Procedia Engineering 15 (2011) 3361 – 3365

1877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2011.08.630

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ProcediaEngineering

Procedia Engineering 00 (2011) 000–000

www.elsevier.com/locate/procedia

Advanced in Control Engineering and Information Science

Hybrid ant colony optimization algorithm for two echelon vehicle routing problem

Wang Meihua*, Tian Xuhong, Chang shan ,Wu Shumin College Informatics, South China Agricultural University , Guangzhou,510642, China

Abstract

A hybrid heuristic ant colony algorithm is proposed and applied to solving the two echelon vehicle routing problem, which combines three heuristic or meta-heuristics. The problem is divided into m+1 CVRP by a separation strategy (distance-based cluster). Then better feasible solutions are built by an improve ant colony optimization with multiple neighborhood descent (IACO_MND), which is taken as the initial solution of the threshold-based local search procedure, two different neighborhood structures, i.e., threshold-based insert and threshold-based swap are successively used. Computational results on the 22 benchmark problems with the size ranging from 20 to 50 show that the proposed hybrid algorithm can find the best known solution for some problems in short time, which indicates that the proposed method outperforms other algorithm in literature. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011]

1. Introduction

Proposed by Dantzig and Ramser [1] in 1959, VRP has made a lot of research achievement in decades. But the researches about the problem more concentrated in single-level scheduling system, and the researches about the multi-level system, such as ME-VRP, are very small. Crainic et al. [2] apply the multi-level system to the practice first time for a City Logistics instance in 2004. Feliu et al. [3,4] integrated the multi-level system into the VRP first time and build the mathematical model for it, they call it Multi-Echelon Vehicle Routing Problem (ME-VRP). With the model and complexity of logistics increase, more and more researchers began to study and discuss the issue. Since VRP is a NP hard problem, so ME-VRP is NP hard, and more complex than the basic VRP. Therefore, the method for solving such problem mainly focused in heuristic and meta-heuristic which can fine the better solution in a relatively short time. And it has achieved some results. Such as Feliu et al. [3,4] use Base-Math heuristic algorithm for solving a number of public instance about 2E-VRP. Crainic et al. [5] proposed a

* * Corresponding author: Wang Meihua (1970-), Association Professor，major：AI and optimization algorithm design Tel. 13928850421; E-mail address: wangmeihua@scau.edu.cn Funds: Science and Technology Planning Project of Guangdong Province under Grant No. 2010B080701070； Natural Science Foundation of Guangdong Province under Grant 9251009001000005；National Natural Science Foundation of China under Grant 11047183

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Clustering-based Heuristic algorithm. Then they simple improved their algorithm, and proposed Multi-Start Heuristic Algorithm [6], it asked that the local search will continue until the object value is not be improved. Crainic et al. [7] analyzed the relationship of the distribution of customers, the system layout and the cost, and compared the multi-level system with the traditional single-level VRP in performance and efficiency, which can shows the feasibility and effectiveness of the Multi-Echelon VRP. Perboli et al. [8,9] derived new families of valid inequalities for the two-echelon vehicle routing problem, in order to more efficiently obtain better strength feasible solution. Therefore, this paper proposed a hybrid ant colony optimization algorithm for 2E-VRP.

2. Problem definition

Defined the central depot set V0={v0}, a set Vs of intermediate depots called satellite and customer set Vc, wherein each customer i∈Vc has a positive demand di associated, the problem consists in minimizing the total transportation costs, calculated by considering arc costs cij for shopping goods from one point to the other of the transportation network, while satisfying the demand of all the customers with a limited fleet of vehicles. Differing the classical VRP, the freight stored in V0 must transit through intermediate depots, called satellites, and then be delivered to the customers. The demand of each customer has to be satisfied by only one satellite, and there are on thresholds on minimum and maximum number of customers served by a single satellite. This assumption induces, for each 2E-VRP feasible solution, a partition of Vc set in, at most, |Vs| subsets, each one referring to a different satellite. Customer-satellite assignments are not known in advance, not allowing to solve the problem by decomposition into |Vs| +1VRPs. Two distinct fleet of vehicles m1 and m2, with different capacity size K1 and K2, are available to serve first and second network level, respectively.

