Benchmarking Algorithms for Dynamic Travelling Salesman Problems

Transcription

1 Benchmaring Algorithms for Dynamic Travelling Salesman Problems Lishan Kang,2 Aimin Zhou 2 Bob McKay 3 Yan Li 4, Zhuo Kang 4 Department of Computer Science & Technology China University of Geosciences, Wuhan,P.R.China State Key Lab of Software Engineering,Wuhan University, Wuhan, P.R.China School of IT & EE University of New South ADFA, Canberra, Australia Computer Center Wuhan University Wuhan, P.R.China ang@whu.edu.cn, amzhou@hotmail.com, b.mcay@adfa.edu.au, llyyan2000@2cn.com, ang_zh@2cn.com Abstract- Dynamic optimisation problems are becoming increasingly important; meanwhile, progress in optimisation techniques and in computational resources are permitting the development of effective systems for dynamic optimisation, resulting in a need for objective methods to evaluate and compare different techniques. The search for effective techniques may be seen as a multi-objective problem, trading off time complexity against effectiveness; hence benchmars must be able to compare techniques across the Pareto front, not merely at a single point. We propose benchmars for the Dynamic Travelling Salesman Problem, adapted from the CHN-44 benchmar of 44 Chinese cities for the static Travelling Salesman Problem. We provide an example of the use of the benchmar, and illustrate the information that can be gleaned from analysis of the algorithm performance on the benchmars. I. INTRODUCTION With the evolution of computing environments from centralized computing through distributed computing to mobile computing, the confluence of Web services, peerto-peer systems and grid computing provide the foundation for Internet distributed computing and mobile computing allowing applications to scale from proximate Ad-hoc networs to planetary-scale distributed systems[3]. Recently, new features such as wire and wireless mixture, smartness, macro and micro-mobility, ultra-scalability, interoperability and invisibility have been added to the world of computing and communications [6,7]. Key research challenges they commonly face are the optimization of dynamic networing, arising from networ planning and design, load-balance routing and traffic management [3,4,5]. They lead to a very important theoretical mathematical model: the Dynamic Travelling Salesman Problem (D-TSP). A wide variety of algorithms have been proposed for the dynamic TSP, such as ant algorithms[9,0,,2], competitive algorithms(on-line algorithms)[3] and dynamic inver-over evolutionary algorithms[8], and hence there arises a need to evaluate and compare them. This comparison, however, is not straightforward. In common with other optimisation problems, the search for good algorithms is a multi-objective problem, trading off algorithm performance against time complexity. However, unlie static optimisation problems, in dynamic optimisation problems the performance/complexity tradeoff is not discretionary. In static optimisation problems, we can always wait longer for a solution. In dynamic optimisation problems, the trade-off between performance and time complexity is determined by the problem difficulty and the relationship between the rate of change and the available computing resources. So it is crucial that there be accurate evaluation of the performance of an algorithm at a given combination of problem complexity, and required computing resources relative to the rate of change. This in turn implies a need for benchmars which permit a fine-grained study of the performance requirements of dynamic optimisation. In this paper, we propose a family of such benchmars for D-TSP, and demonstrate the insights which may be obtained by their application. The paper is organized as follows: we formally define the dynamic TSP in section 2, discuss the evaluation of solvers in section 3, and propose benchmars in section 4. Section 5 discusses some experiments with the proposed benchmars, while section 6 gives our conclusions. II. DEFINING THE DYNAMIC TSP The well-nown travelling salesman problem (TSP) may be formally described as follows: Given n cities {c, c 2,, c n } and a cost(distance) matrix: = { } () D d ij n! n where d ij is the cost(distance) from c i to c j, find a permutation π = (π, π 2,..., π n ), such that: n! i= d," = min " (2) i i+ where π n+ = π. Definition : A dynamic TSP(D-TSP) is a TSP determined by the dynamic cost (distance) matrix as follows: t) = { d ij ( t)} (3) D ( n( t)! n( t)

