A Hybrid Bat Algorithm with Path Relinking for Capacitated Vehicle Routing Problem

Abstract

The capacitated vehicle routing problem (CVRP) is an NP-hard problem with wide engineering and theoretical background. In this paper, a hybrid bat algorithm with path relinking (HBA-PR) is proposed to solve CVRP. The HBA-PR is constructed based on the framework of continuous bat algorithm; the greedy randomized adaptive search procedure (GRASP) and path relinking are effectively integrated into bat algorithm. Moreover, in order to further improve the performance, the random subsequences and single-point local search are operated with certain loudness (probability). In order to verify the validity of the method in this paper, and it's efficiency and with other existing methods, several classical CVRP instances from three classes of CVRP benchmarks are selected to tested. Experimental results and comparisons show that the HBA-PR is effective for CVRP.

1. Introduction

The vehicle routing problem (VRP) is a classical combinatorial optimization problem that was proposed in the late 1950s, and it is still a research hotspot in the field of Operations Research. The capacitated vehicle routing problem (CVRP) is introduced by Dantzig and Ramser in 1959 [1], which designs a set of customer demands that have to be served with a fleet of vehicles from a depot or central node, each vehicle has the uniform capacity, and each customer has a certain demand that must be satisfied at minimal cost. These costs usually represent distances, traveling times, number of vehicles employed, or a combination of these factors.

It is known that the CVRP is an NP-hard problem [2]. Various approaches have been presented to solve the CVRP during the last decades, such as linear programming [3], several metaheuristics [4–6], and many hybrid heuristics with variable neighborhood search or constructive heuristic methods [7–10]. The overview of methods presented in [6] shows that at least 29 different methods for the CVRP exist; all achieve more or less comparable performance. Although several methods can produce good solutions, the computational time is long when the scale of instances is large. Meanwhile, an abundance of methods for the CVRP is a population-based algorithm and the parameter setting of the algorithm, is pretty important; however, the parameter setting of many metaheuristics is not considered in the literature.

Bat algorithm (BA) is a fairly new metaheuristics proposed by Yang [11] in 2010, which inspired by the intelligent echolocation behavior of microbats when they forage. As we know, many new metaheuristics have been widely used and successfully applied to solve the CVRP notwithstanding BA has not yet been applied to solve the CVRP, and BA has been applied to solve other problems with great success. For example, Gandom et al. focus on solving constrained optimization tasks [12]. Yang and Gandom apply bat algorithm to solve many global engineering optimizations [13]. A classification mode is proposed by Mishra et al. using bat algorithm to update the weights of a functional link artificial neural network (FLANN) classifier [14]. Meanwhile, some researchers have improved bat algorithm and applied it to various optimization problems. Xie et al. proposed a DLBA bat algorithm based on differential operator and Lévy flights trajectory to solve function optimization and nonlinear equations [15]. Wang et al. proposed a new bat algorithm with mutation (BAM) to solve the uninhabited combat air vehicle (UCAV) path planning problem [16]. In this paper, we propose a hybrid bat algorithm (HBA-PR) to solve capacitated vehicle routing problem.

The rest of this paper is organized as follows. The problem description of CVRP and original bat algorithm are described in Section 2. The hybrid bat algorithm (HBA-PR) for CVRP is described detailedly in Section 3. The experimental results of the HBA-PR and comparisons with other previous algorithms are shown in Section 4. In the last section, we conclude this paper and point out some future work in Section 5.