3. Hybrid ACO Algorithm

Our method (Hybrid ant colony optimization algorithm) has three phases. First phase, we apply throughout a separation strategy (distance-based cluster) splitting the problem into m+1 routing sub-problems (CVRP). Second phase, we proposed Improved Ant Colony Optimization with Multiple Neighborhood Descent (IACO_MND) to solve the m+1 sub-problem. This phase can obtain better feasible solution. Last phase, we proposed Threshold-Based Local Search (TLS) to improve the better feasible solution. Follow our paper will introduce the three phases detail.

The initial clustering is based on the direct shipment criterion, which assigns a customer to its nearest satellite in Euclidean distance. It can build the correspondence between satellite and customer. Other words, one customer only belong to one satellite, but one satellite can include multiple customers. Moreover, the assignment must be feasible with respect to the satellite’s capacity constraints. If the assignment is not feasible, the customer is assigned to the second nearest satellite, and so on until a feasible assignment is found. It will split the problem into m+1 CVRP.

This phase, a hybrid heuristic algorithm IACO_MND is proposed and applied to solving the capacitated vehicle routing problem, which combines two meta-heuristics, i.e., Ant Colony Optimization [10-13] and Multiple Neighborhood Descent [14]. Several feasible solutions are built by an insertion based IACO solution construction method, which is taken as the initial solution of the MND procedure. During the MND procedure, three different neighborhood structures, i.e., insertion, swap and 2-opt are successively used.

First, we use the distance-based greedy algorithm to construct a better feasible solution s0. And setting initial pheromone according to equation ( )0 0 /f s nτ = , where n is number of the customers.

3363Wang Meihua et al. / Procedia Engineering 15 (2011) 3361 – 3365Wang Meihua, Tian Xuhong, Chang Shan,Wu Shumin/ Procedia Engineering 00 (2011) 000–000 3

IACO_MND use a knapsack-based heuristic method to construct the feasible solution. According to equation ∑

∈

=Vh

ijkijkijkijkijp βαβα μτμτ / , our algorithm select the customer k from the unvisited list V and

inserted it into the current path between customer i and j which have been added into the path. Moreover,

ijkτ is defined as ijkjikijk ττττ 2/)( += and ijkμ is defined as )(0 ijkjikkijk cccc −+−=μ ,

where ijτ is pheromone between path and i j , and is distance between customer and ijc i j .

According to equation 011 )1( τρτρτ +−= ijij

sij ∈

, we make use of the constructing solutions to

update the local pheromone, where τ and 0 1 1ρ< < is evaporation coefficient of pheromone and

0τ initial pheromone.

After each iteration, according to global pheromone updating rule τρτρτ Δ+−= ijij )1( and

)(/1 sbsf=Δτ , where sbsji ∈∀ ),( , we make use of the best counts, best solutions ssb, which are choose from best solutions in history, to update the global pheromone, where the best count is number of selecting solutions.

The main steps of the multiple neighborhood descent are the following. First, we starting it from the initial solution and choose one of the neighborhood structures to local search until you find the local optimal solution. Then restart the search process from the new better initial solution. If any of the neighborhoods cannot continue to improve the current solution, the end of the MND process. In this paper we choose three neighborhood structures: inert, swap and 2-opt.

4. Experiment and analysis

In this Section, we present the experimental results of the proposed algorithm, including the performance analysis of our algorithm, and some comparisons with the literatures in terms of both the effectiveness and efficiency.

