European Journal of Operational Research

Transcription

1 European Journal of Operational Research 196 (2009) Contents lists available at ScienceDirect European Journal of Operational Research journal hoepage: Production, Manufacturing and Logistics An iproved ant colony optiization for vehicle routing proble Yu Bin a, *, Yang Zhong-Zhen a, Yao Baozhen b a College of Transportation and Logistics, Dalian Maritie University, Dalian , PR China b School of Civil Engineering and Architecture, Being Jiaotong University, Being , PR China article info abstract Article history: Received 8 August 2007 Accepted 22 February 2008 Available online 8 March 2008 Keywords: Vehicle routing proble Iproved ant colony optiization Ant-weight strategy Mutation operation The vehicle routing proble (VRP), a well-known cobinatorial optiization proble, holds a central place in logistics anageent. This paper proposes an iproved ant colony optiization (IACO), which possesses a new strategy to update the increased pheroone, called ant-weight strategy, and a utation operation, to solve VRP. The coputational results for fourteen benchark probles are reported and copared to those of other etaheuristic approaches. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Finding efficient vehicle routes is a representative logistics proble which has been studied for the last 40 years. A typical vehicle routing proble (VRP) ais to find a set of tours for several vehicles fro a depot to a lot of custoers and return to the depot without exceeding the capacity constraints of each vehicle at iniu cost. Since the custoer cobination is not restricted to the selection of vehicle routes, VRP is considered as a cobinatorial optiization proble where the nuber of feasible solutions for the proble increases exponentially with the nuber of custoers increasing (Bell and McMullen, 2004). Heuristic algoriths such as siulated annealing (SA) (Chiang and Russell, 1996; Koulaas et al., 1994; Osan, 1993; Tavakkoli-Moghadda et al., 2006), genetic algoriths (GAs) (Baker and Ayechew, 2003; Osan et al., 2005; Thangiah et al., 1994; Prins, 2004), tabu search (TS) (Gendreau et al., 1999; Seet and Taillard, 1993; Renaud et al., 1996; Brandao and Mercer, 1997; Osan, 1993) and ant colony optiization (Doerner et al., 2002; Reiann et al., 2002; Peng et al., 2005; Mazzeo and Loiseau, 2004; Bullnheier et al., 1999; Doerner et al., 2004) are widely used for solving the VRP. Aong these heuristic algoriths, ant colony optiizations (ACO) are new optiization ethods proposed by Italian researchers Dorigo et al. (1996), which siulate the food-seeking behaviors of ant colonies in nature. It has been successfully applied as a solution to soe classic copounding optiization probles, e.g. traveling salesan (Dorigo et al., 1996) quadratic assignent (Gabardella et al., 1997), job-shop scheduling (Colorni et al., * Corresponding author. Tel.: E-ail address: inlfish@yahoo.co.cn (B. Yu). 1994), telecounication routing (Schoonderwoerd et al., 1997), etc. If taking the central depot as the nest and custoers as the food, the VRP is very siilar to food-seeking behaviors of ant colonies in nature. This akes the coding of an ant colony optiization for the VRP is siple. Aong the earliest studies was that of Bullnheier et al. (1997) who presented a hybrid ant syste algorith with the 2-opt and the saving algorith for the VRP. Other researches of the ACOs to the VRP included the work by Bullnheier et al. (1999), Bell and McMullen (2004), Chen and Ting (2006). In the ACOs, the 2-opt exchange was used as an iproveent heuristic within the routes found by individual vehicles and the pheroone updating rules ainly considered the global feature of the solution. This paper proposes an iproved ant colony optiization with a new pheroone updating rule that can integrate the global feature and the local feature, a utation operation and the 2-opt exchange for the VRP. The reainder of the paper is organized as follows. Section 2 presents the atheatical odel for VRP. In Section 3, we present the IACO with ant-weight strategy and the utation operation. Soe coputational results are discussed in Section 4 and lastly, the conclusions are provided in Section Vehicle routing proble The VRP is described as a weighted graph G =(C,L) where the nodes are represented by C ={c 0, c 1,...,c N } and the arcs are represented by L ={(c i, c j ): i j}. In this graph odel, c 0 is the central depot and the other nodes are the N custoers to be served. Each node is associated with a fixed quantity q i of goods to be delivered (a quantity q 0 = 0 is associated to the depot c 0 ). To each arc (c i, c j )is associated a value d i,j representing the distance between c i and c j /$ - see front atter Ó 2008 Elsevier B.V. All rights reserved. doi: /j.ejor