2 where d ij (t) is the cost from city(node) c i to city c j, t is the real world time. This means that the number of cities n(t) and the cost matrix are time-dependent. But first, let us generalize the above to optimisation problems in general. An optimisation problem is defined over a space S of solutions; for any s S, there is a corresponding objective value f(s). In a dynamic optimisation problem, the objective value is also a timedependent function, f(t,s). Definition 2: A dynamic optimisation problem DP is an optimisation problem determined by a dynamic objective function f(t,s), where the objective is to minimize f(t,s) for all values of t. However, this definition is just a theoretical model for mobile ad-hoc networing and similar applications. In practice, we need another approach. For any continuous-time dynamic optimisation problem DP, there is a family of related discrete-time dynamic problems D*P with differing time windows. Can the D*P problems act as tracers for the corresponding DP problem in other words, are the solutions always close to the solutions of the corresponding problem DP? Formally, we may define a D*P as follows: Let t, = 0,, 2,, m, t 0 =0 and t m = T be a sequence of discrete real world time sampling points. Definition 3: D*P is a series of optimisation problems determined by the objective functions f(t,s) = 0,, 2,., m-, with time windows [t, t + ], where { } m i t = 0 is a sequence of real world time sampling points. In particular, for the travelling salesman problem: Definition 4: D*-TSP is a series of TSP determined by the cost matrix: t ) = { d ( t )} (4) D ( ij n( t )! n( t ) = 0,, 2,., m-, with time windows [t, t + ], where { } m i t = 0 is a sequence of real world time sampling points. From a practical perspective, there is an additional constraint: the objective function f(t,s) should change relatively slowly with t ; if the objective function changes too quicly for algorithms to trac the solution, then there is no advantage in considering it as a dynamic optimisation problem. It is best treated as a sequence of independent static optimisation problems. D-TSP and D*-TSP are certainly NP-Hard problems (since any static TSP may be straightforwardly converted into a D-TSP or D*-TSP with a time-invariant cost matrix). There are many difficult open questions in the field of discrete-time dynamic optimization problems, because implicitly they involve a two-objective optimization problem. That is, D*P solvers should be designed as solutions of a two-objective optimization problem. One objective is to minimize the objective f(t,s) at each time interval in particular, for dynamic TSP, to minimize the length of the tour π = (π, π 2,, π n (t )): n( t ) ( d ( t )) =! d", ( ) i " t i+ i= n ( ) +!. " (5)! = where t The other is to minimize the size of the time interval : s = t - t (6) where t t. The two-objective optimization problem has a set of Pareto optimal solvers. Different applications and users impose different biases that lead to different solvers being chosen. That is primarily why it is difficult to evaluate an algorithm for a dynamic optimisation problem. A second difficulty for evaluating an algorithm arises because the exact solution of a dynamic optimisation problem is usually unnown there is limited value in designing an algorithm for problems with a nown solution. This is a general issue for dynamic optimisation, but it arises particularly for dynamic TSP because of its NP-hardness. In the subsequent discussion, we consider only the dynamic TSP. III. DESIGNING AND EVALUATING SOLVERS FOR D*-TSP There are two methods for measuring the performance of D*-TSP solvers: () Online Performance: Denote the error e(t )=d(π(t ))-d(π*( t )) (7) where d(π *( t )) is the length of tour π *( t ), which is the exact solution of the D*-TSP, and π(t ) is the approximate tour got by the D*-TSP solver A in time interval [t, t + ]. The performance of a D*-TSP solver A is defined as: m e ( A) = "! e( t )( t! t T = 0 (2) Offline Performance: Denote the error ) + (8) e t ) = d(! ( t )) " d(! ( t )) (9) ( where! t )) is the best tour got by solver A without ( time restriction (we assume that algorithm A is good enough to solve the static TSP. The assumption is useful in practice for large scale TSP). Then the offline performance of D*-TSP solver A is defined as: m e ( A) = "! e( t )( t! t T = 0 IV. BENCHMARKS FOR D-TSP ) + (0) If D-TSP is implicitly a multi-objective optimization problem, then benchmaring of algorithms implicitly requires evaluation not just at one point, but along the Pareto front. Hence we require not just one benchmar,

