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21Journal of Sustainable Development of Transport and Logistics
journal home page: https://jsdtl.sciview.net
Prokudin, G., Chupaylenko, O., Dudnik, O., Dudnik, A., & Pylypenko, Y. (2018). Traveling salesman problem in the function of freight transport optimization. Journal of Sustainable Development of Transport and Logistics, 3(1), 29-36. doi:10.14254/jsdtl.2018.3-1.3.
Scientific Plafor
ISSN 2520-2979
Traveling salesman problem in the function of freight transport optimization
Georgii Prokudin * , Olexiy Chupaylenko ** , Olexiy Dudnik ** , Alena Dudnik *** , Yuriy Pylypenko ***
* National Transport University,
1, M. Omelianovycha-Pavlenka Str., Kyiv 01010, Ukraine
Dr., Professor, Department of International Road Transportation and Customs Control
** National Transport University,
1, M. Omelianovycha-Pavlenka Str., Kyiv 01010, Ukraine
PhD, Associate Professor, Department of International Road Transportation and Customs Control
*** National Transport University,
1, M. Omelianovycha-Pavlenka Str., Kyiv 01010, Ukraine
OPEN
8
ACCESS
Article history:
Received: January 31, 2017 1st Revision: February 19, 2018
Accepted: March 10, 2018
DOI:
10.14254/jsdtl.2018.3-1.3
Abstract: The use of modern information technology means in solving the traveling salesman problem to optimize the routing of freight transportation in international traffic is motivated in this article. The process of solving the traveling salesman problem is automated by modern information technology means, in particular the Delphi Software and the function "Search Solution" in the Microsoft Office Excel table processor. The existing requirements and restrictions on the specificity and dimension of the problem are considered as well.
Keywords: Freight, logistics, information technology, traveling salesman problem, closed contour, optimization, route, checkpoint.
1. Introduction
Historical reference. In 1859, William Hamilton formulated a problem "Around the World Tour". The problem was focused on finding the shortest route, which would provide one-time visiting of each given settlement and returning to the starting point. The problem gave rise to a new direction in the theory of graphs, known as the search for Hamiltonian cycles in graphs. The Hamiltonian cycle of a graph with n vertices can be represented by the set of pairs of the graph adjacent vertices:
{(i, /2);(/2, /3); ...,(/„_!, in );(Vi, i)}.
The problem of the Hamiltonian cycles in the graph theory gained different generalizations (Kozachenko, Vernygora, & Malashkin, 2015)]. One of these generalizations is the traveling salesman problem, which often occurs in various modifications in transport logistics when planning transportation. The traveling salesman problem is a modified problem of en-route to the destination point; however, in this case, the connection between the points should form a closed cycle.
Corresponding author: Olexiy Chupaylenko E-mail: [email protected]
This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license
The traveling salesman (came from French, commis voyageur) leaves the first city, visits only once each of n cities and returns to the first city. The distances between cities are known. The challenge is to find a route to cities, which ensures the shortest closed cycle of salesman's travel.
2. Literature review
There are several distinct cases of general problem statement, in particular the geometric traveling salesman problem (so-called planar or Euclidean, when the distance matrix reflects the distance between points on a plane), the triangular traveling salesman problem (the triangle inequality occurs on the matrix of values), and the symmetric and asymmetric problems of a traveling salesman. There is also a generalization of the problem, the so-called generalized traveling salesman problem (Kunda, 2008).
Statement of the traveling salesman problem. There are n cities. A matrix of distances C = |c,y| between them is specified. In the general case, C . ^ C . A traveling salesman leaves the first city Ao, and
y J1
then visits the other cities one at a time and returns to the city A0. Therefore, the route of a traveling salesman is a closed cycle without loops. It is necessary to define the order, in which the city can be driven around to minimize the total traveled distance.
Mathematical model of the problem. Let's introduce the variables: x = 1, if a salesman moves
from the city Ai to the city Aj; xy = 0 - vise versa
where i, j = 1,2,..., n; i ^ J . It is necessary to find
n n
min ZZ cj ■ xj, (1)
'J 'J
i=0 J=0
under conditions
n
2Xj = 1 i = 1,2,...,n, (2)
j=o
n
2Xj = 1 j = 1,2,...,n, (3)
i=0
U -u. + n■ Xj < n-1 i, j = 1,2,...,n ; i ^ j . (4)
where u, ,u . - arbitrary integral nonnegative numbers.
