- Email: [email protected]
Self-Organizing Iterative Algorithm for Travelling Salesman Problem
18th IFAC Conference on Technology, Culture and International 18th IFAC Conference on Technology, Culture and International Stability 18th IFAC Conference on Technology, Culture and International Stability 18th Conference Sept on Technology, Culture and International Baku,IFAC Azerbaidschan, 13-15, 2018 Available online at www.sciencedirect.com Stability 18th Conference Sept on Technology, Baku,IFAC Azerbaidschan, 13-15, 2018Culture and International Stability Baku, Azerbaidschan, Sept 13-15, 2018 Stability Baku, Azerbaidschan, Sept 13-15, 2018 Baku, Azerbaidschan, Sept 13-15, 2018
ScienceDirect
IFAC PapersOnLine 51-30 (2018) 268–270 Self-Organizing Iterative Algorithm for Travelling Salesman Problem Self-Organizing Iterative Algorithm for Travelling Salesman Problem Self-Organizing Iterative Algorithm for Travelling Salesman Self-Organizing Iterative Urfat Algorithm Salesman Problem Problem Nuriyev*.for OnurTravelling Ugurlu** Self-Organizing Iterative Urfat Algorithm for Travelling Salesman Problem , Nuriyev*. Onur Ugurlu**
Fidan Nuriyeva*** ,**** Urfat Nuriyev*. Onur Ugurlu** Urfat Nuriyev*. Fidan Nuriyeva*** Onur Ugurlu** ,**** ,**** Fidan Nuriyeva*** Urfat Nuriyev*. Onur Ugurlu** Fidan Nuriyeva*** ,**** * Faculty of Science, Department Mathematics, Ege University, Izmir, Turkey **** Fidan of Nuriyeva*** * Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey Faculty of Science, Department of Mathematics, Ege (e-mail: [email protected]) ** Faculty of Science, Department of Mathematics, EgeUniversity, University,Izmir, Izmir,Turkey Turkey (e-mail: [email protected]) * Faculty of Science, Department of Mathematics, Ege University, Izmir, (e-mail: [email protected]) ** Faculty of Science,(e-mail: Department of Mathematics, Ege University, Izmir,Turkey Turkey [email protected]) ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) *** Faculty of Science, Department [email protected]) Computer Science, Dokuz Eylul University, Izmir, Turkey (e-mail: ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) *** Faculty of Science, Department of Computer Science, Dokuz Eylul University, Izmir, Turkey (e-mail:[email protected]) *** Department of Computer Science, Dokuz Eylul University, Izmir, (e-mail: [email protected]) *** Faculty Faculty of of Science, Science, Department of Computer Science, Dokuz Eylul University, Izmir, Turkey Turkey (e-mail:[email protected]) **** Institute of Control Systems of ANAS, Baku, Azerbaijan (e-mail:[email protected]) *** Faculty of Science, Department of Computer Science, Dokuz Eylul University, Izmir, Turkey **** Institute(e-mail:[email protected]) of Control Systems of ANAS, Baku, Azerbaijan **** of **** Institute Institute(e-mail:[email protected]) of Control Control Systems Systems of of ANAS, ANAS, Baku, Baku, Azerbaijan Azerbaijan **** Institute of Control Systems of ANAS, Baku,algorithm Azerbaijanfor solving Travelling Abstract: This paper presents a self-organizing heuristic iterative Abstract: This paper presents a self-organizing heuristic iterative algorithm for solving Travelling Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied Abstract: This paper presents aa self-organizing heuristic iterative algorithm for solving Travelling Abstract: This paper presents self-organizing heuristic iterative algorithm for solving Travelling Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied solutions are found. After that, the priority values of the edges are updated according to these solutions. Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied Abstract: This paper presents a self-organizing iterative algorithm for solving Travelling Salesman Problem. In the algorithm, firstlyvalues priority values thepriority edges are determined. Then, varied solutions are found. After that, priority ofheuristic the edges are updated according these Then, all the edges are sorted in the descending order according toof the values and into order tosolutions. improve solutions are found. After that, the priority values of the edges are updated according to these solutions. Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied solutions are found. After that, the priority values of the edges are updated according to these solutions. Then, all the edges arealgorithm sorted inisdescending order according to thetopriority values andinineach orderiteration to improve solutions an iterative used. Greedy algorithm is used find the solution and Then, all the edges sorted in descending order according to the priority values and in order to improve solutions areiterative found.are After that, the priority values of the edges are updated according to these solutions. Then, all the edges are sorted in descending order according to the priority values and in order to improve solutions an algorithm is used. Greedy algorithm is used to find the solution in each iteration and the priority values of the edges are updated according to the solution. The proposed algorithm is solutions an iterative algorithm used. Greedy algorithm is used find solution in iteration and Then, all the edges are sorted inis order according to the theto values and orderalgorithm to improve solutions an iterative algorithm isdescending used. Greedy algorithm isto used topriority find the the solution inineach each iteration and the priority values of the edges are updated according solution. The show proposed is compared with Nearest Neighbour and Greedy algorithm. Experimental results that the proposed the priority values of the edges are according the The proposed algorithm is solutions an iterative algorithm is used. Greedy used find the solution in each iteration and the priority values of the edges are updated according to thetosolution. solution. The show proposed algorithm is compared Nearest Neighbour andupdated Greedyalgorithm algorithm.isto Experimental results that the proposed algorithm iswith efficient. compared with Nearest Neighbour and Greedy algorithm. Experimental results show that the proposed the priority values of the edges are updated according to the solution. The proposed algorithm is compared iswith Nearest Neighbour and Greedy algorithm. Experimental results show that the proposed algorithm efficient. algorithm is efficient. compared with Nearest Neighbour and Greedy algorithm. Experimental results show that the proposed © 2018, IFAC (International Federation of Automatic Control) Hosting by ElsevierNP-Hard Ltd. All rights reserved.selfalgorithm is heuristics, efficient. Keywords: artificial intelligence, combinatorial optimization, problems, algorithm isalgorithms. efficient. artificial intelligence, combinatorial optimization, NP-Hard problems, selfKeywords: heuristics, organizing Keywords: algorithms. heuristics, artificial artificial intelligence, intelligence, combinatorial combinatorial optimization, optimization, NP-Hard NP-Hard problems, problems, selfselfKeywords: heuristics, organizing organizing algorithms. Keywords: heuristics, artificial intelligence, combinatorial optimization, NP-Hard problems, self organizing algorithms. organizing algorithms. In this paper, a new self-organizing iterative algorithm is 1. INTRODUCTION In this paper, new the self-organizing iterative algorithm is proposed to aasolve TSP problem. Two heuristic 1. INTRODUCTION In this paper, new self-organizing iterative algorithm is In this paper, a new self-organizing iterative