How To Solve A Large Number Of Problems In A Large Graph With A Divide And Conquer Algorithm

Transcription

1 2011 Fourth International Symposium on Computational Intelligence and Design : A Divide-and-Conquer Based Memetic Algorithm for Capacitated Arc Routing Problem Xiaomeng Chen School of Information Science and Technology University of Science and Technology of China Hefei, China monachen@mail.ustc.edu.cn Abstract The Capacitated Arc Routing Problem (CARP) has attracted much attention during the last few years due to its wide applications in real life. A Memetic Algorithm with Extended Neighborhood Search (MAENS) was developed for solving this kind of problem. A powerful local search operator, the so-called MS operator, was introduced and plays an important role in MAENS. But the main disadvantage is the high computational cost due to the large enumerative number of selecting route groups. In this paper, we propose an improved approach for MAENS with a divide-and-conquer strategy, named, which divides a large graph into small sub graphs in order to decrease enumerative number so as to reduce the computational cost. An adaptive method is used for choosing parameters during dividing. Experimental results show that manages to obtain the same level of solution quality as MAENS with much less computational time. Keywords- CARP; MAENS; ; MS operator; Divide-and-conquer I. INTRODUCTION The Capacitated Arc Routing Problem (CARP) is one of the most typical forms of the arc routing problem with wide applications in real world, such as urban waste collectio post delivery, salting route optimizatio winter gritting, etc. [1]. It is NP-hard, with exact methods only applicable to small-size instances (less than 30 required edges) [2]. As a result, heuristics and meta-heuristics are often considered in the literature [3-7] and some memetic algorithms are also proposed, which combine the genetic algorithm (GA) with local search [8-11]. Tang et al. proposed a Memetic Algorithm with Extended Neighborhood Search (MAENS) for CARP [12]. In their work, a novel local search operator, the Merge-Split (MS) operator, was proposed. It is capable of searching using large step size by merging several routes into an unordered list of tasks and splitting the list to construct several better routes, and it is less likely to be trapped in local optima. MAENS combines the advantages of both the traditional and MS operators in its local search procedure so as to attain a good tradeoff between exploitation and exploration. As demonstrated in [12], the disadvantage of MAENS is the high computational cost. The MS operator would be time-consuming to enumerate all neighbors generated by MS, so MAENS employs a random candidate list strategy to examine MS neighbors. However, this strategy can be inefficient because the MS search potential of different route groups may vary largely. So in MAENS-II [13] proposed by H. Fu et al., a heuristic candidate list strategy samples the neighbors generated by the MS operator to avoid unnecessary calling of the MS operator. In this paper, we propose a divide-and-conquer based strategy to address the high computational cost, and the new algorithm is named. Firstly, divide the route set of each individual into several uncrossed subsets of routes composing sub individuals before MS operation; secondly, apply MS operator to each sub individual to get optimized ones; finally, combine all the optimized sub individuals into a new optimum individual. II. CARP, MAENS AND MS OPERATOR CARP can be described as follows: consider a directed connected graph G = (V, A, E), with the vertex set V, the arc (directed edge) set A and the edge set E. There is a central depot vertex dep V, where a set of vehicles are based. Each arc or edge (v i, v ) has a nonnegative traveling cost i and a non-negative demand i : the edges or arcs with positive demands are called tasks, and those with zero demands are called non-tasks. One must serve in a least cost fashion all tasks, using identical vehicles with capacity Q under the following constraints: Each route starts and ends at the depot vertex dep; Each task is served exactly in one route; The sum of demands on the arcs or edges served by one vehicle does not exceed Q. As a memetic algorithm, MAENS consists of three main phases: initializing, crossover and local search. The pseudo code of MAENS is provided in Fig.1, where psize is the population size, p ls is the probability of carrying out local search and f is the fitness function /11 $ IEEE DOI /ISCID