2 172 B. Yu et al. / European Journal of Operational Research 196 (2009) Each tour starts fro and terinates at the depot c 0, each node c i ust be visited exactly once, and the quantity of goods to be delivered on a route should never exceed the vehicle capacity Q. 3. Iproved ACO for VRP 3.1. Generation of solutions Using ACO whose colony scale is P, an individual ant siulates a vehicle, and its route is constructed by increentally selecting custoers until all custoers have been visited. The custoers, who were already visited by an ant or violated its capacity constraints, are stored in the infeasible custoer list (tabu). The decision aking about cobining custoers is based on a probabilistic rule taking into account both the visibility and the pheroone inforation. Thus, to select the next custoer j for the kth ant at the ith node, the ant uses the following probabilistic forula. 8 s < P a gb j R tabu p ðkþ ¼ s a k hrtabu ih gb ih k ð1þ : 0 otherwise where p (k) is the probability of choosing to cobine custoers i and j on the route, s the pheroone density of edge (i,j), g the visibility of edge (i,j), a and b the relative influence of the pheroone trails and the visibility values, respectively and tabu k is the set of the infeasible nodes for the kth ant Mutation operation Mutation operation referring to genetic algorith (Yu and Yang, 2007; Yu et al., 2007) alters each child at a predefined probability. The operators can help the IACO to reach further solutions in the search space. The idea of the utation operation is to randoly utate the tour and hence produce a new solution that is not very far fro the original one. In this paper, the utation operator is designed to conduct custoer exchanges in a rando fashion. Fig. 2 shows the representation of the parent solution in Fig. 1. The steps for the utation operation are as follows: Step 1. Select the two tours fro the selected parent solution and select the utating point(s) fro the each utating tour. Fig. 3a shows the 9th custoer in the 3rd tour and the 12th custoer in the 4th tour are selected. Step 2. Exchange the custoers in the different tours and generate the child solution (see Fig. 3b). Step 3. Ensure the child solution local optiality. The 2-opt is applied to iprove the utated tours in child solution. Finally, the representation and the tours of the utated child solution is as Figs. 3c and 4. However, the utation operation ay violate vehicle capacity constraints. There are two approaches to deal with this situation. The first one is to assign a very high cost for such candidate solutions and accordingly reduce their probability of being selected in the forthcoing search. The second approach is to try to fix the resultant capacity violations by adjusting the delivery aounts. The advantage of the second approach over the first one is that it is ore suitable in probles that are ore likely to produce vehicle capacity violations and it enables IACO to investigate further points in the search space. Therefore, the second approach is adopted to deal with the vehicle capacity violation situation. Each route of the solution is utated with a certain probability p. Usually, the diversity of the solution is large at the beginning of a run and decreases with the tie. We adapt the utation rate during a run to proote a fast convergence to good solutions during the first generations and to introduce ore diversity for escaping fro local optia during later stages. The utation probability at the generation t is p ðtþ ¼p in where p in þðpax and pax Fig. 2. An exaple of a parent solution. pin Þ1 t=t ð2þ are the lower and the upper utation rates for the beginning and ending, respectively and T and t are the given axiu nuber of generations and the current generation of the iteration, respectively. According to preliinary tests, we suggest to set the lower utation rate to p in ¼ 1=n c, where n c is the nuber of the custoers, and the upper utation rate to p ax ¼ 1=n v, where n v is the aounts of the routes in the solution. Fig. 1. An exaple of the VRP. Fig. 3a. Procedure of utation (Step 1).