algorithm is proposed to solve the TSP problem. Two heuristic INTRODUCTION algorithms, Nearest Neighbour (NN) and Greedy Algorithm 1. INTRODUCTION Traveling Salesman 1. Problem (TSP) is perhaps the most well - In proposed to solve the TSP problem. Two heuristic this paper, a new self-organizing iterative algorithm is proposed to solve the TSP problem. Two heuristic algorithms, Nearest Neighbour (NN) and Greedy Algorithm INTRODUCTION Traveling Salesman 1. Problem (TSP)problem is perhaps are conducted against 19TSP benchmark real world TSP known combinatorial optimization in the themost set ofwell NP--- proposed algorithms, Nearest Neighbour (NN) and Greedy Algorithm to solve the problem. Two heuristic Traveling Salesman Problem (TSP) is perhaps the most well algorithms, Nearest Neighbour (NN) and Greedy Algorithm are conducted against 19 benchmark real world TSP Traveling Salesman (TSP) is perhaps most well known combinatorial optimization problem inofthe the setisofthat NPcollected in TSPLIB95. Results obtained by those Hard (Lawler et al., Problem 1986). The significance TSP it-- problems are conducted against 19 benchmark real world TSP algorithms, Nearest Neighbour (NN) and Greedy Algorithm known combinatorial optimization problem in the set of NPTraveling Salesman Problem (TSP) is perhaps the most well are conducted against 19 benchmark real world TSP problems collected in TSPLIB95. Results obtained by those known combinatorial optimization problem in the set of NPHard (Lawler et al., 1986). The significance of TSP is that it algorithms compared with exactly optimal solutions. includes manyet problems that areproblem naturalinof applications in problems collected in TSPLIB95. Results obtained by those are conducted against 19 benchmark real world TSP Hard (Lawler al., 1986). The significance TSP is that it known combinatorial optimization the set of NPproblems collected TSPLIB95. Results solutions. obtained by those algorithms comparedinwith exactly optimal Hard (Lawler et al., 1986). The significance of TSP is that it includes many problems that are natural applications in computer science and engineering (Appligate et al., 2006; algorithms compared exactly optimal problems collected inwith TSPLIB95. Results solutions. obtained by those includes many problems that are natural applications in Hard (Lawler et al., 1986). The significance of TSP is that it algorithms compared with exactly optimal solutions. includes many that are (Appligate natural applications in computer science and engineering et al., 2006; 2. APPROACHES FORoptimal SOLVING TSP Lenstra and Kan,problems 1975). with exactly solutions. computer science and et includes many that are (Appligate natural applications in algorithms2.compared APPROACHES FOR SOLVING TSP computer science and engineering engineering (Appligate et al., al., 2006; 2006; Lenstra and Kan,problems 1975). 2. APPROACHES FOR SOLVING TSP Lenstra and Kan, computer engineering (Appligate 2006; By literature, many algorithms approaches 2. APPROACHES FORand SOLVING TSPhave been Lenstra andscience Kan, 1975). The problem is 1975). toandfind an optimal tour foreta al., traveling By literature, many algorithms and approaches 2. APPROACHES FOR SOLVING TSPhave been Lenstra and Kan, 1975). proposed to solve the TSP problem. These algorithms and The problem is to find an optimal tour for a traveling literature, many algorithms and approaches have salesman wishing to find visit an eachoptimal of a list of nforcities exactly By The problem is to tour a traveling By literature, many algorithms and approaches have been been proposed to solve the TSP problem. These algorithms and approaches can be classified into Heuristic Algorithms, The problem is to find an optimal tour for a traveling salesman wishing to visit each of a list of n cities exactly proposed to solve the TSP problem. These algorithms and literature, many algorithms and approaches have been once and then return to thean home city. Such optimal tour is By salesman wishing to visit each of a list of n cities exactly proposed to solve the TSP problem. These algorithms and approaches can be classified into Heuristic Algorithms, The problem is to find optimal tour for a traveling Algorithms, Approximate Algorithms and salesman wishing towhose visit of a listSuch of noptimal cities exactly once andtothen return to theeach home city. tour is Metaheuristic approaches can be classified into Heuristic Algorithms, proposed to solve the TSP problem. These algorithms defined be a tour total distance (cost) is minimized once and then return to the home city. Such optimal tour is approaches can be classified into Heuristic Algorithms, Metaheuristic Algorithms, Approximate Algorithms and and salesman wishing towhose visit each of a listSuch of noptimal cities exactly Exact Algorithms (Lawler et. Al., 1985). once and then return to the home city. tour is defined to be a tour total distance (cost) is minimized Metaheuristic Algorithms, Approximate Algorithms approaches can be classified into Heuristic Algorithms, (Appligate et al., 2006). defined be aareturn tour whose distance (cost) is Algorithms, Algorithms and and Exact Algorithms (Lawler et. Approximate Al., 1985). once andto to thetotal home city. Such tour is Metaheuristic defined tothen be tour whose total distance (cost)optimal is minimized minimized (Appligate et al., 2006). Exact Algorithms (Lawler et. Al., 1985). Metaheuristic Algorithms, Algorithms TSP problem possessing noApproximate longer than 20 cities can and be (Appligate et al., 2006). Exact Algorithms (Lawler et. Al., 1985). defined to be a tour whose total distance (cost) is minimized (Appligate et al., 2006). of TSP has been firstly lunched Exact The general formulation TSP problem possessing no longer than heuristic 20 citiesmethods can be Algorithms (Lawler et. Al., 1985). optimally solved by exact methods. The (Appligate et al., 2006). The general formulation of TSP has been firstly lunched problem possessing longer than 20 cities can based on the graph theory of in 1930s. Problem can be lunched defined TSP TSP possessing no longer The than 20 TSP citiesproblems can be be optimally solved byquality exactno methods. heuristic methods The general formulation TSP has been firstly could problem provide high solutions for the The general formulation TSP with has been firstly based on the and graph theory of ingraph 1930s. Problem can(cities). be lunched defined optimally solved by exact methods. heuristic methods TSP problem possessing no longer The than 20 TSP citiesproblems can be as a weighed connected n vertices We optimally solved by exact methods. The heuristic methods could provide high quality solutions for the based on the graph theory in 1930s. Problem can be defined The general formulation of TSP has been firstly lunched possessing large amount number of cities. However, the based the aand graph theoryone ingraph 1930s. Problem can bealldefined as a weighed connected with n vertices (cities). could provide high quality solutions for the TSP problems solved by exact number methods. heuristic methods want toon tour from of these vertices, visit of We the optimally could provide high quality solutions for the TSP problems possessing large amount ofThe cities. However, the as aa weighed connected with nn vertices (cities). We based onstart the aand graph theoryone ingraph 1930s. Problem can bealldefined optimum solution can not be guaranteed (Johnson and as weighed