3 but a whole series of benchmars trading off problem complexity against computational requirements. We propose a family of benchmars based on the wellnown CHN44 benchmar for static TSP, which uses the positions of 44 Chinese cities. The positions of the 44 cities:(xi,yi),i =,2,...,44 as specified in [4] are appended to this paper. In satellite communications, as described in the DARPA Airborne Communications Node Project [], the networ routing configuration problem naturally gives rise to a D-TSP, in which the land-based communications nodes are fixed, but the satellite-based nodes are dynamic, their orbits being described by the equation: (X (t) - X ) 2 + (Y (t) - Y ) 2 2 =R () =, 2,..., M(t), where (X (t), X (t)), =,2,...,M(t) are the positions of the satellites. The positions are dependent on the real time t, as is the number of satellites M(t). In this case, both the D-TSP and D *-TSP are symmetric, that is dij (t)=dji(t). The parameters of the problems are m, M(ti) for t0 ( = 0), t, t2,..., tm (= T), R, and (X, Y ) for =,2,...,M(ti). In general, the above problems are highly dynamic, and possibly too difficult as a benchmar for the current state of the art in dynamic optimisation. However if M(ti)=M =constant and the time windows [ti,ti+]=δt are equal, we obtain a simpler version of the problem highly suited to benchmaring the current state of dynamic TSP. In the proposed benchmar, CHN44+M, there are the original CHN44 cities, plus M satellites. The simplest case is just one satellite node; we tae the center of the orbit as (X,Y ) = (253,906) and the radius is R = 2905 (see Fig.). More complex (and more realistic) CHN44+M problems can be constructed in 3D space, using city locations (x i,y i,z i) satisfying: (x i - x 0) 2 +(y i - y 0) 2 +(z i - z 0) 2 = r 2 (2) where (x 0,y 0,z 0)is the center of the earth and r is the radius. The orbits of the satellites become: (X (t) - X ) 2 + (Y (t) - Y ) 2 + (Z (t) - Z ) 2 = R 2 (3) One can adjust the parameters to meet requirements for different dynamics, such as the frequency, severity and predictability of changes [2]. In addition, there are other advantages of the CHN44+M:. The best route ever nown for CHN44 is given in the appendix A, and the length of the shortest route is , which can be used as a lower bound of CHN44+M for evaluating the global searching ability of the algorithms. 2. The dynamic behavior of the system of CHN44+M can be visualized easily. This is useful for examing the efficiency of the algorithms, as can be seen in figure, which is a screen-shot from a dynamic visualisation of CHN44+. V. EXPERIMENTS WITH CHN44+ In this section, we perform some experiments using the CHN44+ problem. Our aim with these experiments is to understand the behaviour of the the Dynamic Inver-Over Evolutionary Algorithm (DIOEA) [8] in an increasingly dynamic environment. DIOEA is used in all experiments. The environment for the experiments is an Intel P4.4GHz CPU with 256M RAM. We measure the offline error e, together with: Maximum error: e = max { e( t )} (4) m = 0, L, m Minimum error: e = { e( t )} (5) Average error: r e min =, L, 0 m = m! ( e( a m + t t= 0 )) (6) Test : 40 Sampling Points in a Cycle We start with a relatively easy test for DIOEA, in which the satellite orbits relatively slowly, the time for a single revolution (T ) ranging between 8.0s and 40.0s. The optimisation algorithm is updated 40 times per revolution (i.e Δt ranges from 0.2s to.0s). Figure 2 shows the error curve for T = 8.0s, and Table shows the experimental results. Figure : CHN44+ Benchmar

4 B. Test 3: 0 Sampling Points in a Cycle Finally, we sample a highly dynamic environment, in which there are only 0 sample points per revolution, and T ranges from 2.0s to 0.0s (ie Δt again ranges from 0.2s to.0s). Figure 4 shows the error curve for T = 8.0s, and Table 3 shows the experimental results. Figure 2 Error Curve for T = 8:0s(Test) Figure 4: Error Curve for T = 8:0s(Test3) Table : Results of Test A. Test 2: 20 Sampling Points in a Cycle We now sample an increasingly dynamic environment, with increased change between sample points. The optimisation is carried out 20 times per revolution, with T ranging from 4.0s to 20.0s (ie Δt again ranges from 0.2s to.0s). Figure 3 shows the error curve for T = 8.0s, and Table 2 shows the experimental results. Figure 3: Error Curve for T = 8:0s(Test2) Table 2: Results of Test2 table 3: Results of Test3 C. Analysis In each of the sets of experiments, we run the algorithm 0 times. All results are similar, the maximum errors are very large, but the minimum and average errors are quite acceptable, with the average relative errors being about 2%. The maximum errors all come from the beginnings of the runs, at time 0. That is, MIOEA has some difficulty in finding the optimum for the initial static TSP. However, once found, it can trac the moving optimum in a dynamic TSP quite well. Paradoxically, the dynamic objective provides sufficient disturbance to move MIOEA away from the initial local optimum found, and trac the global optimum. MIOEA actually performs better on the dynamic TSP than on a static TSP of equivalent complexity. Equally important, MIOEA behaves relatively well as the frequency and severity of change increases, with a relatively smooth decline in performance. VI. CONCLUSIONS In this paper, we discussed the difficulties in providing benchmars for dynamic optimisation problems, pointing out the implicitly multi-objective nature of the problem, and the concomitant requirement for a family of