3 B. Yu et al. / European Journal of Operational Research 196 (2009) Fig. 3b. Procedure of utation (Step 2). s new ¼ q s old þ XK k Ds k q 2ð0; 1Þ ð3þ 3.3. Local search Fig. 3c. Procedure of utation (Step 3). Fig. 4. Tours of the utated child solution. In the 2-opt exchange, all possible pairwise exchanges of custoer locations visited by individual vehicles are tested to see if an overall iproveent in the objective function can be attained. The ethod has been used in several ACOs (Bullnheier et al., 1997, 1999; Bell and McMullen, 2004; Chen and Ting, 2006) for the VRP Update of pheroone inforation The updating of the pheroone trails is a key eleent to the adaptive learning technique of ACO and the iproveent of future solutions. First, Pheroone updating is conducted by reducing the aount of pheroone on all links in order to siulate the natural evaporation of the pheroone and to ensure that no one path becoes too doinant. This is done with the following pheroone updating equation, where s new is the pheroone on the link (i,j) after updating, s old the pheroone on the link (i,j) before updating, q the constant that controls the speed of evaporation, k the nuber of the route, K the nuber of the routes in the solution and K >0 and Ds k are the increased pheroone on link (i,j) of route k found by the ant. The pheroone increent updating rule uses the ant-weight strategy presented by Yang et al. (2007). Specifically, the strategy is written as: ( Ds k Q ¼ Dk d if link ði; jþ on the kth route KL k D k ð4þ 0 otherwise where Q is a constant, L the total length of all routes in the solution, i.e. L ¼ P k Dk, D k the length of the kth route in the solution, d the length of edge (i,j) and k the nuber of custoers in the kth routes and k >0. The ant-weight strategy updates the increased pheroone in ters of the solution quality and the contribution of each link to the solution, which consists of two coponents: the global pheroone increent and the local pheroone increent. In the antweight strategy, the quantity of the global pheroone increent, Q/(KL), of each route is related to the total length of the solution, while the one of the local pheroone increent (D k d )/( k D k ) of each link is based on the contribution of link (i,j) to the solution. Since the strategy for updating the increased pheroone considered both the global feature and local one of a solution, it can possibly ensure that the assigned increased pheroone is directly proportional to the quality of routes. The ore favorable the link/route is the ore pheroone increent is allocated to it, and the ore accurate directive inforation is provided for later search. Meanwhile, by adjusting the pheroone assigning ethod for the links of current optial path autoatically, the algorith can facilitate ore delicate searches in the next cycle in a ore favorable area, which assist in expanding the learning capacity fro past searches. The paraeters for updating the increased pheroone on the edges in the solution in Fig. 1 are calculated as Fig. 5. Moreover, in order to prevent fro local optiization and increase the probability of obtaining a higher-quality solution, upper and lower liits [s in, s ax ] are fixed to the updating equation., X s in ¼ Q 2d 0i ; ð5þ i, X s ax ¼ Q d 0i ; ð6þ i where d 0i is the distance fro the central depot to the ith custoer.

4 174 B. Yu et al. / European Journal of Operational Research 196 (2009) Fig. 6. The flowchart of IACO. Table 1 Coputational results using IACO and the well-known published results No. n Q Best know Best Worst Average Tie Fig. 5. Paraeters for updating the increased pheroone. C a C a C a C a C b C a C a C a C a C b C a C a C a C a a Taillard (1993). b Rochat and Taillard (1995) Overall procedure The flowchart of our IACO for the VRP is shown in Fig Nuerical analysis The heuristics described in the previous sections is applied to the 14 vehicle routing probles which can be downloaded fro the OR-library (see Beasley, 1990), and which have been widely used as bencharks, in order to copare its ability to find the solution to VRP. The inforation of the 14 probles is shown in coluns 2 4 in Table 1, which consists of the proble size n, the vehicle capacity Q, and the well-known published results (Taillard, 1993; Rochat and Taillard, 1995). The IACO paraeters used for VRP instances are Q = 1000, a = 2, b = 1 and q = 0.8. Then, the IACO were coded in Visual C++.Net 2003 and executed on a PC equipped with 512 MB of RAM and a Pentiu processor running at 1000 MHz. Coluns 5 8 present the results fro IACO including the best solution, the worst solution, and the average solution and average run tie (second). The nubers in bold are the results as the best-known solutions. To evaluate the ant-weight strategy and the utation operation, the two ant colony optiizations with different strategy are constructed. The first one is a standard ant colony optiization