2 Begin Calculate Dikstra matrix recording the shortest distance between tasks. Initialize a population P of solutions. Evaluate the population P. While stopping criterion is not met Set an intermediate population P 1= P. Crossover (P 1), resulting intermediate population P 2. Local search (P 2, p ls), resulting intermediate population P 3. Evaluate the population P 3 P. Use stochastic ranking to sort solutions in P 3 P. Set P = {the first psize solutions in P 3P}. End While End Return the best feasible solution S ever encountered. Figure 1. Pseudo code of MAENS The local search phase in MAENS can be divided into two stages, shown in Fig. 2. The MS operator aiming to improve a given solution by modifying its multiple routes is composed of two steps, i.e. Merge and Split. Given a solutio the Merge step randomly selects r (r > 1) routes from n ones, and combines them together to form an unordered list of tasks. The Split step directly operates on the unordered list generating five solutions with five rules. The best one is the output of the MS operator. Input: An offspring individual S xp 2 generated by crossover in MAENS. Output: Solution S*. Set S 4 = S x. Stage 1: Set S* = S 4. Conduct Single Insertio Double Insertion and Swap local search on S*. Find the best solution S 1, S 2 and S 3 in the corresponding neighborhoods N 1(S*), N 2(S*) and N 3(S*). Set S 4 = the best solution in set {S 1, S 2, S 3}. Until f(s 4) f(s*)) Stage 2: Conduct Merge-Split local search on S*. Find the best solution S 4 in the corresponding neighborhoods N ms (S*). Until f(s 4) f(s*) Return the solution S*. Figure 2. Pseudo code of the local search process in MAENS The main disadvantage of MAENS is the high computational cost. To select r routes from n ones, m= r C n different pairs of routes are formed, and m will be very large if r and n are big enough. Even if r=2, m=(n 2 -n)/2 is also a large number for the MS operator when n is more than 20. So in MAENS, only 100 route groups are randomly selected if m>100. III. DIVIDE-AND-CONQUER BASED MEMETIC ALGORITHM AND Because the MS operator in MAENS conducts local search on all candidate route groups regardless of their capacity of improvement, too much time is wasted on unnecessary MS operations on routes with little associativity. If a current individual can be divided into sub individuals in which routes have high potential to be improved, the efficiency of conducting MS on each sub individual will increase. In, a divide-and-conquer strategy is conducted to divide a larger graph into sub graphs by splitting the original route set of a given solution into subsets according to five rules. The rules are deliberately designed to make the dividing better cluster routes. Experimental results showed that in this way, the running time is reduced and better or comparable results are achieved. The pseudo code of is provided in Fig.3, where AsMatrix is the associative matrix of tasks and two portions printed in boldface are different from MAENS. Begin Initialize a population P of solutions. Evaluate the population P. While stopping criterion is not met Calculate Task Associative matrix based on P. Set an intermediate population P 1 = P. Crossover (P 1), resulting intermediate population P 2. Improved local search (P 2, p ls, AsMatrix), resulting intermediate population P 3. Evaluate the population P 3P. Use stochastic ranking to sort solutions in P 3 P. Set P = {the first psize solutions in P 3P}. End While End Return the best feasible solution S ever encountered. Figure 3. Pseudo code of A. Associativity of Route Pairs To make the dividing process more reasonable, the associativity of each route pairs is estimated after a generation forms. Two steps are conducted i.e. calculating associative matrix of tasks and generating order lists of route-pairs. 1) Calculating the Associative Matrix of Tasks An associative matrix of tasks is calculated at the beginning of evolution iteration. If two tasks appear in one route of a solution individual, they are considered to be associative in the individual. The associativity of two tasks, task i and task, denoted by a i, is an element in the associative matrix of tasks defined as follows: a i, fi,, k (1) S k 1, taski and task in one route of S k f i,, k (2) 0, others where i, are two task IDs, and S k is the k th solution individual of the population. 2) Calculating Order Lists of Route-pairs For a given individual with n routes, two matrices of route-pairs, Route Associative matrix and Route 84