2. Problem Descriptions and Bat Algorithm

2.1. Capacitated Vehicle Routing Problem

The CVRP is considered to be the classical version of the VRP, which designs a set of customer demands that have to be served with a fleet of vehicles from a depot or central node, each vehicle has the uniform capacity, and each customer has a certain demand. The objective makes the expended cost as minimal as possible. Let 𝐺=(𝑉,𝐸) be a complete graph with a set of vertices 𝑉={0,1,…,𝑘}, where the vertex {0} represents the depot and the remaining ones represent the customers. Each edge 𝑒_𝑖𝑗={𝑖,𝑗}∈𝐸 has a nonnegative cost 𝑐_𝑖𝑗, and each customer 𝑖∈𝑉^′=𝑉⧵{0} has a demand 𝑑_𝑖. Let 𝑆=(1,2,…,𝑚) be the set of homogeneous vehicles with capacity 𝑄. The CVRP consists in constructing a set of up to 𝑘 routes in such a way that (1) every route starts and ends at the depot; (2) all demands are accomplished; (3) the vehicle’s capacity is not exceeded; (4) a customer is visited by only a single vehicle; (5) the sum of costs is minimized. The mathematical formulas are defined as follows [10]: minZ=^𝑘𝑘_𝑖=0^𝑚_𝑗=0𝑠=0𝑐_𝑖𝑗𝑒_𝑖𝑗𝑠,s.t.^𝑘_𝑖=0𝑑_𝑖𝑦_𝑖𝑠≤𝑄,𝑠=1,2,…,𝑚,_𝑚𝑠=1𝑦_𝑖𝑠=1,𝑖=1,2,…,𝑘,_𝑘𝑖=0𝑒_𝑖𝑗𝑠=𝑦_𝑖𝑠,𝑗=1,2,…,𝑘;𝑠=1,2,…,𝑚,_𝑘𝑗=0𝑒_𝑖𝑗𝑠=𝑦_𝑖𝑠,𝑖=1,2,…,𝑘;𝑠=1,2,…,𝑚, (1)(2) where 𝑠 denotes the number of vehicles; 𝑒_𝑖𝑗𝑠=1 if vehicle 𝑠 is from 𝑖 to 𝑗, otherwise 𝑒_𝑖𝑗𝑠=0; the 𝑦_𝑖𝑠=1 if vehicle 𝑠 is loading (active), otherwise 𝑦_𝑖𝑠=0.

2.2. Basic Bat Algorithm

The basic bat algorithm (BA) is a metaheuristic first introduced by Yang in 2010. In simulations of BA, under several under idealized rules, the updated rules of bat’s positions 𝑥_𝑖 and velocities 𝑣_𝑖 in a 𝐷-dimensional search space are defined. The new solutions 𝑥^𝑡_𝑖 and velocities 𝑣^𝑡_𝑖 at generation 𝑡 are given by fr_𝑖=fr_min+fr_max−fr_min𝑣𝛽,_𝑡𝑖=𝑣_𝑖^𝑡−1+𝑥^𝑡_𝑖−𝑥_∗fr_𝑖,𝑥^𝑡_𝑖=𝑥_𝑖^𝑡−1+𝑣^𝑡_𝑖, (3) where 𝛽∈[0,1] is a random vector drawn from a uniform distribution and fr_𝑖 denotes frequency of each bat. Generally, the frequency fr_𝑖∈[fr_min,fr_max]. Here 𝑥_∗ is the current global best location (solution) which is located after comparing all the solutions among all the 𝑛 bats.

After the position updating of bat, a random number is generated; if the random number is greater than the pulse emission rate 𝑟_𝑖, a new position will be generated around the current best solutions; it can be represented by 𝑥=𝑥_∗+𝜀×Ld_𝑡, (4) where 𝜀∈[−1,1] is a random number, while Ld_𝑡=⟨Ld^𝑡_𝑖⟩ is the average loudness of all the bats at current generation 𝑡.

Furthermore, the loudness Ld_𝑖 and the pulse emission rate 𝑟_𝑖 will be updated, and a solution will be accepted if a random number is less than loudness Ld_𝑖 and 𝑓(𝑥_𝑖)<𝑓(𝑥_∗). Ld_𝑖 and 𝑟_𝑖 are updated by Ld_𝑖^𝑡+1=𝛼×Ld^𝑡_𝑖,𝑟_𝑖^𝑡+1=𝑟⁰_𝑖×1−exp−𝛾×𝑡, (5) where 𝛼, 𝛾 are constants and 𝑓(⋅) is fitness function. The algorithm repeat until the termination criterion is reached.