Condition (2) means that a salesman enters each city only once, except for the first one. Condition (3) means that carrier leaves each city only once as well. Condition (4) ensures the closure of the route containing n points, and the absence of loops (Lashenyh, & Kuzkin, 2006).
Society informatization is a global social process characterized by the fact that the dominant activity in the social production sphere is the collection, accumulation, production, processing, storage, transfer and use of information based on modern microprocessor and computer technology, and various means of information exchange (Prokudin, Danchuk, Tsukanov, & Tsymbal, 2013).
The importance of information technology application in the transport sector is indisputable. Optimization of freight delivery schemes is very important in the transport industry and logistics (Prokudin, 2006). In most segments of the market, the delivery of goods adds to its value an amount equivalent to the cost of the product itself. In addition, it should be noted that the use of modern information technology for optimization of such delivery leads to minimization of costs, often at least from 5 to 20% of the product total cost.
This study is mainly focused on the use of modern information technology means in solving the salesman problem to optimize the routing of freight transportation in international traffic. The most important factors that need to be considered in solving the task are:
- the distances between points of departure and destination, and customs posts;
- time of service at checkpoints (customs clearance);
- time of loading and unloading operations;
- average speed of the vehicle (V);
- time of rest under the European agreement regarding the work of vehicle crews, which perform international automobile transportation (UTRT).
Based on the real data on the locations of points of departure and destination, and checkpoints (CP) across the state border of Ukraine, the distances between them, and average speeds of vehicles, the necessary calculations were made.
3. Presentation of the main material
The research was conducted in Zhytomyr region. The wood and wood products make up 23% of the total exports of the region. Therefore, based on the data analysis of the State Statistics Service of Ukraine, laws and regulations, and the economic and social situation in our country, wooden pellets were selected as over-the-road freight. In Poland the vehicle will be loaded by wooden furniture.
In Zhytomyr region, 10 points of departure were selected: 1) Dubrivka; 2) Romaniv; 3) Liubar; 4) Malyn; 5) Ovruch; 6) Novograd-Volynsky; 7) Zhytomyr, 8) Korosten; 9) Berdychiv; 10) Radomyshl. These places are known to be the largest producers of wooden pellets in the region.
For further research, 2 automobile checkpoints were selected in Volyn region and 6 in Lviv region (Fig. 1). The determining factors for choosing a CP are the time for customs operations and the distance from the places of freight departure. The list of 8 checkpoints at the Ukrainian border is as follows:
1) Yagodyn; 2) Ustylug; 3) Ugryniv; 4) Rava-Ruska; 5) Grushiv; 6) Krakovets; 7) Shegini; 8) Smilnytsa.
Figure 1: Automobile checkpoints at the Ukrainian-Polish border
The selected checkpoints in Poland are characterized by the same features as the checkpoints in Ukraine (see Fig. 1), namely:
1) Dorogusk; 2) Zosin; 3) Dolgobychuv; 4) Grebenne; 5) Budomyezh; 6) Korchova; 7) Medyca; 8) Krostsenko.
In Poland, the wooden pellets meet a ready market, and the furniture factories import their finished products to Ukraine. That is why, the selected destination points are as follows: 1) Slupsk; 2) Verushuv; 3) Ratsibuzh; 4) Elblong; 5) Morong; 6) Brodnytsia; 7) Warsaw; 8) Keltse; 9) Ostrovets-
Sventokshyskyi; 10) Vengruv. In the Microsoft Access database environment, a database was created with appropriate distances between checkpoints, points of departure and destination (Fig. 2).
