A New Approach to Solving Dynamic Traveling Salesman Problems

Transcription

1 A New Approach to Solving Dynaic Traveling Salesan Probles Changhe Li 1 Ming Yang 1 Lishan Kang 1 1 China University of Geosciences(Wuhan) School of Coputer 4374 Wuhan,P.R.China lchwfx@yahoo.co,yanging72@gail.co,ang_whu@yahoo.co Abstract. The Traveling Salesan Proble (TSP) is one of the classic NP hard optiization probles. The Dynaic TSP (DTSP) is arguably even ore difficult than general static TSP. Moreover the DTSP is widely applicable to real-world applications, arguably even ore than its static equivalent. However its investigation is only in the preliinary stages. There are any open questions to be investigated. This paper proposes an effective algorith to solve DTSP. Experients showed that this algorith is effective, as it can find very high quality solutions using only a very short tie step. 1 Introduction With the developent of coputer science and counication technology, fro highly centralized coputing through distributed coputing to today s highly obile coputing, coputing environents have changed a great deal. Key research challenges they face in coon are the optiization of dynaic networs, arising fro networ planning and designing, load-balance routing and traffic anageent. However, guaranteeing that these systes run effectively and reliably is a difficult proble. It leads to a very iportant theoretical atheatical odel: the Dynaic TSP (DTSP). Because of the characteristics of DTSP itself, the solutions to general static TSPs are usually unsuitable for DTSP. Most cost too uch tie to gain good solutions, so the general algoriths are inefficient. Though a nuber of authors have researched [1][2][3][4] the DTSP, since it was proposed by Psaraftis[5], exploration of the DTSP is still in the preliinary stages, and any open questions need to be discussed. The ultiate (but unobtainable) goal is to find an optiu solution at every oent, as tie progresses. In fact, if you want to get better solutions, the efficiency will be lower, and conversely, if you require quic solutions, their quality will reduce that is to say the two goals (solution quality and tie response)are in conflict. Since we can t get an optiu solution at every instant, we can solve the proble in discrete tie steps, finding good solutions in a tie step as short as possible. In this paper, we will introduce an iproved Inver-Over[6] algorith(gsinver- Over) based on a gene pool. Generally, we find that using a gene pool, which reduces the search space sharply, gives solutions uch ore rapidly, without degradation of

2 the solution quality. Thus it draatically iproves the syste perforance in the cobined objectives. The GSInver-Over algoriths can iprove the individuals using inforation either fro other individuals or fro gene pools. We augent this with elastic relaxation as a local search ethod, which perits the rapid evolution of variants of individuals which were successful in previous situations. Our experients show that these operators can provide highly satisfactory results. In the reainder of this paper, there are five sections: (1) description of DTSP; (2) design of the gene pool; (3) the detailed algoriths; (4) analysis of the results; (5) suary and conclusions. 2 Description of DTSP Definition 1 A dynaic TSP(DTSP) is a TSP deterined by a dynaic cost (distance) atrix as follows: Dt () = { dij()} t (1) nt () nt () where dij ( t) is the cost fro city(node) ci to city c j, and t is the real-world tie. In this definition, the nuber of cities nt () and the cost atrix are tie-dependent. The Dynaic Traveling Salesan Proble is to find a iniu-cost route containing the all the nt () nodes. Definition 2 DTSP: Given all nt () ( P1, P2,..., P nt ( )) points, and the corresponding cost atrix D = { dij ( t)}, i, j = 1,2,..., n( t), find a iniu-cost route containing all the nt () points, where t stands for the oent of tie t; dij ( t) stands for the distance between the objective point Pi and the objective point P j, and dij () t = d ji () t. For exaple: nt () Min( d( T ())) t = in( d ()) t (2) i= 1 T i, Ti+ 1 where T {1, 2,..., n( t)} if i j, then Ti Tj, T = nt () + 1 T1 In definition 1, we dee the change of a DTSP s cost atrix with tie as a continuous process. Practically, we discretize this change process. Thus, A DTSP becoes a series of optiization probles: Dt ( ) = { dij( t )} nt ( ) nt ( (3) ) =,1, 2,..., 1, with tie windows [t, t +1 ], where { t } i= is a sequence of real world tie sapling points.