3. Hybrid Bat Algorithm with Path Relinking for CVRP

3.1. Solution Representation in HBA-PR

Since standard BA is a continuous optimization algorithm, the standard continuous encoding scheme of BA cannot be used to solve CVRP directly. In order to apply BA to solve CVRP, the first step is to devise a suitable representation for the candidate solutions in designing a hybrid bat algorithm for a particular problem. Each individual is a sequence with integer number which is the order of visiting these customers; the 0 represents the depot. For example, if we have the following three routes:(6) the coded individual with integer is 0→3→5→8→4→9→2→6→1→7→0, and the bat individual is represented as 3→5→8→4→9→2→6→1→7.

3.2. Hybrid Bat Algorithm

Aimed at the capacitated vehicle routing problem, based on the idea of bat algorithm, a hybrid bat algorithm is proposed, which integrates greedy randomized adaptive search procedure (GRASP) heuristic and bat algorithm, and the path relinking as an intensification strategy to explore local trajectories connecting elite solutions obtained by the proposed algorithm. The hybrid bat algorithm with path relinking is named as HBA-PR.

GRASP [17, 18] is a heuristic already applied to many optimization problems successfully [19–21]. GRASP consists of a two-phase iterative process: construction phase and local search phase. In the first phase, a greedy randomized solution is built. Due to this solution is not guaranteed to be locally optimal; a local search is performed in the second phase. The final result is simply the best solution found over all iterations.

The construction phase can be described as a process which stepwise adds one element at a time to a partial (incomplete) solution. According to a greedy function, all elements are sorted, and the Restricted Candidate List (RCL) is constructed based on the order, and then from the list randomly select a element. In the second phase, a local search is initialized from these points, this iterative process is repeated until a termination criterion is met, and the best solution found over all iterations is taken as the result.

RCL is created using a parameter 𝛼 to restrict the size of this list. Candidate 𝑒∈𝐶 is sorted according to their greedy function value 𝑓(𝑒). In a cardinality-based RCL, the latter is made up by the 𝑘 top-ranked elements. In a value-based construction, the RCL consists of the elements in the set {𝑒∈𝐶∶𝑓_∗≤𝑓(𝑒)≤𝑓_∗+𝛼×(𝑓^∗−𝑓_∗)}, where 𝑓_∗=min(𝑓(𝑒)∶𝑒∈𝐶), 𝑓^∗=max{𝑓(𝑒)∶𝑒∈𝐶}, and 𝛼∈[0,1]. Since the best value for 𝛼 is often difficult to determine, a random value is often assigned. The values for 𝛼 adopted in the constructive heuristics are set using reactive strategies, which usually leads to better performance than using fixed values [22].

In original bat algorithm, the bat individual randomly selects a certain range of frequency, its velocity is updated according to their selected frequency, and at last a new position is generated using its velocity and its own position. The idea is that the position of bat individual is updated by adjusting its frequency of sonic pulse. In this paper, the position updating of bat adopted GRASP to generate a new position, the frequency is used for restricting the size of CRL, frequency equivalent to parameter 𝛼 in GRASP, and the frequency is variable value. In basic bat algorithm, the loudness Ld and the pulse emission rate 𝑟 are updated according as the iterations proceed. As the loudness usually decreases, while the rate of pulse emission increases, it indicates that the bats approximate their prey (optimum solution). The pulse emission rate 𝑟 is updated by 𝑟(𝑡)=1+exp−5𝑡_max×𝑡−𝑡_max/2⁻¹, (7) where 𝑡 denotes the 𝑡th generation and 𝑡_max is the maximal generation. The rate 𝑟 is similar to Sigmoid function, and the frequency fr is determined according to the pulse emission rate 𝑟, which is represented by ((((fr=1−max0.2,min0.8,𝑟)),rand>𝑟,max0.2,min0.8,𝑟)),rand≤𝑟, (8) where rand is a random number and 0.2 and 0.8 are experience parameters reference [18]. The frequency fr decreases gradually at firstly and then increases gradually, while the generation 𝑡 increases. Figure 1 is the changing curve of rate 𝑟, Figure 2 is an example of frequency fr, and Algorithm 1 shows the pseudocode of greedy randomized construction with frequency fr.