Figure 2: Database of distances between CPs, points of departure and destination
Number - Point of departure - Destination L
1 Dubrivka Romaniv 55
2 Dubrivka Lyubar SI
3 Dubrivka MaKn 187
4 Dubrivka Ovruch 161
5 Dubrivka Novograd-Volynsky 39
6 Dubrivka Zhytomyr 97
7 Dubrivka Korosten 130
8 Dubrivka Berdychiv 127
9 Dubrivka RadomysH 200
10 Dubrivka Yagodin 338
11 Dubrivka Ustyhig 314
12 Dubrivka Ushryniv 337
13 Dubrivka Rava-Ruska 389
For the convenience of further calculations, data on these distances are automatically reformatted into a Microsoft Excel table processor file, as shown in Fig. 3. The software for solving the traveling salesman problem was designed using the Delphi programming algorithmic language (Johnson, 1990).
Figure 3: Distances between CPs, points of departure and destination in Microsoft Excel file
format
A B c D E F G H 1 J K L M N o P Q R s T
1 Town 1 2 3 4 5 Cr 7 8 9 10 11 12 13 14 15 16 17 18
2 1 Dubrivka 0 55 81 187 161 39 97 130 127 200 338 314 337 389 432 417 426 455
3 2 Romaniv 55 0 50 152 182 73 64 136 63 137 381 357 368 443 445 460 457 486
4 3 Lyn bar 81 50 0 174 219 112 84 173 69 157 395 343 329 378 407 393 413 447
5 4 Malin 187 152 174 0 94 146 88 58 128 36 404 405 442 508 537 532 557 591
6 5 Ovruch 161 182 219 94 0 139 133 48 173 127 384 385 422 488 521 505 530 564
7 6 N ovog ra d -Volyn s ky 39 73 112 146 139 0 84 108 125 157 316 292 307 367 406 395 420 454
S 7 Zhytomyr 97 64 84 88 133 84 0 87 41 74 392 368 403 454 483 471 496 530
9 8 Korosten 130 136 173 58 48 108 87 0 127 90 357 357 394 460 489 474 499 533
10 9 Berdychiv 127 63 69 128 173 125 41 127 0 115 433 409 395 431 473 459 484 513
11 10 Radoiriyshl 200 137 157 36 127 157 74 90 115 0 437 437 474 531 559 544 566 600
12 11 Vagodin (PP) 338 381 395 404 384 316 392 357 433 437 0 83 120 212 239 256 283 325
13 12 Ustylug (PP) 314 357 343 405 385 292 368 357 409 437 83 0 60 148 179 196 218 265
14 13 Ushryniv (PP) 337 368 329 442 422 307 403 394 395 474 120 60 0 103 134 150 173 220
15 14 Rava-Ruska (PP) 389 443 378 508 488 367 454 460 431 531 212 148 103 0 32 55 93 162
16 15 Hrushiv (PP) 432 445 407 537 521 406 483 489 473 559 239 179 134 32 0 26 82 151
17 16 Krakovets (PP) 417 460 393 532 505 395 471 474 459 544 256 196 150 55 26 0 33 89
IS 17 Shegini (PP) 426 457 413 557 530 420 496 499 484 566 283 218 173 93 82 33 0 57
19 18 Smilnytsia (PP) 455 486 447 591 564 454 530 533 513 600 325 265 220 162 151 89 57 0
To begin with, a page to entry the input data for further calculations is developed, namely the time characteristics of loading and unloading operations in this route. The process of loading and unloading pallets and furniture is mechanized. The time of service in the CP and the average technical speed of the vehicle are set (Vt = 65 km/h).
Let's consider an example of the traveling salesman problem with 1 point of departure in Ukraine, 4 CPs (two in Ukraine and two in Poland), 4 destination points in Poland. That is, in one of the cities of Zhytomyr region, 20 tons of wooden pellets are loaded. The process of loading is mechanized. In each 4 cities of Poland, 5 tons of freight are unloaded (20 tons in total); and in the last city, 20 cubic meters of furniture are loaded (also approximately 20 tons). Both wooden pellets and furniture belong to the 1st class of goods, that is, the coefficient of static use of the vehicle carrying capacity is equal to (ycm = 1). It
should be noted that the program works in two modes, manual and automatic. That is, CPs can be selected independently, or the program does it automatically, selecting the closest ones to the point of departure.
The program generates a table, which clearly shows the distances between the specified cities and the CPs. It allows finding possible closed routes and choosing the shortest of all variants (Fig. 4).