4.1 Experiment Setup

All experiments were performed on a Windows XP machine with one 1.96 GHz Core2 processor and 2GB memory. And we use Java as programming languages to finish the algorithm. Moreover, all the comparison is on the same experimental environment and all the programmes own same time complexity. To evaluate the performance of our algorithm, we adopt the test data sets which transform from CVRP’s enchmark data build by Elion (http://branchandcut.org/VRP/data/). On how to transform data sets can be found in the literature [3,4], and these data have been many researchers as a test data set about 2E-VRP. Through a large number of experiments to determine the parameters set as follows: ant_num=6,ρ=0.2,ρ1=0.1，α=1，β=2，MaxIter=100+n，best_counts=6，MaxConsNoImprove = MaxIter/4。

4.2 Compare with other literatures

In this section, we compare our algorithm with other literatures. It can illustrate the performance of our method in another way. Compared with the literature [5], our method obtains 6 better solutions and 1 same solution in total 12 instances. Overall, our method increases 1.59% than they are. Compared with the literature [3], our method obtains 12 better solutions and 4 same solutions in total 21 instances. Meanwhile, our algorithm shows better performance in larger-scale. In the instance E-n51-k5, our algorithm increases 14.3%. Overall, our method increases 5.56% than they are. In the instances larger

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than 50, ours increase 4.2%. But in the instances smaller than 50, ours is worse 2.67% that literature [5]. In summary, our algorithm is superior to existing algorithm.

Table 1 our method compare with literature [5]

method ous literature [5] GAP(%)

E-n22-k4-S2_7_18 417.07 417.07 0

E-n22-k4-S2_9_15 384.96 387.84 0.742574

E-n22-k4-S2_10_20 470.42 487.27 3.458042

E-n22-k4-S2_11_15 371.50 383.80 3.204794

E-n22-k4-S2_12_13 444.66 437.351 -1.6712

E-n22-k4-S2_13_17 405.56 524.63 22.696

E-n33-k4-S2_2_10 774.54 733.60 -5.5807

E-n33-k4-S2_3_14 745.40 740.10 -0.71612

E-n33-k4-S2_4_18 801.16 748.43 -7.04542

E-n33-k4-S2_5_6 760.76 835.20 8.912835

E-n33-k4-S2_8_26 756.19 756.65 0.060794

E-n33-k4-S2_15_23 824.58 785.47 -4.97918

Table 2 our method compare with literature [3]

method ous literature [3] GAP(%)

E-n22-k4-S2_7_18 417.07 417.07 0

E-n22-k4-S2_9_15 384.96 384.96 0

E-n22-k4-S2_10_20 470.42 470.42 0

E-n22-k4-S2_11_15 371.50 371.50 0

E-n22-k4-S2_12_13 444.66 427.22 -4.08221

E-n22-k4-S2_13_17 405.56 392.78 -3.25373

E-n33-k4-S2_2_10 774.54 749.36 -3.3602

E-n33-k4-S2_3_14 745.40 751.74 0.843377

E-n33-k4-S2_4_18 801.16 729.91 -9.76148

E-n33-k4-S2_5_6 760.76 851.78 10.68586

E-n33-k4-S2_8_26 756.19 766.94 1.401674

E-n33-k4-S2_15_23 824.58 787.31 -4.73384

E-n51-k5-S2_3_18 626.23 664.96 5.82441

5. Summary

It is a hot research field that combination of the advantages of different heuristic and meta-heuristic to design more effective hybrid heuristic algorithm for combinatorial optimization problems. Combination of greedy algorithm, ant colony optimization and local search algorithm, this paper

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4-5955

639-646

171-176

proposed a hybrid ant colony optimization algorithm to solve the 2E-VRP. Make use of the rapid of the traditional heuristic, the search diversity of ant colony optimization and the strong local search ability of local search to improve the quality of the solution and speed up the convergence of the algorithm. In the experiment, we run our method in 22 different scale benchmark data. The result illustrates that our algorithm can obtain optimal or better feasible solution in a short time. Furthermore, we compared our algorithm with other 2 literatures, and our method increased 1.59%, 5.56% than they are. It shows that our algorithm is superior to existing algorithm.

Acknowledgements

The authors thank Mr. Zhang Zebin for his programming of the algorithms.

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