𝑆←𝑅  rand_Chosen_Vertex(𝑣);  // R is selected at random a vertex as initial solution 𝐶←𝑉⧵𝑣; // The candidate set 𝐶 is initialized While    𝐶≠Ø   do    𝐼𝐶←𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒_Incremental_Costs(𝑆); //  the incremental costs are evaluated    𝑖𝑐min←min({𝑖𝑐∈𝐼𝐶});     𝑖𝑐max←max({𝑖𝑐∈𝐼𝐶});     𝑅𝐶𝐿←𝑒∈𝐶∣𝑖𝑐(𝑒)≤𝑖𝑐min+𝑓𝑟×𝑖𝑐max−𝑖𝑐min;  // 𝑅𝐶𝐿 is created     𝑠←𝑆𝑒𝑙𝑒𝑐𝑡_Element (RCL); // a vertex 𝑠 is randomly selected from 𝑅𝐶𝐿     𝑆←𝑂𝑏𝑡𝑎𝑖𝑛_Min_Solution(𝑆, 𝑠); // the partial tour is updated by inserting the     vertex 𝑠     𝐶←𝐶⧵{𝑠}; // the candidate set 𝐶 is updated end return    𝑆

Algorithm 1 

Greedy_Randomized_Construction (𝑓𝑟).

Figure 1 

Changing curve of rate 𝑟.

Figure 2 

Changing curve of frequency fr.

The local search phase uses the 2-Opt heuristic for exchanging. Algorithm 2 shows the local search procedure. The input parameter is an initial solution 𝑆 obtained by the Greedy_Randomized_Construction procedure. The current route need is divided into 𝑚 subroute according to the load, set the start, and end of suroute said to 0; for example, 𝑆={1,2,3,4,5,6,7}, subroute is 0→1→2→3→4→0, and 0→5→6→7→0.

𝑆′←𝑆; {𝑠𝑟}←𝐶𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡_Sub-route(𝑆′); for each sub-route 𝑠𝑟 do   𝑠𝑟′←2-opt (𝑠𝑟); // carry out the 2-opt operation end 𝑆′←𝐶𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡_Indiviaual 𝑠𝑟′; if 𝑓(𝑆′)<𝑓(𝑆)   then   𝑆←𝑆′ end return    𝑆

Algorithm 2 

Local_Search_Phase (𝑆).

3.3. Hybrid Bat Algorithm with Path Relinking

Path relinking was originally proposed by Glover [23]; Laguna and Martí [24] were the first to use path relinking within a GRASP strategy. Path relinking generates new solutions by exploring trajectories connecting an initial solution 𝑥_𝑠 to an elite guiding solution 𝑥_𝑡. The path relinking procedure consists in selecting moves that introduce attributes contained in the guiding solution 𝑥_𝑡 to the initial solution 𝑥_𝑠 until the initial solution is completely transformed in the guiding solution 𝑥_𝑡. Path relinking may also be viewed as a constrained local search strategy applied to the initial solution 𝑥_𝑠. Furthermore, there are several alternatives that have been considered, which involve tradeoffs between computation time and solution quality. These alternatives include periodical relinking, forward relinking, backward relinking, back and forward relinking, mixed relinking, randomized relinking, and truncated relinking [18].

One important issue in implementing a path relinking technique is the strategy to construct the elite set (ES). We adopted a fixed size elite set, and a solution 𝑥 is inserted into the ES as follows.