3 3 Design of Gene Pool Heuristic rules can draatically affect the efficiency of DTSP algoriths. The TSP is an NP-hard proble, and adding only one node, will increase the candidate search space by n n!. Thus it is ipossible to search all the candidate individuals. One useful heuristic derives fro the fact that ost of the edges in a iniu-cost route will join nearby vertices. So it is generally desirable for the eleents in the gene pool to select fro edges which go to nearby vertices. Unfortunately, this heuristic is often violated: for hard TSPs, a sall proportion of the edges in optial routes will need to connect distant vertices. If the heuristic is too rigidly applied, soe of the edges in the optial route won t exist in the gene pool. So a heuristic ethod based on iniising local distances sees reasonable, but in fact, this ind of restriction usually results in bad perforances. We describe an alternative heuristic for the design of the gene pool. It is based on the concept of α-nearness [7], which derives fro Miniu Spanning Trees (MSTs). The α-length α(i,j) of an edge <vi,v j > can be defined as the difference in length between the true MST, and the length of the 1-tree which is constrained to contain <v i, v j >. + α( i, j) =L T ( i, j) -L T (4) ( ) ( ) where T is an any given MST of length L(T),T + (i,j) is a 1-tree that contain the edge < v i, v j >,that is, given a MST of length L(T),α(i,j) is the increase length of a 1-tree required to contain the edge < v i, v j >. It is easy to see that: α(i,j) and α(i,j)= when the edge < v i, v j > belong to T. It can copute α(i,j) in the following rules: (1) if the edge < v i, v j > belong to T, then α(i,j)=. (2) Otherwise, insert the edge < v i, v j > into T, this will create a circle containing the edge < v i, v j >,then α(i,j) is the length difference between the longest edge of the circle and the edge < v i, v j >. The gene pool is a candidate set of soe ost proising edges. The candidate set ay, for exaple, contain α-nearness edges incident to each node. Generally speaing, the experience value of is 6 to 9.we set =8 in CHN144 proble[8]. Through experients, it can be shown experientally that 5%-8% (i.e the experient result of table 1 of instances in TSPLIB) of the connections in an optial TSP solution are also in the iniu spanning tree. This is a far larger proportion than the proportion of the n shortest edges. Thus we expect that a gene pool constructed based on α-nearness ay perfor better than a gene pool constructed on the distance. It should be possible to use a saller gene pool while aintaining the solution quality. Taing the CHN144 proble for exaple, 76.3% of the connections in the best nown solution are also edges in the iniu spanning tree. If we bias the gene pool based on the α-nearness, we expect that it will better atch the optial solution. That is to say, we expect that a gene pool that probabilistically includes eleents close to ebers of the MST will also include eleents close to ebers of the optiu TSP circuit. The gene pool based on the α-nearness has another rearable advantage that it is independent of instance scale, it eans,when the

