SOME NEW RESULTS ON THE TRAVELLING SALESMAN PROBLEM

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1 Category: prelimiary commuicatio Domiika Crjac Milić 1 Ljiljaka Kvesić 2 SOME NEW RESULTS ON THE TRAVELLING SALESMAN PROBLEM Abstract: The travellig salesma problem (or The sales represetative problem) has bee isufficietly explored so far. Oe of the first results o this issue was provided by Euler i 1759 (The problem of movig a kight o the chess board), Kight's Tour Problem. Papers o this subject were writte by A.T. Vadermode (1771), T. P. Kirkma (1856) ad may others. The sales represetative problem is a major challege due to the applicatio i solvig theoretical ad practical problems such as the quality of algorithms ad of optimizatio methods. This well-kow optimizatio problem has bee extesively studied from several aspects sice 193. I geeral form the study was started by Karl Meger, seekig the shortest route through all poits of a fiite set with kow distaces betwee every two poits. Sice the, there have bee may formulatios of the problem. this paper we shall provide a aalysis of the ature of the commercial represetative problem, ad highlight its complexity ad some ways of its solutio. We shall use graph theory, ad pay particular attetio to the search of Hamiltoia cycle of miimum weight i the weighted graph. Durig the paper developmet we were led by the followig questio: "How to miimize the total distace travelled by a sales represetative i order to visit give locatios exactly oce ad retur to the startig poit?" Keywords: graph, top, edge, trail, algorithm, cycle Author s data: 1 Domiika Crjac Milić, Associate Professor, Faculty of Electrical Egieerig, J.J Strossmayer Uiversity of Osek, Keza Trpimira 2B, 31 Osek, Croatia, domiika.crjac@etfos.hr; 2 Ljiljaka Kvesić, Assistat Professor, Faculty of Sciece ad Educatio, Uiversity of Mostar, 88 Mostar, Bosia ad Herzegovia, ljkvesic@gmail.com Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 UDK : ; DOI IJVA

2 34 Itroductio Although the problem of sales represetative is a well-kow optimizatio problem ad so far it has bee studied from several aspects [6], i this research paper, usig graph theory, we are goig to give our attetio to the search of Hamiltoia cycle of miimum weight i the weighted graph. Lookig at the problem from a ecoomic aspect, the target of the compaies i sellig busiess is to aimate potetial customers withi as short as possible time, o a daily basis ad with the least possible costs, to buy their products. For plaig daily activities of their sales represetatives, it is importat to miimize the total distace that a represetative should pass i order to visit give locatios exactly oce ad retur to the startig poit. The umber of locatios to visit i oe period of time is kow, the distace betwee the locatios visited is give (costat). Costs required for tours, ad time lapse of each tour are iclied to a costat. Suppose that a sales represetative ca pass oly oce through oe place. The problem ca be mathematically formulated as follows: x where 1, usig path i, j, (1), otherwise x are decisio variables (the j -th job is performed or ot performed i the i -th positio), ad x are path costs (or time) spet o the i-th site of the cliet tour i, j. Target Fuctio is mi z c x (2) i1 j1 with restrictios i1 x 1, j 1,2,..., (3) x 1, i 1,2,..., (4) j1 x, x x... x m1, m 2,..., -1 (5) j1 j2 j2 j3 jm j1 Omittig the last restrictio, we obtai a assigmet problem [5]. Defiitios ad Basic Cocepts Defiitio G, where A graph is a ordered triplet V, E, V V G is a oempty set whose elemets are called vertices, E E G is a set whose elemets are called edges ad it is disjuctive with V, ad is a u, v to each edge e fuctio which coects from E, where u, v V. Furthermore, the vertices u, v are said to be adjacet if there is a edge e whose eds are u ad v. The edge e is icidet to vertices u ad v e u, v or e u v. The degree with labeled of vertex v i the graph is the umber of graph G edges icidetal with v. A walk is the sequece W v e v e... v e whose members are k k alterately vertices v ad edges e i the way that i i the eds of e are vertices v i v, i i 1 i i 1,2,..., k. The umber k is said to be a walk legth of sequece W, wherei v is the begiig ad vk the ed of the walk W. A walk is closed whe v v. k If all edges are mutually differet, a walk is called a path. If all the vertices are mutually differet, the path is called a route. A cycle is closed route whose vertices, except the ed vertices, are differet from each other. Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 UDK : ; DOI IJVA