3 Distance matrix, are calculated based on the Task Associative matrix and the Dikstra matrix respectively. They record the average association or distance between each task taken from the i th route and each task taken from the th route. ra i, in Route Associative matrix and rd i, in Route Distance matrix are defined as follows: a rd ra i, i, p, q taskproutei taskqroute N N i d p, q taskproutei taskqroute N N i, i 1,2... 1,2... i, i 1,2... 1,2,... i (3) (4) where N k is the number of tasks in route k, and a p,q and d p,q respectively represent elements in the Task Associative matrix and in the Dikstra matrix. We put ra i, (i = 1,2 = 1,2 i<) in order and form one order list of route-pair association named RList 1. Similarly, we put rd i, (i = 1,2 = 1,2 i<) in order and form an order list of route-pair distance named RList 2. A reverse order list of RList 1, named RList 3 and an order list of nearest task distance between routes, named RList 4, are formed. These four order lists of route-pairs are updated before graph dividing providing reference standards of rule 1 to rule 4 during graph dividing respectively; see section C. B. Select Appropriate Upper Bound and Dividing Rules Before the dividing process, an upper bound u of route number of each sub individual and a dividing rule must be chosen. To find a suitable u and an appropriate dividing rule for a specific individual, selfadaption is adopted. A traditional selection technique, Roulette Wheel Selection [14], is used to decide an upper bound and a dividing rule according to their respective candidate pools. The appropriate upper bound of route number can lead most local search operators on every sub individual to be effective and the probability of success for local search operators on routes from different sub individuals to be small. C. Improved Local Search Process The improved local search process in is shown in detail in Fig. 4. Compared with the local search process in MAENS, the one in modifies the MS operator of MAENS into three components: Divide, MS and Combine; so we call a divide-andconquer based memetic algorithm. Input: an offspring individual S x P 2 generated by crossover in. Output: Solution S*. Set S b = S x. Stage 1: Set S*= S b. Conduct Single Insertio Double Insertion and Swap local search on S*. Find the best solution S 1, S 2 and S 3 in the corresponding neighborhoods N 1(S*), N 2(S*) and N 3(S*). Set S b. = the best solution in set {S 1, S 2, S 3}. Until f(s b ) f(s*) Stage 2: Set S b = S* Select upper bound u of route number from upper bound candidate pool. Select dividing rule rule i from dividing rule candidate pool. Calculate Rlist shown in section if it is necessary. Divide S* into sub solution set SubS*={ S 1 *, S 2 *,, S m *} with upper bound u of each sub individual solution according to rule i. Conduct Merge-Split local search on each element S i*of SubS*, where 1 i m, S i* u, and find the best solution set SubS= { S 1, S 2 S m }. Combine all the elements of SubS together to generate a new solution S*. If f(s*) < f(s b) then Set S b = S* Set S*= S b. Conduct Single Insertio Double Insertion and Swap local search on S*. Find the best solution S 1, S 2 and S 3 in the corresponding neighborhoods N 1 (S*), N 2 (S*) and N 3 (S*). Set S b = the best solution in set {S 1, S 2, S 3 }. Until f(s b) f( S*) Until two consecutive loops without new better solution. Return the solution S*. Figure 4. Pseudo code of the improved local search process in Figure 5. Example of routes and route set dividing The Divide component is developed to divide an individual solution into m sub individuals. A simple example of dividing is shown in Fig.5, in which on the left of the equals sign is the original individual and on the right are two sub individuals. In order to make the process of clustering routes reasonable and optimal, we choose five distinct rules to decide how to divide the routes set: Ensure that the more associative routes are put into one sub individual; Ensure that the closer routes (i.e. routes with smaller distance between them) are put into one sub individual; 85

4 Put routes with less association into one sub individual; Put routes with closer tasks into one sub individual; Divide the routes set randomly. Before dividing, a dividing rule rule and a proper upper bound u is selected using the method in section B. If =1, 2, 3 or 4, Rlist generated before dividing is put in use. If =5, we use Rlist 5, a list of route pairs with random order. Routes are put into sub individuals in order from the head to the tail of Rlist within the limit of the upper bound. The MS component is developed to conduct MS local search operator on each sub individual. The process of MS is similar to the one in section 2. The Combine component combines routes of each sub individual optimized after the MS component into one new individual containing all the tasks as a complete solution. D. Performance Analysis As discussed in section 2, the original MS operator is conducted on min(m=o(n 2 ), 100) different groups of routes of an individual with n routes. In, suppose