A solution 𝑥 is always inserted into ES if it is not full. Otherwise, the generated solution 𝑥 is inserted in ES only if its cost is better than the worst cost solution found in ES, and the worst cost solution is replaced by the solution 𝑥. Algorithm 3 shows the pseudocode for the algorithm to construct and maintain the elite set ES.

if cardinality(𝐸𝑆) ≠ 𝑝𝑠 then  // 𝑝𝑠 is the population size 𝐸𝑆←𝐸𝑆∪{𝑥} else   𝐸𝑆𝑤←𝑡ℎ𝑒 worst solution in 𝐸𝑆   if 𝑓(𝑥)<𝑓(𝐸𝑆𝑤) then        𝐸𝑆←𝐸𝑆⧵{𝐸𝑆𝑤}       𝐸𝑆←𝐸𝑆∪{𝑥}   end end return    𝐸𝑆

Algorithm 3 

Elite_Set (𝑥).

Algorithm 4 shows the pseudocode of path relinking procedure; an instance is 𝑥_𝑠={1,2,3,4,5,6,7}, 𝑥_𝑡={1,3,4,2,5,6,7}, according to process of Algorithm 4, the first different element is position 2 in 𝑥_𝑠; after executing replace operation, the 𝑥₁={1,3,4,2,5,6,7}; the second different element is position 3 in 𝑥_𝑠; after executing replace operation, the 𝑥₂={1,2,4,3,5,6,7}; the third different element is position 4 in 𝑥_𝑠; after executing replace operation, the 𝑥₃={1,4,3,2,5,6,7}; if the 𝑥₂ is better than the 𝑥_𝑡, and the only, then return 𝑥₂.

𝑥𝑠←𝑥; // initial solution 𝑥𝑡←𝐸𝑆∗; //  guiding solution 𝑓(𝑥𝑡)←𝑓(𝐸𝑆∗); Δ←𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒(𝑥𝑠,𝑥𝑡); // find out the difference position between 𝑥𝑠 𝑥𝑡 for 𝑖=1:  cardinality(Δ) do  𝑗←𝐹𝑖𝑛𝑑_Position(𝑥𝑠,𝑥𝑡,Δ𝑖); // find the position of the element of 𝑥𝑡  in Δ𝑖 which in 𝑥𝑠  𝑥𝑖←𝑅𝑒𝑝𝑙𝑎𝑐𝑒(𝑥𝑠,Δ𝑖,𝑗); // replace the 𝑗  position of 𝑥𝑠  with the  element of 𝑥𝑠  in Δ𝑖  𝑥𝑖←𝑅𝑒𝑝𝑙𝑎𝑐𝑒(𝑥𝑠,𝑥𝑡,Δ𝑖);  //replace the Δ𝑖  position of 𝑥𝑠  with the corresponding element of 𝑥𝑡   if    𝑓(𝑥𝑖)<𝑓(𝑥𝑡)   then    𝑥∗←𝑥𝑖;    𝑓(𝑥𝑡)←𝑓(𝑥𝑖);   end end return    𝑥∗

Algorithm 4 

Path_Relinking (𝐸𝑆_∗, 𝑥).

3.4. Subsequence and Single-Point Local Search

In this paper, the loudness Ld_𝑖 of bat individual 𝑖 is relative to its own fitness fit_𝑖, the better fitness, and the lower loudness. The loudness can be described by Ld_𝑖=fit_𝑖−fit_∗+0.1fit^∗−fit_∗+0.1, (9) where constant 0.1 is used for avoiding the denominator is zero, and fit_𝑖 is the fitness of individual 𝑖, fit_∗ and fit^∗ are the minimum and maximum fitness in current population, respectively. In HBA-PR, the loudness reflects the quality of individual. In this algorithm, there are two kinds of local search are embedded into HBA-PR to further improve the performance, random subsequence local search that and single-point local search. The random subsequence local search includes random subsequence inverse and random subsequence insert, and the single-point local search includes single-point insert and single-point swap.

