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APPLICATION OF MATHEMATICAL MODELS AND OPTIMIZATION
APPROACHES IN TOURISM FIRMS ISSUE KOMMIVOYAJYOR
Zilolaxon Xabibulloxonovna Mamatova
Fergana state university , associate professor,
pedagogical sciences doctor of philosophy (PhD)
E-mail:
mamatova.zilolakhon@gmail.com
ORCID ID
Diyora Jamoliddinova Umidjonovna
Fergana state university student 22.11-group
E-mail:
Annotatsiya:
The issue of Kommivoyajer (Travel sales I am the problem — TSP) is one of
the issues expansions for eigenvalues of classic discrete optimization, if the minimum
distance to all the cities with the cost or one-time visit, return to the starting point requires.
The tsp's mathematical formulalashuvi in this article, the main methods or specific
(bruteforce, dynamic programming, tarmoqlash-limiting) and approximate (genetic algorithm,
near methods) approaches will be covered. Also, the practical issue is to be applied and the
complexity is analyzed.TSP his full kombinatorika features and high-level complexity is
popular. Kommivoyajer expansions for the eigenvalues of discrete optimization problems is
one of the classic issues, if the minimum distance to all the cities with the cost or one-time
visit, return to the starting point requires. The tsp's mathematical formulalashuvi in this article,
the main methods or specific (bruteforce, dynamic programming, tarmoqlash-limiting) and
approximate (genetic algorithm, near methods) approaches will be covered. Also, the
complexity of the issue is to be practical and applied analysis this problem is np-complete
problem, that is, find its exact solution, it becomes very complicated as the number of issues
have increased. The computer algorithm is developed to find the solution through the
different, in particular, network search algorithm of genetic algorithm and simulation
annealing methods been.Kommivoyajyor the issue of in practice in many areas, for example,
logistics, transportation, robotics, and is used in the management of different resources. It
also used as a means to increase productivity, optimize and theoretical issues is very
important.
Annotation:
I am the travel your sales problem (TSP) is one of the most well-known and
studied problem in computer science and mathematical optimization. The objective of the
problem is to find the shortest possible travel router that allows a specified set of your sales to
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the city that I visit, your visit each city only once, and return to its ultimate point your lovely
starti. Is famous for its high combinatorial complexity of nature and a full TSP. It is classified
as an np-problem, completed me that technique becomes increasingly difficult as the number
grows of the city that you find an exact solution. Various algorithms have been using
computers to find solutions developer, including network search algorithms, genetic
algorithms, simulated annealing and the metho. I applied in many practical problem is the
travel of your sales area, such as logistics, transportation, robotics, and various resource
management systems. It is also of great theoretical importance for use as a tool for improving
the optimization and premium.
Annotasiya
:kommivoyajera far zada president (TSP) — I v matematicheskoy zada president
has features to track an samix izvestnix izuchennix when I optimizasii computer science.
Zada president who tselsiy — nayti kratchayshiy route, posetit pozvolyayutshiy
kommivoyajeru zadannie much more productive, posetiv kajdiy odin tolko raz gorod, v I v to
a section isxodnuyu tochka konsov kontserni. TSP izvestna I svoystva I polnostyu svoimi
kombinator visokoy slojnostyu. Zada president will far yavlyaetsya NP-polnoy, chto
oznachaet, chto vsyo po naxojdenie tochno cargo resheniya stanovitsya slojnee uvelicheniya
chislo gorodov is merry. Razlichnie kompyuterov razrabotani pomotshyu s naxojdeniya
resheniy beginning of the algorithm, the algorithm vklyuchaya poisk v, a set of algorithm to
simulirovannogo otj I geneticheskie method. Zada president razlichnix oblastyax far
kommivoyajera primenyaetsya v, takix kak logistics, transport, robotics I sistem upravleniya
of resources to me. Teoreticheskoe I znachenie imeet takje kak instrument bolshoy mother
optimizasii povisheniya effektivnosti the beginning.
Keywords:
Kommivoyajer issue, TSP, optimization, kombinatorika, tarmoqlash-limiting,
dynamic programming, heuristik methods, genetic algorithm, either‘lni to reduce, the
calculation complexity, operations research. resources management, simulation made
annealing, genetic algorithm, network search algorithm, the solution you find.
Keywords:
travel I am your sales problem (TSP), optimization, combinatorics, NP-
completed problem, shortest path algorithms, logistics, transportation, robotics, resource
management, simulated annealing, genetic algorithms, network security, algorithms, please
find solution.