and connected graph with vertices (cities). We want to start tour from of these vertices, visit of the possessing large amount number of cities. However, the could provide high quality solutions for the TSP problems other vertices for exactly one time and finally return to start possessing large amount number of cities. However, the optimum solution can not be guaranteed (Johnson and want to start aaand tour from one of these vertices, visit all of the as a weighed connected graph with nfinally vertices (cities). We McGeoch, 1997). want to start tour from one of these vertices, visit all of the other vertices for exactly one time and return to start optimum solution can not be guaranteed (Johnson and possessing large amount number of cities. However, the vertex. This tour called Hamiltonian tour. In the TSP, we optimum solution can not be guaranteed (Johnson and McGeoch, 1997). other vertices for one and finally return to want to start atour tourexactly from one oftime these vertices, visit all ofstart the other vertices for exactly one time and finally return to start vertex. This called Hamiltonian tour. In the TSP, we McGeoch, 1997). optimum solution can not be guaranteed (Johnson and want find atour minimum cost Hamiltonian cycle (Davendra, Heuristic algorithms used to solve the TSP problems in this McGeoch, 1997). vertex. This called In the TSP, we other to vertices for exactly one time andtour. finally to start vertex. This called Hamiltonian Hamiltonian tour. In return the(Davendra, TSP, we McGeoch, want to find life atour minimum cost Hamiltonian cycle Heuristic used to solve TSP in this 1997). Neighbour 2010). Real application areas of TSP include problems work are algorithms Nearest from the Both Endproblems Points (NND) want to find a minimum cost Hamiltonian cycle (Davendra, Heuristic algorithms used to TSP in vertex. This tour called Hamiltonian tour. In the TSP, we want arise toReal findinlife a logistics, minimum costareas Hamiltonian cycle (Davendra, 2010). application of TSP include problems Heuristic used to solve solve the TSP problems in this this work are algorithms Nearest Neighbour from the Both Endproblems Pointsreviewed (NND) that telecommunication networks, data and Greedy Algorithm. These algorithms are briefly 2010). Real life application areas of TSP include problems work are Nearest Neighbour from Both End Points (NND) want to find a minimum cost Hamiltonian cycle (Davendra, Heuristic algorithms used to solve the TSP problems in this 2010). Real life application areas of TSP include problems work are Nearest Neighbour from Both End Points (NND) that arise in logistics, telecommunication networks, data and Greedy Algorithm. These algorithms are briefly reviewed analysis, computational biochemistry, artificial intelligence, as follows. that in logistics, telecommunication networks, data and Greedy Algorithm. These 2010).arise Real application areas of TSP include problems work are Nearest Neighbour from Both are Endbriefly Pointsreviewed (NND) that arise inlife logistics, telecommunication networks, data as andfollows. Greedy Algorithm. These algorithms algorithms are briefly reviewed analysis, computational biochemistry, artificial intelligence, and so on. analysis, computational biochemistry, artificial intelligence, as follows. that arise in logistics, telecommunication networks, data and Greedy Algorithm. These algorithms are briefly reviewed analysis, and so on.computational biochemistry, artificial intelligence, as follows. and so analysis, biochemistry, artificial intelligence, 2.1follows. Nearest Neighbour Algorithm and so on. on. There arecomputational many variations of TSP: Symmetric TSP, as 2.1 Nearest Neighbour Algorithm and so on. There are many variations of TSP: Symmetric TSP, Asymmetric TSP, The MAX TSP, The MIN TSP, TSP TSP, with 2.1 Nearest Neighbour Algorithm There are many variations of TSP: Symmetric 2.1 Nearest Neighbour Algorithm There are many variations of a The TSP:MIN Symmetric TSP, Asymmetric TSP, The MAX TSP, TSP, TSP with This is perhaps the simplest and most straightforward TSP multiple visits (TSPM), TSP with closed tour, TSP with an 2.1 Nearest Neighbour Algorithm Asymmetric TSP, The MAX TSP, The MIN TSP, TSP with There are many variations of TSP: Symmetric TSP, This is perhaps the simplest and most Asymmetric TSP, The MAX TSP, The MIN TSP, TSP with multiple visits (TSPM), TSP with a closed tour, TSP with an heuristic. Always visit the nearest citystraightforward is the key to TSP this open tourvisits andTSP, etc. The (Gutin andwith Punnen, 2002). InTSP this paper, This is perhaps the simplest and most straightforward multiple (TSPM), TSP a closed tour, with an Asymmetric MAX TSP, The MIN TSP, TSP with This is perhaps the simplest and most straightforward TSP heuristic. Always visit the nearest city is the key to TSP this multiple visits (TSPM), TSP with a closed tour, TSP with an open tour and etc. (Gutin and Punnen, 2002). In this paper, algorithm (Appligate et. al., 2006). we consider the symmetric TSP. heuristic. Always the nearest city is this This is perhaps simplest and most open tour and etc. (Gutin and Punnen, 2002). this paper, multiple TSP with a closed tour,In an algorithm heuristic. Alwaysthevisit visit the 2006). nearest citystraightforward is the the key key to to TSP this (Appligate et. al., open tourvisits and etc. (Gutin and Punnen, 2002). InTSP thiswith paper, we consider the(TSPM), symmetric TSP. algorithm et. al., heuristic. (Appligate Always visit the 2006). nearest city is the key to this we consider the symmetric TSP. open tour and etc. (Gutin and Punnen, 2002). In this paper, algorithm (Appligate et. al., 2006). we consider the symmetric TSP. algorithm (Appligate et. al., 2006). we consider the symmetric TSP. Federation of Automatic Control) Copyright © 2018, 2018 IFAC 268Hosting 2405-8963 © IFAC (International by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 268Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 268 Copyright © 2018 IFAC 268 10.1016/j.ifacol.2018.11.299 Copyright © 2018 IFAC 268
IFAC TECIS 2018 Baku, Azerbaidschan, Sept 13-15, 2018
Urfat Nuriyev et al. / IFAC PapersOnLine 51-30 (2018) 268–270
The steps of the algorithm are as following:
269
Part II. In the second part of the algorithm, in order to improve the existing solution an iteration algorithm is used: Update priority values of the edges and then sort them in descending order. After that, run Greedy algorithm.
Step 1. Select a random city. Step 2. Find the nearest unvisited city and go there.
At any step of the iteration, if the best solution is repeated several times or after a certain number of steps, the algorithm ends. Steps of the algorithm is as following:
Step 3. Are there any unvisited cities left? If yes, go to Step 2. Step 4. Return to the first city.
G V , E ,
Let
We can obtain the best result out of this algorithm by starting the algorithm over again for each vertex and repeat it for n times.
V n, E m
Let u 2.2 The Nearest Neighbour Algorithm from Both End Points (NND)
uj
The algorithm starts with a vertex chosen randomly in the graph. Then, the algorithm continues with the nearest unvisited vertex to the starting vertex. We will have two end vertices. We add a vertex to the tour such that this vertex has not visited before and it is the nearest vertex to these two end vertices. We update the end vertices. The algorithm ends after visiting all vertices (Kizilates and Nuriyeva, 2013).