with the ant-weight strategy (denoted by ACO-W) and the other is a standard ant colony optiization with the utation operation (denoted by ACO-M). The coputational results are as shown in Table 2. The nubers in bold are the best solutions aong three algoriths. It can be observed that ACO-M can obtain the sae solutions as IACO in test proble 1, 2, 3, 6, 7, 11, 12 and 14, while ACO-W can only obtain the optiu solutions in test proble 1, 2, 6, 7 and 14. Copared with ACO-W, the ACO-M generally provides better solutions for the 14 probles. This ay be attributed that the introduction of the utation operation can diversify the ant colony, explore new possible solution space and prevent the algorith fro trapping in local optiization. However, while the utation operation iproves the solutions, it also increases the coputation ties. We can see that the ties consued by

5 B. Yu et al. / European Journal of Operational Research 196 (2009) Table 2 Coputational results using IACO, ACO-W and ACO-M No. IACO ACO-W ACO-M Best Worst Average Tie Best Worst Average Tie Best Worst Average Tie C C C C C ,134 C C C C C ,320 C C C C ACO-M are ore than ACO-W. This ay be because that in ACO-W, the ant-weight strategy is used for updating the increased pheroone. It can assign increased pheroone according to the quality the solution. This iproves the learning capacity of the algorith fro past searches, and enhances the efficiency. Furtherore, IACO integrated the ant-weight and the utation operation can provide the best solutions and consue the less coputation ties copared with ACO-W and ACO-M. Table 3 Deviations fro the best known solution of several etaheuristic approaches Prob. RR-PTS G-TS OSM-TS OSM-SA B-AS IACO C C C C C C C C C C C C C C Average Our IACO are copared with five other eta-heuristic approaches in the paper proposed by Bullnheier et al. (1997), which consisted of parallel tabu search algorith (RR-PTS) by Rego and Roucairol, a tabu search algorith (G-TS) by Gendreau et al., tabu search (OSM-TS), a siulated annealing algorith (OSM-SA) by Osan and ant syste algorith (B-AS) by Bullnheier et al. (1997). The coparison of the deviations fro the best known solution is shown in Table 3. The perforance of our IACO is best aong all eta-heuristic approaches, who produces in eleven probles of fourteen test probles and yields the lowest average deviation. Also, copared with OSM-SA, the tabu search approaches are able to provide better solutions. Also, copared OSM-SA, the tabu search approaches can provide better solutions. For a correct evaluation and coparison of the quality of six algoriths the coputing ties ust be taken into account. However, a correct evaluation and coparison of the coputing ties is generally tough due to the enorous variety of coputers available and used by different researchers. A very rough easure of coputers perforance can be obtained using Dongarra s (Dongarra, 2001) tables where the nuber (in illions) of floatingpoint operations per second (Mflop/seconds) executed by each coputer was used, when solving standard linear equations, with LINPACK progra. Regarding coputational ties, Rego and Roucairol used a sun sparc 4 (about 5.7 MFlop/s), Gendreau et al. used a 36 MHz Silicon Graphics (about 6.7 MFlop/s), Osan used a VAX 8600(about 2.48 MFlop/s), Bullnheier et al. used a Pentiu 100 MHz (about 8 MFlop/s). In this research, the Pentiu 1 GHz running IACO has an estiated power of 75 MFlop/s. Table 4 shows Table 4 Coputation ties of several etaheuristic approaches Probability RR-PTS G-TS OSM-TS OSM-SA B-AS IACO Run ties Scaled ties Run ties Scaled ties Run ties Scaled ties Run ties Scaled ties Run ties Scaled ties Run ties C C C C C C C C C , C C C C C Average

6 176 B. Yu et al. / European Journal of Operational Research 196 (2009) the origin coputation ties and the scaled coputation ties, which use Pentiu 1 GHz as the baseline, of six approaches. The perforance of IACO is copetitive when copared with other eta-heuristic approaches, such as SA, and TS. Although the run ties are not favor in IACO, our IACO still sees to be superior in ters of solution quality with an average deviation of 0.14%. Considering the very rough easure, the scaled ties are viewed as the assistant aspect of the perforance. Regarding the coputation efficiency, we find that the IACO can find very good solutions in an acceptable tie. 5. Conclusions The VRP has been an iportant proble in the field of distribution and logistics. Since the delivery routs consist of any cobination of custoers, this proble belongs to the class of NP-hard probles. This paper presents an IACO with ant-weight strategy and a utation operation. The coputational results of 14 benchark probles reveal that the proposed IACO is effective and efficient. Further research on additional odifications of the IACO to extensions of the vehicle routing proble with tie windows or with ore