Klyuchevie slovo:
kommivoyajera far zada president (TSP), optimizasiya, kombinatorika,
NP-zada president polnaya far, put kratchayshiy, algorithm, logistics, transportation, robotics,
resources upravlenii me, I simulirovanniy otj, geneticheskie algorithm, the algorithm in the
set of poisk v, naxojdenie resheniya.
To enter.
The issue of Kommivoyajer (Travel sales I am the problem — TSP) is one of the
most popular and the most studied optimization problems kombinatorika. The essence of the
matter that iboratki, traders visit multiple cities exactly once, and return to the starting city
along the way to a minimum distance or the total cost is required. In practice this issue is
logistics, marshrutlash, chip design, used in many areas such as genetic research.TSP is np-
complex belongs to the class if the exact solution requires very large computing resources to
find the largest sized cases. For this reason, various mathematical models in solving this issue
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visible and near the algorithm developed. In this article, the tsp's theoretical basis is a detailed
analysis of solution methods and their advantages and disadvantages.
Withdrawal methods:
Bruteforce (full review),dynamic programming (Held-Karp
algorithm),the greedy algorithm (Greedy Algorithm),genetic algorithm, simulyatsiyalangan
tavlanish (simulated annealing), linear programming and be chess (Branch and bound).
Purpose:
the purpose of kommivoyajyor issues – given to the city to visit the field-
provides the shortest and most effective way to find mivoyajyorning. Thus, each city must be
visited once and only at the end of the field-mivoyajyor should come back to the starting city.
The main purpose of the issue, a minimum distance to beijing to visit the city, and thus in
reducing the time and costs is. TSPni solving, the goal through the optimization in the real
world, transport, logistics and resource management to increase efficiency in areas such as it
is. Also, the developed optimization algorithm for solving the issues of tspni and many other
methods can also be applied.
Given a short time to visit each city only once with n units(path, cost) turned out for
find sik lee. Thus, with a lot of cycle number
one. This issue with the issue of finding
a long-term cycle gamilton minimal bound. Problem solving Kommivoyajyor "network and
borders"can you apply the method. This method is a cycle and you are bound sirtmoq graf,
and drafting is conducted using tables.
The standard option.
Will put bring the concept of the table. To do this, the table is coming
from the line of earlier, that is, each row of the table of elements of the same row is removed
the small isolated respectively. After that we also do other than follow the column of the table
were the same, and come to the table columns. All are listed on the table are referred to as
rows and columns are listed. The smallest the sum of h elements were defined by the table of
rows and columns, bring it charts the coefficient is called. For example the following
schedule around the world to travel by air transport will review the historical city.
Will put belgilashlar
1. Paris-China-8200
China-Paris-8100
2. Paris-Agri-6550
China-Agri-3900
3. Paris-Egypt-3200
China-Egypt-7500
4. Paris-Berlin-1050
China-Berlin-7300
5. Paris-Rome-1100
China-Rome-9600
6. Paris-Istanbul-2250
China-Istanbul-7800
7. Agri-Paris-6650
Egypt-Paris-3000
8. Agri-China-3700
Egypt-China-7600
9. Agri-Egypt-5000
Egypt-Agri-5100
10. Agri-Berlin-6200
Egypt-Berlin-2900
11. Agri-Rome-6900
Egypt, Rome-2150
12. Agri-Istanbul-6000
Egypt-Istanbul-1300
13. Berlin-Paris-1000
Rome-Paris-1200
14. Berlin-China-7100
Rome-China-9500
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15. Berlin-Agri-6000
Roman-Agri-6800
16. Berlin-Egypt-2800
Roman-Egyptian-2100 From
17. The Rome-Berlin-1180
Rome-Berlin-1080
18. Berlin-Istanbul-1600
Rome-Istanbul-1370
19. Istanbul-Paris-2200 Anniversary
20. Istanbul-China-7900
21. Istanbul-Agri-6100
22. Istanbul-Egypt-1250
23. Istanbul-Berlin-1700
24. Istanbul-Rome-1270
B/S
1
2
3
4
5
6
7
on the line of the
smallest
1
8200
6550
3200
1050
1100
2250
1050
2
8100
3900
7500
7300
9600
7800
3900
3
6650
3700
5000
6200
6900
6000
3700
4
3000
7600
5100
2900
2150
1300
1300
5
1000
7100
6000
2800
1180
1600
1000
6
1200
9500
6800
2100
1080
1370
1080
7
2200
anniversary
7900
6100
1250
1700
1270
1250
1 table.