For random subsequence insert, firstly, randomly selected an origin of subsequence, and then randomly selected a length of subsequence which is less than length of individual 𝑆. Secondly, after determining the subsequence 𝑆1, randomly selected an insert point in remainder subsequence 𝑆2, 𝑆=𝑆1∪𝑆2; the 𝑆1 is inserted into 𝑆2 location in insert point. An example is shown in Figure 2. For random subsequence inverse, firstly, randomly selected a subsequence with random length, and then the inverse operation is performed. An example is shown in Figure 3.

Figure 3 

Random subsequence insert operation.

For single-point swap, choose two different positions from a permutation randomly and swap them. For single-point insert, choose two different positions from a permutation randomly, and the element in first position is insert into the back of second element. Similarly, two instances are shown in Figures 4 and 5.

Figure 4 

Random subsequence inverse operation.

Figure 5 

Single-point swap operation.

In local search part, the random subsequence local search is performed before random single-point local search. The random subsequence insert and random subsequence inverse are preformed according to loudness Ld. In other words, if a random number is greater than the loudness Ld_𝑖, the random subsequence insert is performed; otherwise, the random subsequence inverse is performed. Similarly, the random single-point insert and random single-point swap are performed with the loudness Ld. If a random number is greater than the loudness Ld_𝑖, the insert operation is performed (Figure 6); otherwise, the swap operation is performed. Note that where local search is operated on the current optimal individual ES_∗ in elite set ES, local search is performed for each individual. Algorithms 5 and 6 show the pseudocode of subsequence and single-point local search.

for 𝑖=1:  ps do  if    rand>𝐿𝑑𝑖 then     𝑥←𝑆𝑢𝑏-sequence_Insert(𝐸𝑆∗); // perform random sub-sequence insert operation  else    𝑥←𝑆𝑢𝑏-sequence_Inverse(𝐸𝑆∗); // perform random sub-sequence inverse operation  end end return    𝑥

Algorithm 5 

Sub-sequence_Local_Search (𝐸𝑆_∗).

for 𝑖=1:  ps do   if rand>𝐿𝑑𝑖 then     𝑥←𝑆𝑖𝑛𝑔𝑙𝑒-point_Insert(𝐸𝑆∗);  //  perform random single-point insert operation   else     𝑥← 𝑆𝑖𝑛𝑔𝑙𝑒-point_Swap(𝐸𝑆∗); // perform random ingle point swap operation   end end return    𝑥

Algorithm 6 

Single-point_Local_Search (𝐸𝑆_∗).

Figure 6 

Single-point insert operation.

3.5. HBA-PR Framework for CVRP

We propose in this work, to incorporate greedy randomized adaptive search procedure, path-relinking strategies, subsequence, and single-point local search to the bat algorithm by defining distinct ways to solve capacitated vehicle routing problem. This iterative process is repeated until the termination criterion is met; Algorithm 7 shows the pseudocode of HBA-PR for CVRP.