Figure 4: Result of the program work
'¿jtf The task of traveling salesman
@ Ua-l"
Dubrivka —¥ Ustylug (UA) —»Zosin (PL} —» Warsawa —» Verusuv —» Elblag —» Slupsk —» Dorohusk (PL) —» Yagodin [UA} —» Dubrivka —¥ Ustylug (UA) —»Zosin (PL) —» Warsawa—» Verusuv —» SI upsk —» Elblag —¥ Dorohusk (PL) -¥ Yagodlin (UA) —» Dubrivka —¥ Ustylug UA} —»Zosin [PL —» Warsawa—¥ Elblag -¥ Verusuv—¥ Slupsk —¥ Dorohusk (PL -¥ Yagodlin (UA —» Dubrivka —¥ Ustylug (UA) —»Zosin (PL) -¥ Warsawa—¥ Elblag -¥ SI ups k —¥ Verusuv—¥ Dorohusk (PL) -¥ Yagodlin (UA) —» Dubrivka —¥ Ustylug UA} —»Zosin (PL —» Warsawa—¥ Slupsk —» Verusuv—¥ Elblag —¥ Dorohusk (PL —¥ Yagodin (UA —» Dubrivka -h>-Ustylug (UA)^-Zosin (PL} ^ Warsawa—¥ Slupsk Elblag -h>-Verusuv Dorohusk (PL) Yagodlin (UA) ■ -» Dubrivka -» Ustylug (UA) -»Zosin (PL) —¥ Verusuv —¥ Warsawa —¥ Elblag —¥ Slupsk —¥ Dorohusk (PL) —¥ Yagodin (UA) —» Dubrivka —¥ Ustylug UA) —»-Zosin (PL —» Verusuv —¥ Warsawa —¥ Slupsk —¥ Elblag —¥ Dorohusk (PL —¥ Yagodin (UA —» Dubrivka —¥ Ustylug (UA) —¥ Zosin (PL) —» Verusuv —¥ Elblag —¥ Warsawa —¥ Slupsk —¥ Dorohusk (PL) —¥ Yagodin (UA) —» Dubrivka —¥ Ustylug UA} —¥Zosin [PL —»Verusuv —¥ Elblag —¥Slupsk —» Warsawa—¥ Dorohusk (PL -¥ Yagodin (UA —» Dubrivka —¥ Ustylug (UA) —»Zosin (PL} —¥ Verusuv —¥ SI upsk —» Warsawa -¥ Elblag —» Dorohusk (PL) —¥ Yagodin (UA) —» Dubrivka —» Ustylug (UA) —¥ Zosin (PL) —» Verusuv—» Slupsk-» Elb lag —»Warsawa —» Dorohusk (PL) -—» Yagodin (UA) Dubrivka —¥ Ustylug (UA) —¥Zosin (PL) -¥ Elblag —» Warsawa —¥ Verusuv —¥ Slupsk —¥ Dorohusk (PL) —¥ Yagodin (UA) —» Dubrivka —¥ Ustylug UA} —¥Zosin [PL -¥ Elblag —» Warsawa —»-Slupsk —¥ Verusuv—¥ Dorohusk (PL -¥ Yagodin (UA —» Dubrivka —¥ Ustylug (UA) —¥Zosin (PL) -¥ Elblag —¥ Verusuv —¥ Warsawa —¥ Slupsk —¥ Dorohusk (PL) —¥ Yagodlin (UA) —» Dubrivka —¥ Ustylug (UA) —»Zosin (PL) -¥ Elblag —¥ Verusuv —»Slupsk —» Warsawa—¥ Dorohusk (PL) -¥ Yagodin (UA) —» Dubrivka Ustylug (UAj Zosin (PL} -¥ Elblag —» Slupsk Warsawa^ Verusuv^ Dorohusk (PL -»Yagodin (UA Dubrivka —¥ Ustylug (UA) —¥Zosin (PL) -¥ Elblag —¥Slupsk —¥ Verusuv—» Warsawa—¥ Dorohusk (PL) -¥ Yagodlin (UA) —» Dubrivka —¥ Ustylug UA} —¥ Zosin [PL —»Slupsk —¥ Warsawa—»Verusuv—¥ Elblag —¥ Dorohusk (PL —»Yagodin (UA —» Dubrivka —¥ Ustylug (UA) —¥ Zosin (PL) —»Slupsk —¥ Warsawa —»- Elblag —¥ Verusuv—¥ Dorohusk (PL) —»Yagodin (UA) —» Dubrivka —¥ Ustylug UA} —¥ Zosin (PL —»Slupsk —¥ Verusuv—¥ Warsawa—» Elblag —¥ Dorohusk (PL —»Yagodin (UA —» Dubrivka —¥ Ustylug (UA) —»Zosin (PL) —»Slupsk —¥ Verusuv—¥ Elblag —» Warsawa—¥ Dorohusk (PL) —»Yagodlin (UA) —¥ Dubrivka —¥ Ustylug (UA) —» Zosin [PL} —»Slupsk —¥ Elblag —¥ Warsawa—»Verusuv—» Dorohusk (PL} —»Yagodin (UA} —¥ Dubrivka —¥ Ustylug (UA) —¥ Zosin (PL)—»Slupsk —¥ Elblag —¥ Verusuv—»- Warsawa—¥ Dorohusk (PL) —»Yagodin (UA)—¥