3 Note that paths ad cycles cotaiig all the vertices of the graph are iterestig for our problem i the study. Defiitio 2 For a path (cycle) that cotais all the graph vertices is said to be Hamiltoia path (cycle) i the graph. Oe of the most difficult algorithmic problems is to aswer the questio whether the graph has got Hamiltoia cycle or ot, especially Hamiltoia cycle of miimum weight [1]. Theoretically, i the fial graph it is possible to fid a Hamiltoia cycle i a fiite umber of steps. We are iterested i quick ad efficiet algorithms. Therefore, the iterval of algorithm executio is importat. Time algorithm essetiality is observed usig the fuctio f : ( is the set of atural umbers), where f, is the umber of elemetary operatios that are ecessary i algorithm. The time course of the algorithm is measured by the total umber of operatios, such as arithmetic operatios, compariso, etc. Furthermore, if f : is the fuctio, it is said that f O q there are c, such, that for each, f cq, q : q it is said to be upper limit for if ad for f. I the evet that f O q tha algorithm has time complexity Cosequet to O q. q, the most frequetly used algorithms are polyomial algorithms ad expoetial algorithms. A algorithm is polyomial if there is a polyomial solutio for the algorithm. Time for solvig algorithm is icreased very slowly compared to the iput data. It is ucertai that there is polyomial algorithm for the solutio of may problems. Defiitio 3 It is said that the ordered pair G, w is G V, E is a graph ad weighted, wherei w: E is a fuctio, (o-egative real umbers). The umber w e is called the edge weight e. Note that the weight of a give edge could mea ay measure which characterizes a edge i ecoomic terms, such as cost, profit, route legth, etc. Some mathematical models a) The trasport model has a very importat role i the maagemet of supply chais. The task is to miimize the total trasportatio costs ad improve service. Suppose there are m storage areas of ai capacities at i locatios for goods to be trasported to j locatios that have a demad for the goods b. j The task is to miimize trasport costs betwee i place ad j place if the uit cost of trasport betwee the two destiatios c are kow. The quatity to be trasported from place i to place j is x. form mit Mathematical formulatio ca be writte i the m c x, (6) i1 j1 wherei T is the fuctio represetig the total cost, x the quatity to be trasported, ad c the uit cost of trasportatio. 35 Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 UDK :: ; DOI IJVA

4 36 It is ecessary to fid the miimum of a T fuctio, subject to certai restrictios: x a, i 1,2,3..., m (7) i j1 m x b, j 1,2,3..., (8) j i1 x, za i 1,2,... m, j 1,2,..., (9) m a i b (1) j i1 j1 b) Assigmet model is a special case of trasport model. The core of the problem is i the distributio or i the assigmet of tasks ad duties to locatios, people, etc., subject to correspodece. I other words, oe job is assiged to oly oe employee,etc. Each positio ca perform some or all of possible tasks for a certai time (with certai costs). It is ecessary allocate tasks such that every positio performs oly oe job ad the total time (or the total costs) required for the performace of all operatios is miimal. Mathematically, it is a ijective projectio [4]. The goal is to fid the optimum usig a measure of idividual success. Of course, this ca be applied to may maagemet issues i ecoomic practice. The aforemetioed ca be mathematically formulated as follows. A square matrix of A type is give with elemets x i, j 1,2,..., 3. a, for It is ecessary to determie the square matrix with elemets i1 j1 subject to X x x a 1 (11) mit x a x (12) i1 j1 Note that it is useful to defie biary variables. 