5 benchmars able to compare algorithms along the Pareto front. We proposed one such family for the Dynamic Travelling Salesman Problem, the CHN44+M family of benchmars, a family of problems of tunable difficulty. Taing one subfamily of the benchmars, the CHN44+ problems, we investigated the performance of a current algorithm, MIOEA, varying the frequency and severity of change. Using the benchmars, we were able to gain an understanding of the behaviour of MIOEA, both of its overall behaviour, and of its performance degradation with frequency and severity of change. Monma and G.Namhauser(eds.). North-Holland, Amsterdam, 995. [5] H.N.Psaraftis. Dynamic Vehicle Routing Problems. In Vehicle Routing: Methods and Studies, B.L.Golen and A.A.Assad(eds.), Elsevier Science Publishers, , 988. ACKNOWLEDGEMENTS This wor was supported by National Natural Science Foundation of China (No and No ). REFERENCES [] Airborne Communications Node at DARPA. [2] J.Brane.Evolutionary Optimization in Dynamic Environments.Norwell:Kluwer Academic Publishers,2002. [3] D.W.Corne, M.J.Oates and G.D.Smith eds. Telecommunications Optimization: Heuristic and Adaptive Techniques. Chichester:John Wiley&Sons LTD,2000. [4] L.S.Kang, Y.Xie,S.Y.You,etc. Nonnumerical Parallel Algorithms:Simulated Annealing Algorithm. Being:Science Press,997. [5] M.Milenovic, S.H.Robinson, R.C.Knauerbase, etc. Toward Internet Distributed Computing. Computer,36(5),pp.38 46,2003. [6] D.Saba and A.Muberje. Pervasive Computing:A Paradigm for the 2th Century. Computer,36(3),pp.25 3,2003. [7] Y.C.Tseng, C.C.Shen and W.T.Chen. Integrating Mobile IP with Ad Hoc Networs. Computer,36(5),pp.48 55,2003. [8] A.M.Zhou, L.S.Kang and Z.Y.Yan. Solving Dynamic TSP with Evolutionary Approach in Real Time. Proceedings of the Congress on Evolutionary Computation, Canberra, Austrilia, 8-2, December 2003, IEEE Press, ,2003 [9] C.J.Eycelhof. Ant System for Dynamic Problem, the TSP Case Ant Canght in a Traffic Jam. Master s thesis, Unversity of Twente, the Netherlands, August, 200. ( [0] M.Guntsch, J.Brane, M.Middendorf and H.Schemc. ACO Strategies for Dynamic TSP. In M.Dorrigo et al.,editor, Abstract Proceedings of ANTS 2000, 59-62, [] M.Guntsch and M.Middendorf. Pheromone Modification Strategies for Ant Algorithms Applied to Dynamic TSP. in Applications of Evolutionary Computing, Lecture Notes in Computer Science, Vol.2037, , 200 [2] M.Guntsch, M.Middendorf and H.Schemc. An Ant Colony Optimization Approach to Dynamic TSP. Proceedings of the GECCO 200. San Francisco, USA, 7- July, 200, Morgan Kaufmann, , 200. [3] G.Ausiello, E.Feuestein, S.Leonardi, L.Stougie, and M.Talamo. Algorithms for the On-line Traveling Salesman. Algorithmica, Vol. 29, No.4, , 200. [4] W.Power, P.Jaillet and A.Odoni. Stochastic and Dynamic Networs and Routing. Networ Rougting, M.Ball, T.Maganti, C.

6 APPENDIX A: LOCATIONS OF CHN44 Table 4: Locations of CHN44 Cities index x y index x y index x y index x y The best route ever nown for CHN44 is( ) and the length is