Dubrivka - 2608 km Dubrivka = 2424 km Dubrivka = 2S23 km Dubrivka = 2390 km Dubrivka = 2914 km Dubrivka - 2565 km Dubrivka = 2640 km Dubrivka =2631 km Dubrivka = 3130 km Dubrivka =2565 km Dubrivka = 2346 km -» Dubrivka = 2390 km Dubrivka = 2996 km Dubrivka = 2353 km Dubrivka = 3171 km Dubrivka = 292.1 km Dubrivka = 2638 km Dubrivka = 2431 km Dubrivka = 3179 km Dubrivka = 3145 km Dubrivka = 3004 km Dubrivka =2938 km Dubrivka = 2655 km Dubrivka = 2623 km
= The shortest closed route of the 24 possible -Dubrivka —» Ustylug (UA) —»Zosin (PL) —»Warsawa -
36.76 years for transportation
45 minutes passing the 1st checkpoint - Ustylug (UA)
40 minutes passing the 1st checkpoint - Zosin (PL)
35 minutes passing the 1st checkpoint - Dorohusk (PL)
40 minutes passing the 1st checkpoint - Yagodin (UA)
750 minutes for unloading / loading
51.93 hours for the entire route
< I
> Elb lag —> Slupsk -^Verusuv -^Dorohusk (PL) -^Yagodin (UA) -^Duhrivka = 2390 km
Based on the combinatorial method (Prokudin, 2014), the program automatically calculates the total distance of the route and selects the one where the distance between the cities is the shortest. Map of the specified route is presented in Fig. 5.
Figure 5: Map of a defined closed route
Thus, the program identified 24 possible routes; the route Dubrivka (Ukraine) ^ Ustylug (CP, Ukraine) ^ Zosin (CP, Poland) ^ Warsaw (Poland) ^ Elblong (Poland) ^ Slupsk (Poland) ^ Verushuv
(Poland) ^ Dorogusk (CP, Poland) ^ Yagodyn (CP, Ukraine) ^ Dubrivka (Ukraine) was considered the most effective one. The length of the route is 2390 km. The total time of transportation is 36.76 h, the time of service at the first CP (Ustylug) - 45 min, at the second CP (Zosin) - 40 min, in the third CP (Dorogusk) - 35 min, in the fourth CP (Yagodyn) - 40 min. The time expenditure for loading and unloading - 760 min. Based on the above timing data, the total time to complete the route with exception of time characteristics for the driver's sleep is 52.10 h (according to UTRT).
A detailed analysis of applying the model of the optimal purpose to solve the traveling salesman problem has shown that in this model, in addition to n! Hamiltonian (full) contours, there are also many incomplete (isolated) contours that cover only certain groups of cities. This fact greatly complicated the solving of the traveling salesman problem and made the researchers look for other more effective methods of its solution.