1, meas that i applicat should be assiged to j job x, i applicat is ot assiged to j job Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 (13) There are several differet methods for solvig this method, out of which we emphasise the method of trasformig the above model i the correspodig etwork model whose solutio is reduced to the problem of determiig the shortest path i the etwork [5]. Three subtypes of the sales represetative problem usually appear i practice: 1) Symmetric sales represetative problem, which was previously described i this paper, ad where there is a udirected weighted graph. 2) Asymmetric sales represetative problem. If there is at least oe pair of places u, v E G the path legth (edge weight) has uequal values, depedig o the directio of the tour; the this is a asymmetric sales represetative problem. I this case, a directed graph is used as a model. Directed graph is usually called digraph ad recorded as ordered triplet D V, A,, wherei V is o-empty set of vertices, amely V Ø. Cross sectio of A ad V sets is empty, amely A ad V are disjuctive. The elemets of A set are called arcs, wherei a fuctio jois u, v, u, v V to a ordered pair of vertices each arc a A [7]. We ca geeralize the aforemetioed to more sales represetatives. Suppose that m umber of sales represetatives departs from the same startig place, visit a set of places ad retur to where they started. It is ecessary to determie the tours for all sales represetatives, so that each place is visited oly oce with miimal total cost. Note that the cost ca be caused by the distace of places, by the time spet travellig, ad by the UDK : ; DOI IJVA

5 cost of trasportatio. For the applicatio purpose we will list some variatios of multiple travellig salesma problem, but they wo't be aalyzed i this paper because of their extesiveess. 1. Let a certai umber of represetatives start off from each of few startig places. After the tour the sales represetatives retur, either each of them i their startig place or i ay of the startig places o the coditio that i the ed the umber of sales represetatives i every startig place is equal as i the begiig. 2. Suppose that the umber of sales represetatives is ot fixed. Out of the available m sales represetatives, we desire to make a selectio of sales represetatives to participate i the tour. Clearly, each sales represetative has her/his ow fixed costs that are take ito cosideratio whe decidig how may sales represetatives to be activated i order to miimize the total cost. 3. Suppose that there is a demad for a sales represetative to visit some places i a give time itervals. This problem occurs i the orgaizatio of air trasport, maritime trasport, trasport by roads ad the like. 4. It is possible to itroduce various restrictios, such as a limited umber of places that a sales represetative should visit, miimum distace, maximum distace etc. Some methods of solvig the travellig salesma problem 1. Exact methods These are the methods that give exact algorithms. Their disadvatage is a prologed performace time. I this place, we might add the method of brachig ad restrictios, where you ca estimate all possible solutios ad reject the adverse oes o the basis of pre-set certai criteria. 2. Approximate methods Algorithms that provide approximate solutios are used here. These are lower time complexity algorithms. a) Nearest eighbour method This method is highly developed ad most simple approximate method. It is about visitig the ext earest eighbourig places which were ot visited. Whe places are visited, it is ecessary to retur to the startig place. b) Greedy algorithms The route is always built by addig the shortest possible edge. Durig this process, either a cycle whose legth is less tha the umber of places, or a vertex with degree higher tha 2 may occur, ie. a place may ot be visited more tha oce, ad the same edge may ot be added multiple times. I addressig the above cocers, the questio is how to fid all the possible tours, compute their legths, ad choose the best of them. I order to simplify the solutio of travellig salesma problem, may approximate methods are developed today ad cotiue to develop, which will provide acceptable solutios. Effective methods have bee developed so far, which eable solvig the problem for a couple of millio places withi quite reasoable amout of time. [2] a) Method of isertig 37 Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 UDK :: ; DOI IJVA