depots, are of interest. References Baker, B.M., Ayechew, M.A., A genetic algorith for the vehicle routing proble. Coputers & Operations Research 30, Beasley, J.E., OR-Library: distributing test probles by electronic ail. Journal of the Operational Research Society 41, Bell, J.E., McMullen, P.R., Ant colony optiization techniques for the vehicle routing proble. Advanced Engineering Inforatics 1 (8), Brandao, J., Mercer, A., A tabu search algorith for the ulti-trip vehicle routing and scheduling proble. European Journal of Operational Research 100, Bullnheier, B., Hartl, R.F., Strauss, C., Applying the ant syste to the vehicle routing proble. In: Second Metaheuristics International Conference, MIC 97, Sophia-Antipolis, France. Bullnheier, B., Hartl, R.F., Strauss, C., An iproved ant syste algorith for the vehicle routing proble. Annals of Operations Research 89, Chen, C.H., Ting, C.J., An iproved ant colony syste algorith for the vehicle routing proble. Journal of the Chinese Institute of Industrial Engineers 23 (2), Chiang, W.C., Russell, R., Siulated annealing eta-heuristics for the vehicle routing proble with tie windows. Annals of Operations Research 93, Colorni, A., Dorigo, M., Maniezzo, V., Trubian, M., Ant syste for job-shop scheduling. Jorbel Belgian Journal of Operations Research Statistics and Coputer Science 34 (1), Doerner, K.F., Gronalt, M., Hartl, R.F., Reiann, M., Strauss, C., Stuer, M., Savings ants for the vehicle routing proble. Applications of Evolutionary Coputing. Springer, Berlin. Doerner, K.F., Hartl, R.F., Kiechle, G., Lucka, M., Reiann, M., Parallel ant systes for the capacitated vehicle routing proble. In: Evolutionary Coputation in Cobinatorial Optiization: 4th European Conference, EvoCOP 2004, LNCS 3004, pp Dongarra, J., Perforance of various coputer using standard linear equations software. Report CS-89-85, University of Tennessee. Dorigo, M., Maniezzo, V., Colorni, A., Ant syste: optiization by a colony of cooperating agents. IEEE Transactions on Systes, Mans, and Cybernetics 1 (26), Gabardella, L., Taillard, E., Dorigo, M., Ant Colonies for the QAP, Technical Report 97-4, IDSIA, Lugano, Switzerland. Gendreau, M., Laporte, G., Musaraganyi, C., Taillard, E.D., A tabu search heuristic for the heterogeneous fleet vehicle routing proble. Coputers & Operations Research 26, Koulaas, C., Antony, S., Jaen, R., A survey of siulated annealing applications to operations research probles. Oega 22 (1), Mazzeo, S., Loiseau, I., An ant colony algorith for the capacitated vehicle routing. Electronic Notes in Discrete Matheatics 18, Osan, I.H., Metastrategy siulated annealing and tabu search algoriths for the vehicle routing proble. Annals of Operations Research 41, Osan, M.S., Abo-Sinna, M.A., Mousa, A.A., An effective genetic algorith approach to ultiobjective routing probles (orps). Applied Matheatics and Coputation 163, Peng, W., Tong, R.F., Tang, M., Dong, J.X., Ant colony search algoriths for optial packing proble. ICNC 2005, LNCS 3611, pp Prins, C., A siple and effective evolutionary algorith for the vehicle routing proble. Coputers & Operations Research 31, Reiann, M., Stuer, M., Doerner, K., A savings based ant syste for the vehicle routing proble. In: Langdon, W.B. et al. (Eds.), GECCO 2002: Proceedings of the Genetic and Evolutionary Coputation Conference. Morgan Kaufann, San Francisco. Renaud, J., Laporte, G., Boctor, F.F., A tabu search heuristic for the ulti-depot vehicle routing proble. Coputers & Operations Research 23 (3), Rochat, Y., Taillard, R.E., Probabilistic diversification and intensification in local search for vehicle routing. Journal of Heuristics 1, Schoonderwoerd, R., Holland, O., Bruten, J., Rothkrantz, L., Ant-based load balancing in telecounications networks. Adaptive Behavior 5 (2), Seet, F., Taillard, E.D., Solving real-life vehicle routing probles efficiently using taboo search. Annals of Operations Research 41, Taillard, R.E., Parallel iterative search ethods for vehicle routing probles. Networks 23, Tavakkoli-Moghadda, R., Safaei, N., Gholipour, Y., A hybrid siulated annealing for capacitated vehicle routing probles with the independent route length. Applied Matheatics and Coputation 176, Thangiah, S.R., Osan, I.H., Sun, T., Hybrid genetic algorith, siulated annealing and tabu search ethods for vehicle routing probles with tie windows. Technical Report 27, Coputer Science Departent, Slippery Rock University. Yang, Z.Z., Yu, B., Cheng, C.T., A parallel ant colony algorith for bus network optiization. Coputer-Aided Civil and Infrastructure Engineering 22, Yu, B., Yang, Z.Z., A dynaic holding strategy in public transit systes with real-tie inforation. Applied Intelligence, (accepted for publication), doi: /s Yu, B., Yang, Z.Z., Cheng, C.T., Optiizing the distribution of shopping centers with parallel genetic algorith. Engineering Applications of Artificial Intelligence 20 (2),