1-row table to bring his will write out the smallest element to the right side of the
corresponding row and the row of the element following it isolated from 2-you will be able to
schedule.
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2-table
The yield of 2-the column of the table in order to bring themselves under the table to
fit the column and the smallest element is isolated from the column element've written, as a
result, the following 3-table is formed.
h = come. row + come. column
1-picture.
3-the table is listed, there is at least one element in every row and column zero. The
following charts coefficient h equal to the number of people you bring
h=1050+3900+3700+1300+1000+1080+1250+20+6*0=13300
c
32
= 0
(7400)
=> c
23
= ∞
B/S
1
2
3
4
5
6
7
1
7150
5500
2150
0
50
1100
2
4200
0
3600
3400
5700
60
3
860
0
1300
2500
3200
2300
4
1700
6300
3800
1600
850
0
5
0
6100
5000
1800
180
600
6
120
8420
5720
1020
0
290
7
950
6650
4850
0
450
20
on the column
of the smallest
0
0
0
0
0
20
0
B/s
1
2
3
4
5
6
7
1
7150
5500
2150
0
(30)
30
1100
2
4200
0
(3900)
3600
3400
5680
60
3
860
0
(7400)
1300
2500
3180
2300
4
1700
6300
3800
1600
per 830
0
(1120)
5
0
(280)
6100
5000
1800
160
600
6
120
8420
5720
1020
0
(1of 20)
290
7
950
6650
4850
0
(1020)
450
0
(30)
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h
'
= h + 7400 = 13300 + 7400 = 20700
In general, the network consists of two important stages of the method of limits and is from:
1) tarmoqlash;
2) determine the lower limit.
Both stages is conducted in parallel for solving the issue. For the implementation of this
stage, the following sequence must complete the work. A) primary come to the table; B) to
determine h koeffitsenti to come; C) to determine the level of zero elements in the table given;
D) on the basis of this level tarmoqlash implementation; E) determine the lower limits of
tarmoqlanish the results of the cycle components; f) reduce the size of the table to one; and g)
the full cycle of harvest remain to keep from; H) this process (2x2) continue until the table is
formed; I) determine the final results corresponding to a cycle network; (J) all limits (ho ba)
to compare; K) is a necessity the results can restore the table to the minimum boundary
tarmoqlash fit to continue.
During the application of this method, all the calculations are conducted using the table
given, and its results will go far in showing graf configured separately. Perfect at the end of
this process(lowest cost) cycle is determined.
If doiracha graf consists of combined mutual, each one of them will determine a certain
xossali collection cycle. This doiracha written by the border-while the number of the same
circle indicates that belongs to a cycle corresponding to the lower limit of the costs. Part 1 of
the primary graf-picture is in view. Thus, the first cycle of primary package which includes
all doiracha define the number of cycle costs go on a voluntary means that h is small. Seen in
the example above, h=13300 it was, therefore, a cycle that is small costs from 13300 it's not.
Located on the zero line, which is great most of the level
i
and superior
j
are being
found,
( , )
i j
on there. Mabodo, if nollar is a multiple senior level, one of them optional is
selected. Thus, the transition from city I to city j doiracha in the right side and it means a
collection of cycles covering all
( , )
i j
is set, doiracha in the left side, while on the contrary,
the transition from city j to city I did not include the route that it means the collection of
( , )
i j
is set.
The most senior levels of zero elements 7400
c
32
= 0
7400
, therefore, tarmoqlanish
grafi1-rasmko'rinishidabo'ladi.Bring the lowest costs Chapdoirachayoniga coefficient
h =
13300
20700 formed by adding zero to the number of the biggest 7400 level is recorded.
(h
1
'
)
in the right side to determine the lower limit of costs doiracha compatible 3-remove the 2nd
row and the 3rd column of the table(off)will be sent(therefore, the size of the table is reduced
to one). Thus,it should be noted that this particular, of course, the city preserved the order of
the numbers(and written)qolishishart,aksholdachalkashliklarkelib out. After that,all he
prohibited the harvest of the full cycle,the issue
i → j → I(I → j
sign-shahardanj-means to the
city) is the loss of to do this
c
ji
element
∞
change icon will be recorded in two,
c
23
= ∞
).