the upper bound of route number of a sub individual is u (u=3, 4, ), the number of sub individuals is n/u, where n is the route number of the original individual, then the number of times of calling the MS operator for an individual is n r m C u (5) u If r = 2, then n 2 n( u 1) m Cu O( nu) (6) u 2 The complexity of O(nu) ensures that computational cost of will increase slowly with n compared with MAENS and MESNS-II if n is sufficiently larger than u. IV. EXPERIMENTS AND ANALYSIS To evaluate the efficacy of the MS with divide-andconquer and, two sets of experiments are carried out. First, we compared results of and MAENS on aspects of CPU time cost and solution cost. Second, performance of and MAENS is analyzed from the aspect of evolution speed. A. Experimental Set-up is designed to reduce the high computational cost of MAENS which presents only in large instances, and the performance analysis of in section III-D indicates that shows no strength when route number n is comparable with upper bound u, or even operates for longer time with additional dividing and combination process. Thus, only the results from large instances are meaningful. In present available test sets for CARP, Brandão and Eglese s set [15] has such a feature. In G1 instances 347 edges from the 375 are required with tasks, and in G2 instances all edges are required. From instance A to instance E in both G1 and G2, the capacity of one vehicle is decreased and the vehicle number is increased. TABLE I. PARAMETER SETTING OF MAENS AND Name Description Value psize Population size 30 ubtrial Maximum trials for generating initial solutions 50 opsize Number of offspring generated in each 6*psize generation P ls Probability of carrying out local search 0.2 (mutation) P Number of routes involved in Merge-Split 2 operator G m Maximum number of generations 500 In the experiments, MAENS and were both coded in C language and run using an Intel(R) E5620. Their differences only appear during the local search process. MAENS adopted the local search operator shown in Fig. 2 while adopted that showed in Fig. 4. All the experiments used the same parameters and were conducted for 30 independent runs. Table I summarizes the parameter settings of MAENS and. B. Results and Comparison Table II presents the solution cost and the CPU time cost of MAENS and. A brief description of the contents in the tables is given below. The columns headed V, R, E, capacity and vehicle indicate the number of vertices, required edges, total edges, capacity of one vehicle and total vehicle number, respectively. The columns headed average solution cost and average CPU time indicate results from solution cost or CPU time from 30 independent experiments. The columns headed win in Wilcoxon test indicate which algorithm presents a better performance according to the Wilcoxon rank-sum test [16]. means that the results have no obvious difference. Values in the row headed mean are attained by calculating the average values for their columns. Results show that the average CPU time of MAENS is times of that of and has an obvious strength on this aspect. The average solution cost of MAENS is times of that of and the difference of the two methods is unnoticeable. The advantage of to MAENS in CPU time has a decreasing trend from for G1_A to for G1_E. This phenomenon results from the predetermined and fixed maximum route pair number 100 of MS in MAENS. Because the capacity of one vehicle is decreased and vehicle number is increased from instance A to instance E in both G1 and G2 it is an 86

5 inevitable outcome that route pairs in one individual are more than 100 resulting from increasing route number. When individuals are divided as in this paper, the extra route pairs without MS are less than the original method for the same vehicle number, because we conduct MS on smaller sub individuals with the same maximum route pair number. So it explains the MAENS strength on computational time decreases. Similarly, results from instance G2_A to G2_E present the same features which can be explained in a similar way. TABLE II. AVERAGE SOLUTION COSTS AND CPU TIME OVER 30 RUNS ON THE BRANDÃO AND EGLESE S SET FOR BOTH MAENS AND name V R E capacity vehicle MAENS average solution cost MAENS/ win in Wilcoxon test MAENS average CPU time MAENS/ win in Wilcoxon test G1_A G1_B G1_C G1_D G1_E G2_A G2_B G2_C G2_D G2_E mean C. Performance Analysis of Fig.6 shows the process of evolution of MAENS and for the ten instances in Brandão and Eglese s set in the form of evolution lines. The gradients of the evolution line show the speed of convergence. In all the graphs in Fig 6, we can obtain the following conclusions: Evolution lines of are lower than those of