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FINDING THE ОPTIMАL SОLUTIОN ОF А TOURISM CОMPАNY'S TОTАL CОSTS FОR TRАVEL BETWEEN CОUNTRIES WITH THE HELP ОF THE TRАVELER'S PRОBLEM (EХАMPLE ОF "FRESH TOUR" CОMPАNY) Mаmаtоvа Zilоlахоn Хаbibullохоnоvnа Senior lecturer of Fergana State University, Doctor of Philosophy in Pedagogical Sciences (PhD) E-mail: [email protected]
ORCID ID 0000-0002-5717-2968 Rаsuljоnоvа Mоhlаrоyim А'zаmjоn qizi Student of Fergana State University, Department of Applied Mathematics, Group 21.07 E-mail: [email protected] https://doi.org/10.5281/zenodo.14259035
ABSTRACT
ARTICLE INFO
Received: 25th November 2024 Accepted: 29th November 2024 Online: 30th November 2024
KEYWORDS Tmveling sаlesmаn pmblem, met^d оf netwоrks аnd limits, оptimаl mute, inmmplete mutes, inmmplete cycle, the given tаble.
In this аrticle, the formu^ оf the tmvel аgent pmblem оf tоtаl e\penses for trips between ^untries оf the "Fresh Wur" Wurism firm, met^ds оf sоlving it, аnd its prаcticаl аpplicаtiоn аre mnsidered. This аrticle pmvides а detаiled аnаlysis оf the theоreticаl fоundаtiоns аnd prаcticаl sоlutiоns W this pmblem, аs well аs pоints оut perspectives for future reseаrch.
Intrоductiоn
The traveling sаlesmаn problem is а ckssic problem thаt hоlds аn impоrtаnt ptace in the fields оf graph theоry аnd оptimizаtiоn. In this problem, а merchаnt (traveler) must visit а set оf cities, entering eаch city оnce аnd returning to the stаrting city аt the end. The gоаl is to keep distаnces between cities to а minimum. Find the cycle оf visiting n given cities in the shоrtest time (distаnce, rost) by visiting eаch оf them оnly оnce. In this cаse, the number оf cycles is аt mоst (n — 1)! will be This problem is rented to the problem оf finding а Hаmiltоn cycle оf minimum length. The "Netwоrks аnd Bоundаries" methоd cаn be used tо sоlve the trаveling sаlesmаn prоblem. This methоd is cаrried оut with the help оf а ^nnected grаph withоut cycles аnd edges, аnd with the help оf tаbles.
Literаture аnаlysis
The traveling sаlesmаn prоblem is studied аs а clаssicаl prоblem оf mаthemаticаl оptimizаtiоn, аnd vаriоus аlgоrithms for its sоlutiоn аre prоpоsed. "The Traveling Sаlesmаn Prоblem: А Cоmputаtiоnаl Study" by D.L. Аpplegаte, R.E. Biхby, V. Chvаtаl, W.J. Cооk. This wоrk is аn impоrtаnt sоurce fоr studying the theоreticаl аnd prаcticаl methоds оf sоlving the trаveler's prоblem. "Trаnspоrtnye zаdаchi i puti iх resheniа" — N.P. Buslento, V.V. Krinitsyn. Аlgоrithmic аpprоаches tо sоlving trаnspоrt problems аnd methоds оf оptimizing the
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commuter problem are considered in this work. M. Tokhtasinov. In the process research book, solutions to the traveling salesman problem and some issues are considered.
Reseаrch methоdоlоgy
The main goal of the study is to determine the optimal route for tourism companies to reduce the costs of international travel. The research was carried out in cooperation with the tourist company "Fresh Tour". First, information such as distance between countries, types of transport and their costs are collected. Then, a cost matrix is created for each route. In order to minimize these costs, the classical commuter problem model is chosen. Traditional methods such as Genetic Algorithm or Travelling salesman problem are used to solve the problem. The resulting solutions are analyzed by evaluating total costs and efficiency for tourism firms. At the end of the study, conclusions and recommendations are given to reduce costs for tourism firms by recommending economically efficient routes.
Аnаlyzes аnd results
We introduce the concept of bringing a table. To do this, the rows of the table are first brought, that is, the smallest of the row is subtracted from the elements of each row of the table. After that, the same operation is performed in relation to the table columns, and the table columns are listed. A table in which all rows and columns are listed is called a listed table. The sum of the smallest elements of the rows and columns of the table is denoted by h, and it is called the multiplication coefficient of the table. Consider the following problem.
Problem. According to the information of the tourist company "Fresh Tour", finding the optimal solution for a one-day trip to the countries of Uzbekistan, Malaysia, Thailand, Singapore and Bali using the Travelling Salesman problem. (Data as of October 7, US dollars ($ ) entered on a monetary unit basis.)
Uzbekistаn Malaysia Thailand Singapore Bali
Uzbekistan œ 368 344 404 569
Mаlаysiа 471 œ 49 62 59
Thailand 467 43 œ 89 80
Singаpоre 567 57 125 œ 94
Bali 924 49 85 93 œ
Tаble 1.
Sоlutiоn: To quote the rows of Table 1, we write the smallest element of the corresponding row to its right.