Further, an example of the traveling salesman problem solution for n = 9 (1 city-supplier in Ukraine, 2 CPs in Ukraine, 2 CPs in Poland, 4 city-consumers in Poland) in the Microsoft Office Excel spreadsheet using the Search Solution function is presented (Kuzmychov & Medvediev, 2005). We select the same CPs and cities, as in the previous example.
In Fig. 6, an Excel spreadsheet with source data (distances) between cities is shown. However, to solve the traveling salesman problem for any (and fully oriented) graphs, the absence of an arc between nodes in the transport correspondence matrix should be designated by the infinity V =Xi. It means that
the numbers of 2-3 orders of magnitude larger than the maximum distance should be entered in these cells, namely, in our case, the number is assumed to be equal to 99999.
Figure 6: Starting matrix of distances between cities
A B C D E F G H 1 J
1 Dubrivka Yagodin Ustylug Dorohusk Zosin Warsawa Verusuv Elblag Slupsk
2 Dubrivka 99999 338 314 99999 99999 99999 99999 99999 99999
3 Yagodin 338 99999 99999 7 99999 99999 99999 99999 99999
4 Ustylug 314 99999 99999 99999 5 99999 99999 99999 99999
5 Dorohusk 99999 7 99999 99999 99999 270 47S 539 734
6 Zosin 99999 99999 5 99999 99999 307 515 5S3 786
7 Warsawa 99999 99999 99999 270 307 99999 313 2S5 471
8 Verusuv 99999 99999 99999 473 515 313 99999 461 472
9 Elblag 99999 99999 99999 539 583 235 461 99999 134
10 Slupsk 99999 99999 99999 734 786 471 472 134 99999
The process of calculating the traveling salesman problem in Excel includes the following steps: inputting the initial data (see Fig. 6); forming a matrix, where the sum of elements in rows and columns is calculated; forming a constraint matrix of consistency and a target function; forming a model of the optimization problem; obtaining the final result (Fig. 7).
Figure 7. Obtaining the final result
12 Dubrivka Yagodin Ustylug Dorohusk Zosin Warsawa Verusuv Elblag Slupsk Out
13 Dubrivka 0 1 0 0 0 0 0 0 0 1
14 Yagodin 0 0 0 1 0 0 0 0 0 1
15 Ustylug 1 0 0 0 0 0 0 0 0 1
IS Dorohusk 0 0 0 0 0 0 1 0 0 1
17 Zosin 0 0 1 0 0 0 0 0 0 1
IS Warsawa 0 0 0 0 1 0 0 0 0 1
19 Verusuv 0 0 0 0 0 0 0 0 1 1
20 Elblag 0 0 0 0 0 1 0 0 0 1
21 Slupsk 0 0 0 0 0 0 0 1 0 1
22 In 1 1 1 1 1 1 1 1 1
36 Target function 2390,0
Consequently, based on the calculations in the Excel table processor, an optimal closed route of freight transportation is obtained (Fig. 8).
Figure 8: Presentation of the optimal transportation route on the map
The length of the calculated route (Dubrovka ^ Yagodyn ^ Dorogusk ^ Warsaw ^ Elblong ^ Slupsk ^ Verushuv ^ Zosin ^ Ustylug ^ Dubrovka) is 2390 km. It should be noted that in two variants of solving this task (using the Delphi software environment and the Excel spreadsheet), the same length of the route 2390 km is calculated, although the sequence of passing cities in both variants is different.
4. Conclusion
During this study, the use of the Delphi Software and the function "Search Solution" in the Microsoft Office Excel table processor in solving the traveling salesman problem to optimize the routing of freight transportation in international traffic is motivated in this article. The existing requirements and restrictions on the specificity and dimension of the problem are considered as well.
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in the online version, at https://jsdtl.sciview.net
Funding
The authors received no direct funding for this research. Citation information
Prokudin, G., Chupaylenko, O., Dudnik, O., Dudnik, A., & Pylypenko, Y. (2018). Traveling salesman problem in the function of freight transport optimization. Journal of Sustainable Development of Transport and Logistics, 3(1), 29-36. doi:10.14254/jsdtl.2018.3-1.3.
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