4 instance scale increases,the size of gene pool won t increase. This character of gene pool is uch suitable to DTSP. Table 1. Instances in TSPLIB CHN144 a28 pr439 u574 u724 rl % 75.7% 79.1% 75.6% 75.2% 89.2% 4 Introducing the Algoriths Operator design is the ey to solving TSPs. A vast range of operators have already been proposed (for exaple: λ-opt[9], LK[1], Inver-Over), and we anticipate that this trend will continue. Based on perforance, any of these local-search inspired operators are superior to the traditional utation, crossover and inversion operators. In this paper, we adopt a for of iproved Inver-Over operator. We propose a highly efficient dynaic evolution algorith based on elastic. 4.1 GSInver-Over Operator Inver-Over is a high-efficiency search operator based on inversion, and having a recobination aspect. It can fully utilize inforation fro other individuals in the population to constantly renew itself. This gives it better search ability than any other operators (within a certain range of probles), yet the coplexity of algorith is low. We can say Inver-Over is a highly adaptable operator which has very effective selection pressure. However Inver-Over operator has its own constrains, experients show that reducing the inversion ties can sharply increase the convergence, this is favourable to DTSP. We set the axial inversion ties Max_tie in the GSInver-Over operator, when the inversion ties surpass Max_tie, end the algorith. The search environent has increased and it can get useful heuristic inforation not only fro other individuals, but also fro the gene pool. The choice of atching individuals is not rando, but biased toward better individuals in the population. This reduces the probability of incorporating bad inforation that daages the individual. The inversion operator is ore rationally designed, incorporating nowledge about the directionality of the route, further iproving the perforance of the algorith. Based on the above iproveents, the GSInver-Over perfored substantially better than the original algorith. the algorith for our GSInver-Over operator is as follows: GSInver-Over Operator: 1. Select a gene g fro individual S randoly and set S =S; 2. If the nuber p generated randoly is less than p1, then select the gene g fro the gene pool of g; 3. Else if p<p2 select an individual randoly fro soe best individuals and g is the gene that is next to g in the selected individual; 4. Else select next gene g randoly fro other genes;

5 5. If the next gene or the previous gene of g in S is g, then go to step 9; 6. Inverse the section fro the next gene of g to g in S ; 7. counter++, if counter > Max_tie, then go to step 9; 8. g= g and go to step 2; 9. If the fitness of S is better than S, then replace S with S ; 4.2 The Dynaic Elastic Operator The changes of a node include three cases: the node disappears, the node appears and the position of the node changes. If the node disappears, it will be deleted directly, then lin the two adjacent nodes of the deleted node. if the node appear, find the nearest node to the appearing node, then insert it to the tour that iniize the total length. if the node position changes, it can be seen as the cobination of the two forer cases. The dynaic-elastic operator is very siple in concept, but we find it is an effective local search operator. Dynaic Elastic Operator 1. Delete the node c and lin the cities adjacent to c; 2. Find the nearest node c* to c in the current individual; 3. Insert c next to C*, on the side that iniize the total length; 4.3 Main Progra Loop In the ain progra loop, we use a difference list Dlist to store the inforation of changed nodes. Note that T is the tie step. 1. Initialize the population; 2. If Dlist is not epty goto 3, else goto 5; 3. Update gene pool; 4. Dynaic-elactic (); 5. For each individual in the population,do GSInver-Over (); Optiizing(); 6. If the T> goto 5; 7. If not terination condition goto 2; When soe nodes change at tie t, it needs to update the gene pool, that is to say, create a new MST of the new nodes topology of tie t, then construct a gene pool according to α-nearness. 5 Experients with CHN146+3 We tested our algorith in a relatively difficult dynaic environent, adapted fro the well-nown CHN145[11] static TSP benchar. The proble has 146 static locations (145 Chinese cities, plus a geo-stationary satellite) and three obile locations, two satellites in polar orbit and one in equatorial orbit (fig. 1).