MAENS, which means that can get better solutions within same given running time than MAENS. This ensures attains a better solution when the total computation time is strictly limited. During the first half of the evolution process, gradients of evolution lines of are larger than those of MAENS, which means that the convergence speed of is faster than MAENS. This illustrates has a stronger ability to approach the optimum solutio especially during the first half of the evolution process. V. CONCLUSION In this paper, we investigated CARP within the framework of MAENS. We developed MS by conducting a divide-and-conquer strategy aiming to overcome the weakness of high computational cost in MAENS. On the one hand, unlike MAENS, which conducts MS on every route pair, has a considerate dividing system to cluster routes with higher improvement potential. On the other hand, takes MS as the basic operator during local search, absorbing the great ability of avoiding being trapped in local optimal of MS. Based on our experimental studies, two main conclusions can be drawn. First, can significantly reduce the computational cost, especially for large instances. Second, runs well in maintaining the capability of achieving better solutions in MAENS and, in most situations of large instances, comes up with less solution cost compared with MAENS. Divide-and-conquer strategy is widely used in resolving different kinds of problems to simplify the solving process when sub problems can be solved in exactly the same way as solving the primary problem. When used into CARP, it must be adapted because the interaction among sub problems leads the dependence of solutions on all sub problems. In, we manage to decrease the relation by using inherited information from former generations. ACKNOWLEDGMENT The author would like to thank Dr. Ke Tang for providing the subect and experiment environment for this paper and also thank Dr. Malcolm Keech and the anonymous reviewers for the helpful comments and criticisms. REFERENCES [1] M. Dror, Ed., Arc Routing: Theory, Solutions and Applications. Bosto MA: Kluwer, [2] R. Hirabayashi R, Y.aruwatari, and N.Nishida, Tour construction algorithm for the capacitated arc routing problem, Asia-pacific Journal of operational Research, no, 3, 1992, pp [3] B. L. Golde J. S. DeArmo and E. K. Baker, Computational experiments with algorithms for a class of routing problems, Comput. Oper. Res., vol.10, no.1, 1983, pp

6 [4] G. Ulusoy, The eet size and mix problem for capacitated arc routing, Eur. J. Oper. Res., vol.22, no.3,, 1985pp [5] A. Hertz, G. Laporte, and M. Mittaz, A tabu search heuristic for the capacitated arc routing problem, Oper. Res., vol.48, no.1, 2000, pp [6] A. Hertz and M.Mittaz, A variable neighborhood descent algorithm for the undirected capacitated arc routing problem, Transport. Sci., vol.35, no.4, 2001, pp [7] P. Beullens, L. Muyldermans, D. Cattrysse, and D. V. Oudheusde A guided local search heuristic for the capacitated arc routing problem, Eur. J. Oper. Res., vol.147, no.3, 2003, pp [8] P. Lacomme, C. Prins, and W. Ramdane-Cherif, Competitive memetic algorithms for arc routing problem, Ann. Oper. Res., vol.131, no.1 4, 2004, pp [9] H. Handa, D. Li L. Chapma and X. Yao, Robust solution of salting route optimization using evolutionary algorithms, in Proc. IEEE Congr. Evol. Comput. Vancouver, BC, Canada, 2006, pp [10] H. Handa, L. Chapma and X. Yao, Robust route optimization for gritting/salting trucks: A CERCIA experience, IEEE Comput. Intell. Mag., vol.1, no.1, Feb.2006, pp.6 9. [11] J. Brandao and R. Eglese, A deterministic tabu search algorithm for the capacitated arc routing problem, Comput. Oper. Res., vol.35, no.4, 2008, pp [12] K. Tang, Y. Mei, and X. Yao, Memetic algorithm with extended neighborhood search(maens) for capacitated arc routing problem, IEEE Trans. On Evol. Comput., vol.13, no.5, Oct. 2009, pp [13] H. Fu, Y. Mei, K. Tang and Y. Zhu, Memetic Algorithm with Heuristic Candidate List Strategy for Capacitated Arc Routing Problem, WCCI 2010 IEEE World Congress on Computational Intelligence-CCIB, Barcelona, Spai July. 2010, pp [14] Baker, J. E, Reducing Bias and Inefficiency in the Selection Algorithm, In Proc. of the Second International Conference on Genetic Algorithms and their Applicatio Hillsdale, New Jersey, USA: Lawrence Erlbaum Associates, 1987, pp [15] J.Brandão and R. Eglese, A deterministic tabu search algorithm for the capacitated arcrouting problem, Comput.Oper. Res., vol.35, no.4, pp , Apr [16] Wilcoxon. F, Individual comparisons by ranking methods, Biometrics Bulletin 1 (6), 1945, pp (a) G1_A (c) G1_B (e) G1_C (g) G1_D (b) G2_A (d) G2_B (f) G2_C (h) G2_D (i) G1_E () G2_E Figure 6. Evolution of MAENS and 88