(1)
j
V v1 , v 2 , , v n , n n 1 2
E e1 , e 2 , , e m ,
.
j 1, 2 , , m be a priority value for the edge e j .
dd d ( e j )
, where dd is the length of the longest edge.
Let us show the best solution which is found with X and the value of the objective function with F ( X ) . The steps of the proposed algorithm are as following. Step 1. i 1 . Step 2. Sort the edges in ascending order.
The steps of the algorithm are as following:
Step 3. Run NND algorithm by starting from the any vertex of the edge with a smallest weight.
Step 1. Choose an arbitrary vertex in the graph.
Let us show founded solution with X 1 and the appropriate value of the objective function with F 1 .
Step 2. Visit the nearest unvisited vertex to this vertex. Step 3. Visit the nearest unvisited vertex to these two vertices and update the end vertices.
X X
1
and F F 1 . The priority values are updated as
below:
Step 4. Is there any unvisited vertex left? If yes, go to Step 3. Step 5. Go to the end vertex from the other end vertex.
uj
(1)
2.3 Greedy Algorithm
u (1) 1, j (1) , u j
if the edge e j is in X
(i)
otherwise
Step 4. i i 1 .
Greedy heuristic gradually constructs a tour by repeatedly selecting the shortest edge and adding it to the tour as long as it does not create a cycle with less than N edges, or increase the value of any node by more than 2. We must not add the same edge twice of course (Appligate et. al., 2006). The steps of the algorithm are as following:
Step 5. The NND algorithm is used by starting from any vertex of the edge which is added to the solution at the end in Step 3. X i and F i are determined and parameter t is calculated by the formula t F F i . The priority values are updated as below:
Step 1. Sort all edges in descending order.
Step 5.1. If t 1 then
Step 2. Select the shortest edge and add it to our tour if it does not violate any of the above constraints.
uj
Step 3. Do we have n edges in our tour? If no, go to Step 2.
(i )
X X
u ( i 1) t , j ( i 1) u , j
i
and
F F
i
if the edge e j is in X
(i)
otherwise
Step 6. i i 1 .
3. THE PROPOSED ALGORITHM
Step 7. The NND algorithm is used by starting from other vertex of the edge, which is added to the solution at the end in Step 3. Parameters X i , F i and t are determined. The priority values are updated as below:
The proposed algorithm consist of the following parts: Part I. In the first part of the algorithm, the priority values of the edges are determined and initial solution is found. Then NND algorithm is used from selected vertices and the priority values of the edges are updated by considering how many times an edge is used in a solution. 269
IFAC TECIS 2018 270 Baku, Azerbaidschan, Sept 13-15, 2018
Step 7.1 If t 1 then
uj
(i )
X X
u ( i 1) t , j ( i 1) , u j
i
and
Urfat Nuriyev et al. / IFAC PapersOnLine 51-30 (2018) 268–270
F F
i
Table 1. Computational Experiments
if the edge e j is in X
(i)
otherwise
Step 8. i i 1 . Step 9. Sort edges in descending order by priority values. Then, run Greedy algorithm and find X i , F i and t . The priority values are updated as below: Step 9.1 If t 1 then
uj
(i )
u ( i 1) 1, j ( i 1) , u j
Step 9.2 If t 1 then
uj
(i )
u ( i 1) t , j ( i 1) u , j
if the edge e j is in X
(i)
or X
otherwise
X X
i
and
F F
i
if the edge e j is not in X
(i)
uj
NN
Greedy
eil51 berlin52 st70 eil76 rat99 kroA100 kroB100 kroC100 kroD100 kroE100 rd100 eil101 lin105 pr107 ch130 kroA150 kroB150 rat195 kroA200
429.983 7544.365 678.597 545.387 1211 21236.951 22141 20750.762 21294.290 22068 7910.396 642.309 14382.995 44303 6110.860 26524 26130 2323 29368
505.774 8182.192 761.689 612.656 1369.535 24698.497 25882.973 23566.403 24855.799 24907.022 9427.333 736.368 16939.441 46678.154 7198.741 31482.020 31320.340 2628.561 34547.691
481.518 9954.062 746.044 617.131 1528.308 24197.285 25815.214 25313.671 24631.533 24420.355 8702.605 789.112 16479.785 48261.816 7142.045 31442.994 31519.083 2957.176 37650.812
Proposed Algorithm 470.330 7959.568 735.852 611.418 1309.027 23438.084 24296.193 22686.925 24395.024 24377.734 8492.701 731.161 15275.108 46304.295 6751.505 29001.383 28588.199 2448.075 33285.976
REFERENCES
otherwise
u ( i 1) t , j ( i 1) , u j
Optimal
or X Appligate, D.L., Bixby, R.E., Chavatal, V., Cook, W.J., (2006). The Travelling Salesman Problem, A Computational Study, Princeton University Press, Princeton and Oxford, 593p. Davendra, D. (2010). Travelling Salesman Problem. Theory and Applications, InTech. Gutin, G., Punnen, A. (eds.). (2002). The Traveling Salesman Problem and Its Variations, Combinatorial Optimization, vol. 12, Kluwer, Dordrecht. Hubert, L.J., Baker, F.B. (1978). Applications of Combinatorial Programming to Data Analysis. The Traveling Salesman and Related Problems. Psychometrika, 43(1), pp. 81–91. Johnson, D.S., McGeoch, L.A. (1997). The Traveling Salesman Problem: A Case Study, Local Search in Combinatorial Optimization, John Wiley & Sons., pp. 215–310. Kızılateş G., Nuriyeva F. (2013). On the Nearest Neighbour Algorithms for Travelling Salesman Problem. Advances in Computational Science. Engineering and Information Technology, Springer, pp. 111-118. Lawler, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G., Shmoys, D.B. (1986). The Travelling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley & Sons. Lenstra, J., Kan, A.R. (1975). Some simple applications of the travelling salesman problem. Journal of the Operational Research Society, 26(4), pp. 717–733. TSPLIB95:https://www.iwr.uni-heidelberg.de/groups/comopt /software/TSPLIB95/
Step 9.3 If t 1 then (i )
G
if the edge e j is X and not in X
(i)
otherwise
Step 10. If i m then go to Step 8. Step 11. Print X , F . Step 12. STOP. 4. COMPUTATIONAL EXPERIMENTS This section presents the results of the computational experiments for the proposed iterative algorithm. The computer program of the proposed algorithm has been coded in C++. The sample problems and the optimum solutions for each of these problems are taken from (https://www.iwr.uniheidelberg.de/groups/comopt/software/TSPLIB95). The proposed algorithm has been compared with NN and Greedy algorithms. Table 1 shows the length of the tour computed by NN, Greedy heuristic and the proposed iterative heuristic algorithm. In Table 1, selected cells show the best results that algorithms have found. 5. CONCLUSIONS In this paper, we have proposed a self-organizing iterative heuristic algorithm for solving TSP. The proposed algorithm based on Nearest Neighbour Algorithm from Both End Points (NND) and Greedy Algorithm. As it is seen in Table 1, comparing with Nearest Neighbour and Greedy Algorithm, the proposed algorithm gives far better solutions.