6 38 We start with the shortest tour of give places. Most usually it is a triagle. A follower of the previous place is a added place. It is closest to ay of the precedig visits ad is added to the optimum positio i the tour. The process is repeated util all the places are added. b) Optimizatio method by at coloies Scietists ofte solve complex problems by watchig ad imitatig atural processes. By studyig ad imitatig the movemet of at coloies, we fid solutios to the problems of a small umber of places. Namely, durig the search for ew areas ats leave a trail of pheromoes that leads the other ats to ew places (places of food). Let s observe a group of ats i differet places. They do ot retur to the places where they were, ad they visit all the remaiig places. The at that uses the shortest route leaves the most itese pheromoe trail, iversely proportioal to the legth of path. Naturally, the rest of ats will follow the peak itesity of the pheromoe ad follow its trail. The procedure is repeated util the shortest tour [1]. To improve previously studied approximate algorithms, ad to fid optimal algorithms, the followig ideas are useful: 2- optimal algorithm: We arbitrary choose two edges i the cycle, which we remove ad joi two ewlycreated paths. Coectig is doe so that visitig coditios are maitaied. The tour is still used if it is shorter tha the trasitioal. The process is cotiued util the impossibility of improvemet. 3-optimal algorithm: We arbitrarily choose 3 edges ad remove them. Two ways of recoectig occur by removig those three edges. We choose a path that has a shorter tour. Note that the 3-optimal tour is the two-optimal oe. k -optimal algorithm: This algorithm improves as the previous two do, but the work is doe with k edges, k 3. Of course, the greater k requires more computatioal time. Oe result of the travellig salesma problem optimizatio To fid the shortest path you eed to fid the path of least weight that coects two give vertices u ad v. For the purpose of simplicity, istead of path weight p let s itroduce the term path legth p, wherei p we, ad the least path e p weight u, v is a distace from w to v, which d u, v. is writte I order to fid the shortest path, we provide the followig Algorithm which fids the shortest path u, v ; what is more, all the shortest paths from u to all other vertices i G. Suppose that S V, so that w S, S V S. If p w... wv is the shortest path from u to S, the u S ad a u, u part of p has to be the shortest u, u Out of this,, path. d u v d u u w uv, (14) distace from u to S is d u, S mi d u, u w uv. (15) us, vs Let s start from the set S u ad costruct a icreasig rage of subsets out of Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 UDK : ; DOI IJVA

7 V, S, S1,..., Sv 1, so that i the ed of i - th step, the shortest path from u to all vertices out of Si are kow. The first step is fidig a vertex which is closest to vertex u, which is obtaied by computig, u1 S d u S ad selectig vertex so that, 1 d u, S d u u, (16) as follow out of (14) results that d u, S mi d u, u w uv mi w u v us, vs vs I additio, suppose S1 u, u1 u v, so it is the shortest u, u 1. (17), p1 - path 1 - path. For the purpose of geeralizatio, suppose that Sk u, u1,..., uk ad suppose that the shortest paths u, u k, p,..., 1 pk have already bee determied, the with (15) we, k ca compute d u S ad select vertex u 1 S k that, k 1 d u, S k d u u k such. (18) Notice that accordig to (15), k1, j j k 1 d u u d u u w u u, (19) for some j k, ad so we obtai the shortest u, uk 1 - path by addig edge u j, uk 1 p. j to path Note that i each step these shortest paths together form a coected graph without cycles. These graphs are