
-d
s
a
rj
/a
rs
, for column r , 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
The system is
rewritten with (2) lost . The result made system canonical to look has If W<0 , the solution is
no .
Stage 2 obtained in Stage 1 The system is optimized using the algorithm.
Below simplex method structural structure scheme shown :
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 1194
A confectionery factory produces 4 different products - Cake (A), Pie (B), Biscuits
(C), Sweet Bread (D) . Production release flour , sugar , butter and worker power with
limited . The enterprise purpose maximum benefit to take .
Given information :
Product
Profit ( mln)
soum )
Flour
requirement (kg)
Sugar
demand (kg)
Butter demand
( kg )
Labor force
(hours)
Cake (A)
8
3
2
2
4
Cake (B)
6
2
3
1
3
Cookies
(C)
5
4
1
2
2
Sweet
bread (D)
7
5
2
3
5
Tenglamaning
standart
ko’rinishdagi yozuvi
r ni tanlash
Tenglamalarn
i o’zgartirish
Sun’iy
o’zgaruvchilar
S ni tanlash
Sun’iy
o’zgaruvchilar
Tenglamalarn
i kanonik
d
i
0
W > 0
Yo’l qo’yilgan
yechimlar
Simpleks
usulning
Yo’q
Yo’q
xа
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 1195
The enterprise's resources are limited as follows:
1.
Flour: not more than 40 kg.
2.
Sugar: not more than 25 kg.
3.
Butter : not more than 20 kg.
4.
Worker Power : not more than 50 hours .
Linear programming model :
Variables :
x
1
-Cake maker release number
x
2
- Cake working release number
x
3
- Cookies working release number
x
4
-Make sweet bread release number
1. Formulation of the issue
max Z=8x
1
+6x
2
+5x
3
+7x
4
Limitations :
3x
1
+ 2x
2
+ 4x
3
+ 5x
4
= 40
2x
1
+ 3x
2
+ x
3
+ 2x
4
= 25
2x
1
+ x
2
+ 2x
3
+ 3x
4
= 20
4x
1
+ 3x
2
+ 2x
3
+ 5x
4
= 50
Elementary Simplex table
Bazis
�
�
�
�
�
�
�
�
Right side
�
�
3
2
4
5
40
�
�
2
3
1
2
25
�
�
2
1
2
3
20
�
�
4
3
2
5
50
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ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
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page 1196
←
↑
Pivot column choice module according to best
big negative value selectively we
will get
x
1
(
most big negative value -8)
.
Free numbers support column to the elements let's be and the most the youngest we
will get
Pivot element =
2
(row 3, column 1)
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 i.e. the only negative the value was -2 for learn we will get
Free numbers support column to the elements let's be and the most the youngest we
will get
Pivot element =
2
(row 2, column 2)
�
-8
-6
-5
-7
0
Simplex table
�
�
�
�
�
�
�
�
Right side
�
�
0
-1
0
2.5
1 0
�
�
0
2
- 1
0
5
�
�
1
0.5
1
1.5
10
�
�
0
1
0
-1
10
�
0
-2
3
5
80
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page 1197
Score doer column and lines place will be replaced .
Pivot column my choice for the most big negative value there is This is not the
optimal solution . that indicates
x
1
= 8.75 x
2
= 2.5 x
3
= 0 x
4
= 0
Z
max
= 85
That is, optimal performance release plan
x
1
-Cake maker release quantity - 8.75 pieces
x
2
- Cake working release quantity -2.5 pieces
x
3
- Cookies working release quantity -0 pieces
x
4
-Make sweet bread release quantity – 0 pieces
Maximum profit -85 million soums
Confectionery factory for Simplex method optimal performance through release plan
General Conclusion
This issue is linear. programming from the methods one was
Simplex method
through confectionery factory working release plan to optimize Factory
cake (
x
1
),
pastry (
x
�
), cookies (
x
�
) and sweet bread (
x
�
) such as products working produces .
release resources limited divided into flour , sugar , butter and worker from the strength
consists of .
Each
product how much benefit to bring and him/her working release for how
much resource requirement
indicated . Purpose
profit maximum to do happened
for
goal function written . Resources limitedness restriction equations through expressed .
Equations additional
variables with strengthened , initial table was formed . Each pivot
Simplex table
�
�
�
�
�
�
�
�
Right side
�
�
0
0
-1.5
-2.5
12.5
�
�
0
1
-0.5
0
2. 5
�
�
1
0
1.25
1.5
8.75
�
�
0
0
0.5
-1
7.5
�
0
0
2
5
85
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 1198
element in step selected , table transformation done . Maximum benefit to take for which
from products how much working release need clearly was given .
Simplex method from resources the most effective use and maximum profit to take
the optimal strategy for to find help This method not only confectionery factory , maybe
logistics , production release and business planning such as also used in other fields possible .
In real business such optimization methods expenses reduce and working release efficiency
to increase service does .
References :
1. L. Kantorovich - " Mathematician programming and economic analysis " (1959).
Production release optimal plan in processes to compose and resources distribution
methods statement done .
2. G. Dantzig - " Linear programming and his/her applications " (1963). Simplex method
and make it real release to the conditions application issues illuminated .
3. R. Dorfman, P. Samuelson, R. Solow - " Linear programming and economic analysis "
(1958). Optimal planning , constraints and goal function based on decision acceptance to
do discussion made .
4. IG Bashmakov - "Optimal working release systems " (2005). Modern working release
processes to optimize related theoretical and practical approaches showing given .
5. GN Nemchinov - " Linear " economic models " (1972). Economic in systems linear
programming and analysis methods to use dedicated .