270
ScienceDirect
IFAC PapersOnLine 51-30 (2018) 268–270 Self-Organizing Iterative Algorithm for Travelling Salesman Problem Self-Organizing Iterative Algorithm for Travelling Salesman Problem Self-Organizing Iterative Algorithm for Travelling Salesman Self-Organizing Iterative Urfat Algorithm Salesman Problem Problem Nuriyev*.for OnurTravelling Ugurlu** Self-Organizing Iterative Urfat Algorithm for Travelling Salesman Problem , Nuriyev*. Onur Ugurlu**
Fidan Nuriyeva*** ,**** Urfat Nuriyev*. Onur Ugurlu** Urfat Nuriyev*. Fidan Nuriyeva*** Onur Ugurlu** ,**** ,**** Fidan Nuriyeva*** Urfat Nuriyev*. Onur Ugurlu** Fidan Nuriyeva*** ,**** * Faculty of Science, Department Mathematics, Ege University, Izmir, Turkey **** Fidan of Nuriyeva*** * Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey Faculty of Science, Department of Mathematics, Ege (e-mail: [email protected]) ** Faculty of Science, Department of Mathematics, EgeUniversity, University,Izmir, Izmir,Turkey Turkey (e-mail: [email protected]) * Faculty of Science, Department of Mathematics, Ege University, Izmir, (e-mail: [email protected]) ** Faculty of Science,(e-mail: Department of Mathematics, Ege University, Izmir,Turkey Turkey [email protected]) ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) *** Faculty of Science, Department [email protected]) Computer Science, Dokuz Eylul University, Izmir, Turkey (e-mail: ** Faculty of Science, Department of Mathematics, Ege University, Izmir, Turkey (e-mail: [email protected]) *** Faculty of Science, Department of Computer Science, Dokuz Eylul University, Izmir, Turkey (e-mail:[email protected]) *** Department of Computer Science, Dokuz Eylul University, Izmir, (e-mail: [email protected]) *** Faculty Faculty of of Science, Science, Department of Computer Science, Dokuz Eylul University, Izmir, Turkey Turkey (e-mail:[email protected]) **** Institute of Control Systems of ANAS, Baku, Azerbaijan (e-mail:[email protected]) *** Faculty of Science, Department of Computer Science, Dokuz Eylul University, Izmir, Turkey **** Institute(e-mail:[email protected]) of Control Systems of ANAS, Baku, Azerbaijan **** of **** Institute Institute(e-mail:[email protected]) of Control Control Systems Systems of of ANAS, ANAS, Baku, Baku, Azerbaijan Azerbaijan **** Institute of Control Systems of ANAS, Baku,algorithm Azerbaijanfor solving Travelling Abstract: This paper presents a self-organizing heuristic iterative Abstract: This paper presents a self-organizing heuristic iterative algorithm for solving Travelling Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied Abstract: This paper presents aa self-organizing heuristic iterative algorithm for solving Travelling Abstract: This paper presents self-organizing heuristic iterative algorithm for solving Travelling Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied solutions are found. After that, the priority values of the edges are updated according to these solutions. Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied Abstract: This paper presents a self-organizing iterative algorithm for solving Travelling Salesman Problem. In the algorithm, firstlyvalues priority values thepriority edges are determined. Then, varied solutions are found. After that, priority ofheuristic the edges are updated according these Then, all the edges are sorted in the descending order according toof the values and into order tosolutions. improve solutions are found. After that, the priority values of the edges are updated according to these solutions. Salesman Problem. In the algorithm, firstly priority values of the edges are determined. Then, varied solutions are found. After that, the priority values of the edges are updated according to these solutions. Then, all the edges arealgorithm sorted inisdescending order according to thetopriority values andinineach orderiteration to improve solutions an iterative used. Greedy algorithm is used find the solution and Then, all the edges sorted in descending order according to the priority values and in order to improve solutions areiterative found.are After that, the priority values of the edges are updated according to these solutions. Then, all the edges are sorted in descending order according to the priority values and in order to improve solutions an algorithm is used. Greedy algorithm is used to find the solution in each iteration and the priority values of the edges are updated according to the solution. The proposed algorithm is solutions an iterative algorithm used. Greedy algorithm is used find solution in iteration and Then, all the edges are sorted inis order according to the theto values and orderalgorithm to improve solutions an iterative algorithm isdescending used. Greedy algorithm isto used topriority find the the solution inineach each iteration and the priority values of the edges are updated according solution. The show proposed is compared with Nearest Neighbour and Greedy algorithm. Experimental results that the proposed the priority values of the edges are according the The proposed algorithm is solutions an iterative algorithm is used. Greedy used find the solution in each iteration and the priority values of the edges are updated according to thetosolution. solution. The show proposed algorithm is compared Nearest Neighbour andupdated Greedyalgorithm algorithm.isto Experimental results that the proposed algorithm iswith efficient. compared with Nearest Neighbour and Greedy algorithm. Experimental results show that the proposed the priority values of the edges are updated according to the solution. The proposed algorithm is compared iswith Nearest Neighbour and Greedy algorithm. Experimental results show that the proposed algorithm efficient. algorithm is efficient. compared with Nearest Neighbour and Greedy algorithm. Experimental results show that the proposed © 2018, IFAC (International Federation of Automatic Control) Hosting by ElsevierNP-Hard Ltd. All rights reserved.selfalgorithm is heuristics, efficient. Keywords: artificial intelligence, combinatorial optimization, problems, algorithm isalgorithms. efficient. artificial intelligence, combinatorial optimization, NP-Hard problems, selfKeywords: heuristics, organizing Keywords: algorithms. heuristics, artificial artificial intelligence, intelligence, combinatorial combinatorial optimization, optimization, NP-Hard NP-Hard problems, problems, selfselfKeywords: heuristics, organizing organizing algorithms. Keywords: heuristics, artificial intelligence, combinatorial optimization, NP-Hard problems, self organizing algorithms. organizing algorithms. In this paper, a new self-organizing iterative algorithm is 1. INTRODUCTION In this paper, new the self-organizing iterative algorithm is proposed to aasolve TSP problem. Two heuristic 1. INTRODUCTION In this paper, new self-organizing iterative algorithm is In this paper, a new self-organizing iterative algorithm is proposed to solve the TSP problem. Two heuristic INTRODUCTION algorithms, Nearest Neighbour (NN) and Greedy Algorithm 1. INTRODUCTION Traveling Salesman 1. Problem (TSP) is perhaps the most well - In proposed to solve the TSP problem. Two heuristic this paper, a new self-organizing iterative algorithm is proposed to solve the TSP problem. Two heuristic algorithms, Nearest Neighbour (NN) and Greedy Algorithm INTRODUCTION