called trees (wood) ad the previous algorithm is called the process of tree growth [3]. The idea for the previous algorithm origiates from Edsger Dkstra Wybe (1959) who, by meas of his algorithm, maaged to determie the distace of idividual poits (destiatios) to other poits cosidered to be importat, but he did ot maage to determie the shortest distace i this way. Kowig that the task of a sales represetative is to visit some busiess destiatios ad retur after the job, provided that each destiatio was visited exactly oce, the questio is how to make the itierary, ad to travel as short as possible? Accordig to the aforemetioed (the shortest path problem) Hamiltoia cycle of miimum weight, which is called the optimal cycle should be foud i the complete weighted graph. Here we will provide a "approximate" approach, cosistig of fidig a Hamiltoia cycle, ad the search for aother cycle of less weight that is slightly modified. If c v1v 2... v V v1, the, for all i, j, 1 i 1 j we ca fid a ew Hamiltoia cycle c v v... v v v... v v v... v v (2) 1 2 i j j1 i1 j1 j2 V 1 where we removed edges vivi 1, v jv j 1, ad added edges viv j ad vi 1v j 1. Furthermore, if for some i, j, we ca apply i j i j i i j j w v v w v v w v v w v v the the cycle , c is a improvemet of c. (21) By cotiuig likewise we shall reach the cycle that caot be improved aymore by this method. It is clear that the fial cycle is ot optimal. I order to achieve greater accuracy, the procedure ca be repeated several times, startig with differet cycles. Coclusio 39 Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 UDK :: ; DOI IJVA

8 4 This paper is a result of aalytical research of the theoretical basis of the travellig salesma problem. Numerous formulatios of the travellig salesma problem are kow i the literature, ad most of them remaied uresolved to date. Here, as cofirmatio of prior thought we metio a ope problem of graph theory: Fidig ecessary ad sufficiet coditio for a graph to have a Hamiltoia cycle. I this research paper, we poited out the variatios of the travellig salesma problem. Some solvig methods were poited out, ad certai improvemets were provided. The paper gives a approximate result of travellig salesma problem optimizatio by usig a Hamiltoia cycle with the aim of cotributig to solvig the travellig salesma problem, which, to this day ad with great effort of scietists, has ot yet bee resolved. Refereces [1] D. L. Applegate (26). The Travelig Salesma Problem: A Computatioal Study, Priceto Uiversity Press, ISBN , Priceto, New Jersey, pp.593 [2] T.Bektas (25). The multiple travelig salesma problem: a overview of formulatios ad solutio procedures, The Iteratioal Joural of Maagemet Sciece Omega, Volume 34, Issue 3, Jue 26, pp , ISSN , The Uiversity of Michiga [3] D.Crjac Milić (29). Stablo (drvo) logičkih mogućosti i jegova primjea u logistici, IX međuarodi zastvei skup Busiess logistics i moder maagemet, Segetla Z., Karić M. (Ed.), pp , ISBN , Ekoomski fakultet u Oseku, , Sveučilište Josipa Jurja Strossmayera u Oseku, Osek [4] M.Crjac, D.Jukić, R.Scitovski (1994). Matematika, Ekoomski fakultet u Oseku,ISBN: X,Osek [5] L. Čaklović (21). Geometra liearog programiraja, Elemet, ISBN: , Zagreb [6] Rajesh Matai, Surya Sigh ad Murari Lal Mittal (21). Travelig Salesma Problem: a Overview of Applicatios, Formulatios, ad Solutio Approaches, Travelig Salesma Problem, Theory ad Applicatios, ISBN: , ITech [7] G. Sierksma (22). Liear ad Iteger Programmig. Theory ad Practice, 2rd Editio, Marcel Dekker, New York, [8] M. Siger (1997). Itroductio to the Theory of Computatio, PWS Pub.co., Bosto [9] M.Sipser ( 212). Itroductio to the Theory of Computatio, Thomso Course Techology,(Ed.: 3 rd ), ISBN , Bosto, Massachusetts, Uited States of America [1] D. Velja (21). Kombiatora i diskreta matematika, Algoritam, ISBN , Zagreb [11] H.S.Wilf (1986). Algorithms ad Complexity, Pretice-Hall, Eglewood Cliffs [12] S.G. Williamso (1985). Combiatorics for Computer Sciece, Computer Sciece Press, ISBN , Rockville Iteratioal Joural - VALLIS AUREA Volume 1 Number 2 Croatia, December 215 UDK : ; DOI IJVA