6 In dynaic optiisation experients, since the results represent a trade-off between solution quality, coputational cost and proble dynaics, it is iportant to specify the coputational environent in which experients were conducted. Our experiental environent consisted of the following: CPU: Intel C4 1.7GHz, RAM:256MB. We easure the offline error ē and μ as our quality etric: e t ) = d( π( t )) d( π( t )) (5) ( μ t ) = e( t ) / d( π( t )) (6) ( where d( π ( t )) is the best tour obtained by a TSP solver (which is assued to be good enough to find an optial solution for the static TSP) d( π ( t )) is the best tour obtained by our DTSP solver. Together with: Maxiu error: e = ax { e( t )} μ = ax { μ( t )} (7) =, L, =, L, Miniu error: e r = in { e( t )} μ = in { μ( t )} (8) =, L, r =, L, Average error: e = 1 ( e( )) = 1 a + 1 t t= (9) μ ( μ( )) a t + 1 t= Fig. 1. Experients with CHN saple tie-points in the period of the satellites were perfored for the experients. The results are given in table 2 with ΔT ranging fro.59s to 1.3s. Fig. 2 to fig. 5 are error curves respectively for ΔT=.59s, ΔT=.326s, ΔT=.579s and ΔT=.982s. Fro the experients, we can see that, when T is very sall, the result is relatively poor. As T increases, the axial and average errors decrease, and the solution quality iproves showing the stability of the algorith, and its rapid convergence. The experients also deonstrate the conflict between the two DTSP goals, requiring a coproise through the choice of T. With the exception of the shortest tie interval, the data in table 2 are generally acceptable, with the average errors being under 1%, and the axial errors less than 2%.

7 Table 2. Error of experients ΔT(s) e () μ (%) e r () μ r (%) e a () μ a (%) errors() saple points errors() saple points Fig. 2. Error Curve for ΔT =.59s Fig. 4. Error Curve for ΔT =.579s errors() saple points Fig. 3. Error Curve for ΔT =.326s errors() saple points Fig.5. Error Curve for ΔT =.982s 6 Conclusions In this paper, we analyze the DTSP which can be seen as a two-objective proble by trading off the quality of the result and the reaction tie. We propose a solution based on a gene pool, which greatly reduces the search space without degradation of the solution quality. By adding the iproved GSInver-Over Operator, we were able to significantly iprove the efficiency of the algorith. Adding the elastic relaxation ethod as a local search operator iproves the syste s real-tie reaction ability.

8 7 Acnowledgeents This wor was supported by National Natural Science Foundation of China (NO ). We would lie to than Bob McKay, of Seoul National University, Korea, for contributions to the presentation of this wor. 8 References 1. C.J.Eycelhof and M.Snoe. Ant Systes for a Dynaic TSP -Ants caught in a traffic ja. In3rd International Worshop on Ant Algoriths(22) 2. Z.C.Huang,X.L.Hu and S.D.Chen. Dynaic Traveling Salesan Proble based on Evolutionary Coputation. In Congress on Evolutionary Coputation(CEC 1),IEEE Press (21) Allan Larsen. The Dynaic Vehicle Routing Proble. Ph.D theis,departent of MatheaticalModelling (IMM) at the Technical University of Denar(DTU)(21) 4. A.M.Zhou, L.S.Kang and Z.Y.Yan. Solving Dynaic TSP with Evolutionary Approach in Real Tie. Proceedings of the Congress on Evolutionary Coputation, Canberra, Austrilia, 8-12, Deceber 23, IEEE Press(23) H.N.Psaraftis. Dynaic vehicle routing probles. In Vehicles Routing: Methods and Studies,B.L.Golden and A.A.Assad(eds),Elsevier Science Publishers(1988) 6. T.Guo and Z.Michalewize. Inver-Over operator for the TSP. In Parallel Proble Sovling fro Nature(1998) 7. Keld Helsgaun,An Effective Ipleentation of the Lin-Kernighan Traveling Salesan Heuristic,Departent of Coputer Science Rosilde University. 8. L.S.Kang, Y.Xie,S.Y.You,etc. Nonnuerical Parallel Algoriths:Siulated Annealing Algorith. Being:Science Press(1997) 9. S. Lin, Coputer Solutions of the Traveling Salesan Proble,Bell Syste Tech. J., 44, (1965) S. Lin & B. W. Kernighan,An Effective Heuristic Algorith for the Traveling-Salesan Proble(1973) 11. Lishan Kang, Aiin Zhou, Bob McKay,Yan Li Zhuo Kang,Bencharing Algoriths for Dynaic Travelling Salesan Probles, Bencharing Algoriths for Dynaic Travelling Salesan Probles, Proceedings, Congress on Evolutionary Coputation, Portland, Oregon(24)