Traveling Salesman 1. Problem (TSP)problem is perhaps are conducted against 19TSP benchmark real world TSP known combinatorial optimization in the themost set ofwell NP--- proposed algorithms, Nearest Neighbour (NN) and Greedy Algorithm to solve the problem. Two heuristic Traveling Salesman Problem (TSP) is perhaps the most well algorithms, Nearest Neighbour (NN) and Greedy Algorithm are conducted against 19 benchmark real world TSP Traveling Salesman (TSP) is perhaps most well known combinatorial optimization problem inofthe the setisofthat NPcollected in TSPLIB95. Results obtained by those Hard (Lawler et al., Problem 1986). The significance TSP it-- problems are conducted against 19 benchmark real world TSP algorithms, Nearest Neighbour (NN) and Greedy Algorithm known combinatorial optimization problem in the set of NPTraveling Salesman Problem (TSP) is perhaps the most well are conducted against 19 benchmark real world TSP problems collected in TSPLIB95. Results obtained by those known combinatorial optimization problem in the set of NPHard (Lawler et al., 1986). The significance of TSP is that it algorithms compared with exactly optimal solutions. includes manyet problems that areproblem naturalinof applications in problems collected in TSPLIB95. Results obtained by those are conducted against 19 benchmark real world TSP Hard (Lawler al., 1986). The significance TSP is that it known combinatorial optimization the set of NPproblems collected TSPLIB95. Results solutions. obtained by those algorithms comparedinwith exactly optimal Hard (Lawler et al., 1986). The significance of TSP is that it includes many problems that are natural applications in computer science and engineering (Appligate et al., 2006; algorithms compared exactly optimal problems collected inwith TSPLIB95. Results solutions. obtained by those includes many problems that are natural applications in Hard (Lawler et al., 1986). The significance of TSP is that it algorithms compared with exactly optimal solutions. includes many that are (Appligate natural applications in computer science and engineering et al., 2006; 2. APPROACHES FORoptimal SOLVING TSP Lenstra and Kan,problems 1975). with exactly solutions. computer science and et includes many that are (Appligate natural applications in algorithms2.compared APPROACHES FOR SOLVING TSP computer science and engineering engineering (Appligate et al., al., 2006; 2006; Lenstra and Kan,problems 1975). 2. APPROACHES FOR SOLVING TSP Lenstra and Kan, computer engineering (Appligate 2006; By literature, many algorithms approaches 2. APPROACHES FORand SOLVING TSPhave been Lenstra andscience Kan, 1975). The problem is 1975). toandfind an optimal tour foreta al., traveling By literature, many algorithms and approaches 2. APPROACHES FOR SOLVING TSPhave been Lenstra and Kan, 1975). proposed to solve the TSP problem. These algorithms and The problem is to find an optimal tour for a traveling literature, many algorithms and approaches have salesman wishing to find visit an eachoptimal of a list of nforcities exactly By The problem is to tour a traveling By literature, many algorithms and approaches have been been proposed to solve the TSP problem. These algorithms and approaches can be classified into Heuristic Algorithms, The problem is to find an optimal tour for a traveling salesman wishing to visit each of a list of n cities exactly proposed to solve the TSP problem. These algorithms and literature, many algorithms and approaches have been once and then return to thean home city. Such optimal tour is By salesman wishing to visit each of a list of n cities exactly proposed to solve the TSP problem. These algorithms and approaches can be classified into Heuristic Algorithms, The problem is to find optimal tour for a traveling Algorithms, Approximate Algorithms and salesman wishing towhose visit of a listSuch of noptimal cities exactly once andtothen return to theeach home city. tour is Metaheuristic approaches can be classified into Heuristic Algorithms, proposed to solve the TSP problem. These algorithms defined be a tour total distance (cost) is minimized once and then return to the home city. Such optimal tour is approaches can be classified into Heuristic Algorithms, Metaheuristic Algorithms, Approximate Algorithms and and salesman wishing towhose visit each of a listSuch of noptimal cities exactly Exact Algorithms (Lawler et. Al., 1985). once and then return to the home city. tour is defined to be a tour total distance (cost) is minimized Metaheuristic Algorithms, Approximate Algorithms approaches can be classified into Heuristic Algorithms, (Appligate et al., 2006). defined be aareturn tour whose distance (cost) is Algorithms, Algorithms and and Exact Algorithms (Lawler et. Approximate Al., 1985). once andto to thetotal home city. Such tour is Metaheuristic defined tothen be tour whose total distance (cost)optimal is minimized minimized (Appligate et al., 2006). Exact Algorithms (Lawler et. Al., 1985). Metaheuristic Algorithms, Algorithms TSP problem possessing noApproximate longer than 20 cities can and be (Appligate et al., 2006). Exact Algorithms (Lawler et. Al., 1985). defined to be a tour whose total distance (cost) is minimized (Appligate et al., 2006). of TSP has been firstly lunched Exact The general formulation TSP problem possessing no longer than heuristic 20 citiesmethods can be Algorithms (Lawler et. Al., 1985). optimally solved by exact methods. The (Appligate et al., 2006). The general formulation of TSP has been firstly lunched problem possessing longer than 20 cities can based on the graph theory of in 1930s. Problem can be lunched defined TSP TSP possessing no longer The than 20 TSP citiesproblems can be be optimally solved byquality exactno methods. heuristic methods The general formulation TSP has been firstly could problem provide high solutions for the The general formulation TSP with has been firstly based on the and graph theory of ingraph 1930s. Problem can(cities). be lunched defined optimally solved by exact methods. heuristic methods TSP problem possessing no longer The than 20 TSP citiesproblems can be as a weighed connected n vertices We optimally solved by exact methods. The heuristic methods could provide high quality solutions for the based on the graph theory in 1930s. Problem can be defined The general formulation of TSP has been firstly lunched possessing large amount number of cities. However, the based the aand graph theoryone ingraph 1930s. Problem can bealldefined as a weighed connected with n vertices (cities). could provide high quality solutions for the TSP problems solved by exact number methods. heuristic methods want toon tour from of these vertices, visit of We the optimally could provide high quality solutions for the TSP problems possessing large amount ofThe cities. However, the as aa weighed connected with nn vertices (cities). We based onstart the aand graph theoryone ingraph 1930s. Problem can bealldefined optimum solution can not be guaranteed (Johnson and as weighed and connected graph with vertices (cities). We want to start tour from of these vertices, visit of the possessing large amount number of cities. However, the could provide high quality solutions for the TSP problems other vertices for exactly one time and finally return to start possessing large amount number