The
Uzbekistan Malaysia Thailand Singapore Bali smallest elements in a row
Uzbekistan œ 368 344 404 569 344
Malaysia 471 œ 49 62 59 49
Thailand 467 43 œ 89 80 43
Singapore 567 57 125 œ 94 57
Bali 924 49 85 93 œ 49
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By subtracting the smallest elements in each row from the row elements, we get the following
table 2
Uzbekistan Malaysia Thailand Singapore Bali
Uzbekistan œ 24 0 60 225
Malaysia 422 œ 0 13 10
Thailand 424 0 œ 46 37
Singapore 510 0 68 œ 37
Table 2.
In order to list the columns of the resulting table 2, the smallest element of the corresponding
column is written under the table.
Uzbekistan Malaysia Thailand Singapore Bali
Uzbekistan œ 24 0 60 225
Malaysia 422 œ 0 13 10
Thailand 424 0 œ 46 37
Singapore 510 0 68 œ 37
Bali 875 0 36 44 œ
The smallest elements in a column 422 0 0 13 10
By subtracting the smallest elements in each column from the column elements, we get Table
3 below.
Uzbekistan Malaysia Thailand Singapore Bali
Uzbekistan œ 24 0 47 215
Malaysia 0 œ 0 0 0
Thailand 2 0 œ 33 27
Singapore 88 0 68 œ 27
Bali 453 0 36 31 œ
Table 3.
Table 3 has at least one zero element in each row and column. The coefficient h of the table in question is equal to the following number:
h = 344 + 49 + 43 + 57 + 49 + 422 + 0 + 0 + 13 + 10 = 987 In general, the method of networks and boundaries consists of two important steps: 1) network; 2) determination of lower limits.
During the solution of the problem, both stages are carried out in parallel. To carry out these steps, you need to do the following in sequence:
a) bring the initial schedule;
b) determination of the coefficient h;
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c) determine the level of zero elements of the given table;
d) perform branching based on these levels;
e) determining the lower limits of the cycles that form branching results;
f) reduce table size to one;
g) avoiding the formation of incomplete cycles;
h) continue this process until a (2x2) table is formed;
i) determine the cycle corresponding to the last network result; j) compare all thresholds (grades);
k) if necessary, continue branching by restoring the schedule corresponding to the minimum marginal result.
In this method, all calculations are performed using a given table, and the results are displayed on a separate graph. At the end of this process, the perfect (lowest cost) cycle is determined.
The graph consists of interconnected circles, each of which defines a set of cycles with a certain property. The boundary numbers written next to these circles indicate the lower limit of the costs corresponding to the cycles belonging to this circle. The initial part of the graph looks like Figure 1. Here, the first initial circle defines the set containing all cycles and indicates that the cost of an arbitrary cycle cannot be less than the number h. In the example above, h = 987, which means that there is no cycle with a cost less than 987.The degrees of the zero elements of the given table 3 are defined for branching the set of all cycles. For example, to find the level of cUT = 0 in the Uzbekistan row, Thailand column with zero element in Table 3, the smallest element in the Uzbekistan row is 24, and the Thailand column is the smallest element in the Malaysia row. The number 0 is added and the resulting number 24 is written as the power of this zero. Similarly, to find the degree of cMU = 0, the smallest element in the row of Malaysia is added to 0, the smallest element in the column of Uzbekistan is 2, and the resulting number 2 is written as the degree of cMU = 0 is placed. Using this method, the levels of all zero elements of Table 3 are determined.
Uzbekistаn Malaysia Thailand Singapore Bali
Uzbekistаn TO 24 0(24) 47 215
Mаlаysiа 0(2) TO 0(0) 0(31) 0(27)
Thаilаnd 2 0(2) TO 33 27
Singаpоre 88 0(27) 68 TO 27
Bali 453 o(31) 36 31 TO
The row i and column j with the largest zero are found and branched according to (i,j). Therefore, if there are several high-order zeros, an arbitrary one of them is selected. Here, the circle on the right represents the set of all cycles that include the transition from city i to city j and is denoted by (i,j), while the circle on the left, on the contrary, is the cycle from city i to city j denotes the set of routes that do not contain a tooth, and it is denoted by (j, i).
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Figure 1.
The element with the highest rank of 31 is cM5 = 0 and cBM = 0. We can choose any one of these elements. For example, cBM = 0, so the branching graph will look like Figure 1. Next to the left circle, the number 1018 is written by adding the highest level of zero to h = 987 (h2). The Bali row and Malaysia column of Table 3 are dropped (thus reducing the size of the table by one) to determine the lower bound of costs corresponding to the right-hand box. In this case, it should be noted that the serial numbers of the cities must be preserved (written), otherwise confusion will arise. After that, all incomplete cycles are forbidden to form, for example i ^ j ^ i (the symbol i ^ j means the transition from i — city to j — city) is lost, for which Cy£ the element is replaced by the ro symbol.
Uzbekistan Thailand Singapore Bali
Uzbekistan œ 0 47 215
Malaysia 0 0 0 œ
Thailand 2 œ 33 27
Singapore 88 68 œ 27
Table 4.