Again he made the statements we can continue our work.
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B/S
1
3
4
5
6
7
on the line of the
smallest
1
5500
2150
0
30
1100
0
2
4200
3600
3400
5680
3900
3400
4
1700
3800
1600
per 830 0
0
5
0
5000
1800
160
600
0
6
120
5720
1020
0
290
0
7
950
4850
0
450
0
0
B/S
1
3
4
5
6
7
1
5500
2150
0
30
1100
2
4200
3600
3400
5680
3900
4
1700
3800
1600
830 per
0
5
0
5000
1800
160
600
6
120
5720
1020
0
290
7
950
4850
0
to 450
0
on the column
most small of
0
3800
0
0
0
h
1
= h + 3400 + 3800 = 20500
B/s
1
3
4
5
6
7
1
1700
2150
0
30
1100
2
800
200
0
(200)
2280
500
4
1700
0
(1050)
1600
per 830
0
(290)
5
0
(280)
1200
1800
160
600
6
120
1920
1020
0
(120)
290
7
950
1050
0
(200)
450
0
(30)
Have a delete on the 3 line 4 on the column.
h
1
1
= h
1
+ 1050 = 20500 + 1050 = 21150
C
43
= 0
1050
= C
34
=
C
24
=
B/S
1
4
5
6
7
on the line
of
the
smallest
1
2150
0
30
1100
0
2
800
0
2280
500
0
5
0
1800
160
600
0
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6
120
1020
0
290
0
7
950
0
450
0
0
kich column
0
0
0
0
290
h
2
= h
1
+ 290 = 20790
B/s
1
4
5
6
7
1
2150
0
(30)
30
810
2
800
0
(800)
2280
210
5
0
(280)
1800
160
310
6
120
1020
0
(0)
0
(210)
7
950
0
(1020)
450
0
(30)
On line 7 column 4 in a row off.
C
74
= 0
1020
= C
47
=
h
2
'
= h
2
+ 1020 = 20790 + 1020 = 21810
B/s
1
5
6
7
on the line
of
the
smallest
1
0
30
810
0
2
800
0
2280
210
0
5
0
160
310
0
6
120
0
0
0
column kich
0
0
30
0
h
3
= h
2
+ 30 = 20790 + 30 = 20870
B/s
1
5
6
7
1
0
(0)
0
(130)
810
2
800
0
(800)
2250
5
0
(250)
130
310
6
120
0
(0)
0
(310)
On row 2 and column 5 in a row off.
C
25
= 0
800
= C
52
=
h
3
'
= h
3
+ 800 = 20820 + 800 = 21620
B/s
1
6
7
of the smallest
on line
1
0
810
0
5
0
130
0
6
120
0
0
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page 565
column kich
0
0
0
C
57
=
h
4
= h
3
+ 0 = 20820
B/s
1
6
7
1
0
(940)
810
5
0
(250)
130
6
120
0
(930)
Toe the line
‘1 column 6 in a row on the citizens and off.
h
4
'
= h
4
+ 940 = 20820 + 940 = 21760
C
16
= 0
(940)
= C
61
=
B/s
1
7
of the smallest on
line
5
0
0
6
0
0
column kich
0
0
C
61
=
h
5
= h
4
+ 0 = 20820
h
5
'
= h
5
+ 0 = 20820 + 0 = 20820
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page 566
The shortest way (optimal route):
1 6 7
4 3 2 5 1
® ® ® ® ® ® ®
Minimum distance: 20820 unit.
Conclusion.
Kommivoyajyor issues of mathematical modeling of many real-life
problems and used for the optimization of a classic issue. Due to the complexity of much of
the calculation of the optimal solution found, in many cases the approximate algorithm or
approach heuristik apply.
Used literature:
1. Papadimitriou, C. H., & Steiglitz, K. (1998). Combinatorial optimization: Algorithms and
complexity. Publication Is Not The Dove.
2. Applegate, D., Bixby, R., Chvátal, V., & Cookie, W. (2006). Travel I Am Your Sales
Problem: A Computational Study. Princeton University Press.
3. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1985). The
problem I am your sales travel: a guide tour of combinatorial optimization. Wiley.
4. From Guti, G., & Punnen, A. P. (2002). I am the travel of your sales problem and its
Variations. Spring.
5. Reinelt, G. (1994). I am your sales travel: Computational solutions for TSP Applications.
Spring.