of cities. However, the optimum solution can not be guaranteed (Johnson and want to start aaand tour from one of these vertices, visit all of the as a weighed connected graph with nfinally vertices (cities). We McGeoch, 1997). want to start tour from one of these vertices, visit all of the other vertices for exactly one time and return to start optimum solution can not be guaranteed (Johnson and possessing large amount number of cities. However, the vertex. This tour called Hamiltonian tour. In the TSP, we optimum solution can not be guaranteed (Johnson and McGeoch, 1997). other vertices for one and finally return to want to start atour tourexactly from one oftime these vertices, visit all ofstart the other vertices for exactly one time and finally return to start vertex. This called Hamiltonian tour. In the TSP, we McGeoch, 1997). optimum solution can not be guaranteed (Johnson and want find atour minimum cost Hamiltonian cycle (Davendra, Heuristic algorithms used to solve the TSP problems in this McGeoch, 1997). vertex. This called In the TSP, we other to vertices for exactly one time andtour. finally to start vertex. This called Hamiltonian Hamiltonian tour. In return the(Davendra, TSP, we McGeoch, want to find life atour minimum cost Hamiltonian cycle Heuristic used to solve TSP in this 1997). Neighbour 2010). Real application areas of TSP include problems work are algorithms Nearest from the Both Endproblems Points (NND) want to find a minimum cost Hamiltonian cycle (Davendra, Heuristic algorithms used to TSP in vertex. This tour called Hamiltonian tour. In the TSP, we want arise toReal findinlife a logistics, minimum costareas Hamiltonian cycle (Davendra, 2010). application of TSP include problems Heuristic used to solve solve the TSP problems in this this work are algorithms Nearest Neighbour from the Both Endproblems Pointsreviewed (NND) that telecommunication networks, data and Greedy Algorithm. These algorithms are briefly 2010). Real life application areas of TSP include problems work are Nearest Neighbour from Both End Points (NND) want to find a minimum cost Hamiltonian cycle (Davendra, Heuristic algorithms used to solve the TSP problems in this 2010). Real life application areas of TSP include problems work are Nearest Neighbour from Both End Points (NND) that arise in logistics, telecommunication networks, data and Greedy Algorithm. These algorithms are briefly reviewed analysis, computational biochemistry, artificial intelligence, as follows. that in logistics, telecommunication networks, data and Greedy Algorithm. These 2010).arise Real application areas of TSP include problems work are Nearest Neighbour from Both are Endbriefly Pointsreviewed (NND) that arise inlife logistics, telecommunication networks, data as andfollows. Greedy Algorithm. These algorithms algorithms are briefly reviewed analysis, computational biochemistry, artificial intelligence, and so on. analysis, computational biochemistry, artificial intelligence, as follows. that arise in logistics, telecommunication networks, data and Greedy Algorithm. These algorithms are briefly reviewed analysis, and so on.computational biochemistry, artificial intelligence, as follows. and so analysis, biochemistry, artificial intelligence, 2.1follows. Nearest Neighbour Algorithm and so on. on. There arecomputational many variations of TSP: Symmetric TSP, as 2.1 Nearest Neighbour Algorithm and so on. There are many variations of TSP: Symmetric TSP, Asymmetric TSP, The MAX TSP, The MIN TSP, TSP TSP, with 2.1 Nearest Neighbour Algorithm There are many variations of TSP: Symmetric 2.1 Nearest Neighbour Algorithm There are many variations of a The TSP:MIN Symmetric TSP, Asymmetric TSP, The MAX TSP, TSP, TSP with This is perhaps the simplest and most straightforward TSP multiple visits (TSPM), TSP with closed tour, TSP with an 2.1 Nearest Neighbour Algorithm Asymmetric TSP, The MAX TSP, The MIN TSP, TSP with There are many variations of TSP: Symmetric TSP, This is perhaps the simplest and most Asymmetric TSP, The MAX TSP, The MIN TSP, TSP with multiple visits (TSPM), TSP with a closed tour, TSP with an heuristic. Always visit the nearest citystraightforward is the key to TSP this open tourvisits andTSP, etc. The (Gutin andwith Punnen, 2002). InTSP this paper, This is perhaps the simplest and most straightforward multiple (TSPM), TSP a closed tour, with an Asymmetric MAX TSP, The MIN TSP, TSP with This is perhaps the simplest and most straightforward TSP heuristic. Always visit the nearest city is the key to TSP this multiple visits (TSPM), TSP with a closed tour, TSP with an open tour and etc. (Gutin and Punnen, 2002). In this paper, algorithm (Appligate et. al., 2006). we consider the symmetric TSP. heuristic. Always the nearest city is this This is perhaps simplest and most open tour and etc. (Gutin and Punnen, 2002). this paper, multiple TSP with a closed tour,In an algorithm heuristic. Alwaysthevisit visit the 2006). nearest citystraightforward is the the key key to to TSP this (Appligate et. al., open tourvisits and etc. (Gutin and Punnen, 2002). InTSP thiswith paper, we consider the(TSPM), symmetric TSP. algorithm et. al., heuristic. (Appligate Always visit the 2006). nearest city is the key to this we consider the symmetric TSP. open tour and etc. (Gutin and Punnen, 2002). In this paper, algorithm (Appligate et. al., 2006). we consider the symmetric TSP. algorithm (Appligate et. al., 2006). we consider the symmetric TSP. Federation of Automatic Control) Copyright © 2018, 2018 IFAC 268Hosting 2405-8963 © IFAC (International by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 268Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 268 Copyright © 2018 IFAC 268 10.1016/j.ifacol.2018.11.299 Copyright © 2018 IFAC 268
IFAC TECIS 2018 Baku, Azerbaidschan, Sept 13-15, 2018
Urfat Nuriyev et al. / IFAC PapersOnLine 51-30 (2018) 268–270
The steps of the algorithm are as following:
269
Part II. In the second part of the algorithm, in order to improve the existing solution an iteration algorithm is used: Update priority values of the edges and then sort them in descending order. After that, run Greedy algorithm.
Step 1. Select a random city. Step 2. Find the nearest unvisited city and go there.
At any step of the iteration, if the best solution is repeated several times or after a certain number of steps, the algorithm ends. Steps of the algorithm is as following:
Step 3. Are there any unvisited cities left? If yes, go to Step 2. Step 4. Return to the first city.
G V , E ,
Let
We can obtain the best result out of this algorithm by starting the algorithm over again for each vertex and repeat it for n times.
V n, E m
Let u 2.2 The Nearest Neighbour Algorithm from Both End Points (NND)
uj
The algorithm starts with a vertex chosen randomly in the graph. Then, the algorithm continues with the nearest unvisited vertex to the starting vertex. We will have two end vertices. We add a vertex to the tour such that this vertex has not visited before and it is the nearest vertex to these two end vertices. We update the end vertices. The algorithm ends after visiting all vertices (Kizilates and Nuriyeva, 2013).
(1)
j
V v1 , v 2 , , v n , n n 1 2
E e1 , e 2 , , e m ,
.
j 1, 2 , , m be a priority value for the edge e j .
dd d ( e j )
, where dd is the length of the longest edge.