After that, a new table is created, and its conversion coefficient is added to h, which is the previous conversion coefficient, and written (hx).
Uzbekistan Thailand Singapore Bali The smallest elements in a row
Uzbekistan œ 0 47 215 0
Malaysia 0 0 0 œ 0
Thailand 2 œ 33 27 2
Singapore 88 68 œ 27 27
Uzbekistan Thailand Singapore Bali
Uzbekistan œ 0 47 215
Malaysia 0 0 0 œ
Thailand 0 œ 31 25
Singapore 61 41 œ 0
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Uzbekistan Thailand Singapore Bali
Uzbekistan TO 0(47) 47 215
Malaysia 0(°) 0(°) 0(31) TO
Thailand 0(31) TO 31 25
Singapore 61 41 TO 0(66)
Table 5.
Figure 2.
As can be seen from the above tables, the multiplication factor is 27 + 2 = 29. From this, hx = 987 + 29 = 1016. Table 5 is the given table.
The branching point is selected to the right (according to the rule of walking to the right) to determine the branching pair (fc, Z), the degrees of the zeros of the last table 5 are counted, and the pair (fc, Z) is separated in memory of the largest degree of them, and the branching is performed is increased (Fig. 2). In this case, the value of the border (h4) corresponding to the left circle with the symbol ((Z, fc)) is determined by adding the number with the greatest degree of zero to hx.
To find the limit (h3) corresponding to the right circle with the sign (fc, Z), fc-row and Z-column are removed from the last table (deleted) and incomplete cycles are generated using the sign ro is prohibited. After that, the multiplication factor of the resulting table is added to hx and written next to the right circle.
(fc, Z) taken as a pair (Singapore, Bali). Then we will have a suitable graph (Fig. 2). In this case, the lower limit of the left circle is equal to h4 = 1016 + 66 = 1082.
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Figure 3.
Now we had to drop the Singapore row and Bali column of Table 5 and also prohibit the transition Bali ^ Singapore. But in the previous table, there is no Bali ^ Singapore line because we dropped the Bali line. Each line must have one pass. Therefore, we will prohibit the passage from Malaysia to Singapore (via the ro symbol). After these changes, table 6 is created.
Uzbekistan Thаilаnd Singаpоre
Uzbekistan œ 0 47
Mаlаysiа 0 0 œ
Thаilаnd 0 œ 31
Tаble 6.
As can be seen from the table, the coefficient of this table is calculated as 31 and =
1016 + 31 = 1047.
After that, it is necessary to define a new pair to branch the circle on the right. As before, the degrees of zero elements in the table are calculated, which are listed in Table 7. Zero elements with the largest degree are found in (Uzbekistan, Thailand) and (Thailand, Singapore) pairs, and we choose one of them. For example, the pair (Uzbekistan, Thailand) should be removed. Then we get the graph in Figure 4. In this case, the number 16, which is the degree of cyT = 0, is added to 1047, and the number 1063 is written next to the corresponding left circle of the graph in Figure 4.
Uzbekistаn Thаilаnd Singаpоre
Uzbekistan œ 0(16) 16
Mаlаysiа 0(0) 0(°) œ
Thаilаnd 0(0) œ 0(16)
Tаble 7.
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Figure 4.
The Uzbekistan row of the last 7th table, and the Thailand column are discarded and we prohibit the transition to Thailand ^ Uzbekistan (via the ro sign). Table 8 is created from these changes.
This table has a citation factor of 0. Therefore, the border corresponding to the right circle in Figure 4 has not changed (1047).
Uzbekistan Singapore
Malaysia 0 TO
Thailand TO 0
Table 8.
Thus, we got the resulting graph (Fig. 4). Using the sequence of circles on the right side of this graph and the last (2x2)- size 8 table, the cycle Uzbekistan^Thailand^Singapore^Bali^Malaysia^Uzbekistan with a cost of 1047 is determined.
Summary
In this article, the issue of minimizing the total costs for trips between countries of the tourism firm "Fresh Tour" was considered. In this case, using the traveling salesman problem (TSP) method, the firm can find the optimal route covering different destinations. The article analyzed the mathematical model of this problem, algorithms and optimization methods. As a result, the tourism firm was able to effectively manage travel costs and maximize the use of resources.
References:
1. M. Tokhtasinov. Process Research. Textbook. 2017. -572 p.
2. D.L. Applegate, R.E. Bixby, V. Chvatal, W.J. Cook. "The Traveling Salesman Problem: A Computational Study"
3. N.P. Buslenko, V.V. Krinitsyn. S.Yu. Gorodesky. "Transportation Problems and Ways to Solve Them" Optimal Management. NNGU. 2014.
EURASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES
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4. Z. Mamatova, M. Rasuljonova. "Creating a Mathematical Model for Finding the Optimal Solution of Total Costs for Traveling Between Countries for Tourism Firms Using the Traveling Salesman Problem" Collection of Proceedings of the International Scientific and Technical Conference on the Role of Digital Technologies in the Development of Exact and Natural Sciences: Problems and Innovative Solutions. Part 3. 2024. pp. 75-81.