Let us show the best solution which is found with X and the value of the objective function with F ( X ) . The steps of the proposed algorithm are as following. Step 1. i 1 . Step 2. Sort the edges in ascending order.
The steps of the algorithm are as following:
Step 3. Run NND algorithm by starting from the any vertex of the edge with a smallest weight.
Step 1. Choose an arbitrary vertex in the graph.
Let us show founded solution with X 1 and the appropriate value of the objective function with F 1 .
Step 2. Visit the nearest unvisited vertex to this vertex. Step 3. Visit the nearest unvisited vertex to these two vertices and update the end vertices.
X X
1
and F F 1 . The priority values are updated as
below:
Step 4. Is there any unvisited vertex left? If yes, go to Step 3. Step 5. Go to the end vertex from the other end vertex.
uj
(1)
2.3 Greedy Algorithm
u (1) 1, j (1) , u j
if the edge e j is in X
(i)
otherwise
Step 4. i i 1 .
Greedy heuristic gradually constructs a tour by repeatedly selecting the shortest edge and adding it to the tour as long as it does not create a cycle with less than N edges, or increase the value of any node by more than 2. We must not add the same edge twice of course (Appligate et. al., 2006). The steps of the algorithm are as following:
Step 5. The NND algorithm is used by starting from any vertex of the edge which is added to the solution at the end in Step 3. X i and F i are determined and parameter t is calculated by the formula t F F i . The priority values are updated as below:
Step 1. Sort all edges in descending order.
Step 5.1. If t 1 then
Step 2. Select the shortest edge and add it to our tour if it does not violate any of the above constraints.
uj
Step 3. Do we have n edges in our tour? If no, go to Step 2.
(i )
X X
u ( i 1) t , j ( i 1) u , j
i
and
F F
i
if the edge e j is in X
(i)
otherwise
Step 6. i i 1 .
3. THE PROPOSED ALGORITHM
Step 7. The NND algorithm is used by starting from other vertex of the edge, which is added to the solution at the end in Step 3. Parameters X i , F i and t are determined. The priority values are updated as below:
The proposed algorithm consist of the following parts: Part I. In the first part of the algorithm, the priority values of the edges are determined and initial solution is found. Then NND algorithm is used from selected vertices and the priority values of the edges are updated by considering how many times an edge is used in a solution. 269
IFAC TECIS 2018 270 Baku, Azerbaidschan, Sept 13-15, 2018
Step 7.1 If t 1 then
uj
(i )
X X
u ( i 1) t , j ( i 1) , u j
i
and
Urfat Nuriyev et al. / IFAC PapersOnLine 51-30 (2018) 268–270
F F
i
Table 1. Computational Experiments
if the edge e j is in X
(i)
otherwise
Step 8. i i 1 . Step 9. Sort edges in descending order by priority values. Then, run Greedy algorithm and find X i , F i and t . The priority values are updated as below: Step 9.1 If t 1 then
uj
(i )
u ( i 1) 1, j ( i 1) , u j
Step 9.2 If t 1 then
uj
(i )
u ( i 1) t , j ( i 1) u , j
if the edge e j is in X
(i)
or X
otherwise
X X
i
and
F F
i
if the edge e j is not in X
(i)
uj
NN
Greedy
eil51 berlin52 st70 eil76 rat99 kroA100 kroB100 kroC100 kroD100 kroE100 rd100 eil101 lin105 pr107 ch130 kroA150 kroB150 rat195 kroA200
429.983 7544.365 678.597 545.387 1211 21236.951 22141 20750.762 21294.290 22068 7910.396 642.309 14382.995 44303 6110.860 26524 26130 2323 29368
505.774 8182.192 761.689 612.656 1369.535 24698.497 25882.973 23566.403 24855.799 24907.022 9427.333 736.368 16939.441 46678.154 7198.741 31482.020 31320.340 2628.561 34547.691
481.518 9954.062 746.044 617.131 1528.308 24197.285 25815.214 25313.671 24631.533 24420.355 8702.605 789.112 16479.785 48261.816 7142.045 31442.994 31519.083 2957.176 37650.812
Proposed Algorithm 470.330 7959.568 735.852 611.418 1309.027 23438.084 24296.193 22686.925 24395.024 24377.734 8492.701 731.161 15275.108 46304.295 6751.505 29001.383 28588.199 2448.075 33285.976
REFERENCES
otherwise
u ( i 1) t , j ( i 1) , u j
Optimal
or X Appligate, D.L., Bixby, R.E., Chavatal, V., Cook, W.J., (2006). The Travelling Salesman Problem, A Computational Study, Princeton University Press, Princeton and Oxford, 593p. Davendra, D. (2010). Travelling Salesman Problem. Theory and Applications, InTech. Gutin, G., Punnen, A. (eds.). (2002). The Traveling Salesman Problem and Its Variations, Combinatorial Optimization, vol. 12, Kluwer, Dordrecht. Hubert, L.J., Baker, F.B. (1978). Applications of Combinatorial Programming to Data Analysis. The Traveling Salesman and Related Problems. Psychometrika, 43(1), pp. 81–91. Johnson, D.S., McGeoch, L.A. (1997). The Traveling Salesman Problem: A Case Study, Local Search in Combinatorial Optimization, John Wiley & Sons., pp. 215–310. Kızılateş G., Nuriyeva F. (2013). On the Nearest Neighbour Algorithms for Travelling Salesman Problem. Advances in Computational Science. Engineering and Information Technology, Springer, pp. 111-118. Lawler, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G., Shmoys, D.B. (1986). The Travelling Salesman Problem: A Guided Tour of Combinatorial Optimization, John Wiley & Sons. Lenstra, J., Kan, A.R. (1975). Some simple applications of the travelling salesman problem. Journal of the Operational Research Society, 26(4), pp. 717–733. TSPLIB95:https://www.iwr.uni-heidelberg.de/groups/comopt /software/TSPLIB95/
Step 9.3 If t 1 then (i )
G
if the edge e j is X and not in X
(i)
otherwise
Step 10. If i m then go to Step 8. Step 11. Print X , F . Step 12. STOP. 4. COMPUTATIONAL EXPERIMENTS This section presents the results of the computational experiments for the proposed iterative algorithm. The computer program of the proposed algorithm has been coded in C++. The sample problems and the optimum solutions for each of these problems are taken from (https://www.iwr.uniheidelberg.de/groups/comopt/software/TSPLIB95). The proposed algorithm has been compared with NN and Greedy algorithms. Table 1 shows the length of the tour computed by NN, Greedy heuristic and the proposed iterative heuristic algorithm. In Table 1, selected cells show the best results that algorithms have found. 5. CONCLUSIONS In this paper, we have proposed a self-organizing iterative heuristic algorithm for solving TSP. The proposed algorithm based on Nearest Neighbour Algorithm from Both End Points (NND) and Greedy Algorithm. As it is seen in Table 1, comparing with Nearest Neighbour and Greedy Algorithm, the proposed algorithm gives far better solutions.
270