(1) Initialize the 𝑝𝑠,  bat population and other parameters; (2) Evaluate fitness for each individual; (3) 𝐸𝑆←𝐸𝑙𝑖𝑡𝑒_Set(𝑥); (4) while  𝑡≤𝑡max  do (5)  Compute pulse emission rate by (7); (6)  Determine frequency 𝑓𝑟 by (8); (7)  for 𝑖=1:  ps do (8)    𝑥←𝐺𝑟𝑒𝑒𝑑𝑦_Randomized_Construction(𝑓𝑟); (9)    𝑥←𝐿𝑜𝑐𝑎𝑙_Search_Phase (𝑥); (10)  end (11)  Evaluate fitness for each individual 𝑥; (12)   𝐸𝑆←𝐸𝑙𝑖𝑡𝑒_Set(𝑥); (13)   𝐸𝑆∗←𝑆𝑒𝑙𝑒𝑐𝑡_Best_Elite(𝐸𝑆); // select the best individual in elite set 𝐸𝑆 (14)  for 𝑖=1:   𝑝𝑠 do (15)    𝑥←𝑃𝑎𝑡ℎ_Relinking(𝐸𝑆∗,  𝑥); (16)  end (17)  Evaluate fitness for each individual 𝑥; (18)   𝐸𝑆←𝐸𝑙𝑖𝑡𝑒_Set(𝑥); (19)   𝐿𝑑𝑖←𝐶𝑜𝑚𝑝𝑢𝑡𝑒_Loudness(𝑓(𝑥)); //Compute loudness of each individual by (9); (20)   𝐸𝑆∗←𝑆𝑒𝑙𝑒𝑐𝑡_Best_Elite(𝐸𝑆); (21)   𝑥←𝑆𝑢𝑏-sequence_Local_Search (𝐸𝑆∗)  //carry out random sub-sequence local    search (22)  Evaluate fitness for each individual 𝑥; (23)   𝐸𝑆←𝐸𝑙𝑖𝑡𝑒_Set(𝑥); (24)   𝐸𝑆∗←𝑆𝑒𝑙𝑒𝑐𝑡_Best_Elite(𝐸𝑆); (25)   𝑥←𝑆𝑖𝑛𝑔𝑙𝑒-point_Local_Search (𝐸𝑆∗)  //carry out random single-point local search (26)  Evaluate fitness for each individual 𝑥; (27)   𝐸𝑆←𝐸𝑙𝑖𝑡𝑒_Set(𝑥); (28)  𝑡=𝑡+1 (29) end (30) Output result and plot

Algorithm 7 

HBA-PR.

4. Numerical Simulation Results and Comparisons

To test the performance of the proposed HBA-PR which is extensively investigated by a large number of experimental studies computational simulations are carried out with some well-studied problems taken from the web http://www.branchandcut.org/, a reference site which contains detailed information regarding a large number of benchmark instances. In this paper, 12 instances from three classes of benchmarks are selected. The first class is Augerat et al. Set A instances, the second class is Augerat et al. Set P instances, and the third class is Augerat et al. Set E instances. So far, these problems have been widely used as benchmarks to certify the performance of algorithms by many researchers.

All computational experiments are conducted with MATLAB 2012a, and in our simulation, numerical experiments are run on a PC with AMD Athlon (tm) II X4 640 Processor 3.0 GHz and 2.0 GB memory. In the experiment, the termination criterion is set as maximum generation 𝑡_max=200. Each instance independently run15 times for comparison.

4.1. Parameter Analysis

In the subsection, parameters of HBA-PR are determined by experiments, and the impact of each parameter is analyzed. In particular, the HBA-PR has few parameters; we only need to test population size (𝑝𝑠) in HBA-PR. A small 𝑝𝑠 which may lead to insufficient population information is provided, and the diversity cannot be guaranteed. On the other side, a large one which indicates diversity is sufficient, but the computing time will increase and the precision of optimal solution may have lesser improvement. In order to evaluate the sensitivity of parameters 𝑝𝑠, three benchmarks selected from different benchmark set are chosen to run 10 times. These benchmarks are A_n33_k5, E_n23_k3 and P_n19_k2, and the statistical result and convergence curves are shown in Figures 7, 8, 9, and 10.

Figure 7 

Box-and-whisker diagram of selected three instances.

Figure 8 

Convergence curve for A_n33_k5.

Figure 9 

Convergence curve for E_n23_k3.

Figure 10 

Convergence curve for P_n19_k2.

The ordinate normalized fitness (log) in Figures 8–10 is logarithm of normalized fitness; the aim is to show the convergence curves clearly. The normalization formula of fitness is (fit−fit_∗)/(fit^∗−fit_∗), where fit_∗ is the best-known solution, and fit^∗ is the initial fitness.

Figure 7 represents the relative error of test case A_n33_k5, E_n23_k3, and P_n19_k2 after 10 times independent running, which shows the sensitivity of parameter 𝑝𝑠. From the three test cases, the parameter 𝑝𝑠 can be determined. For A_n33_k5 and P_n19_k2, the performance of HBA-PR is better when 𝑝𝑠=30. For E_n23_k3, that 𝑝𝑠 is equal to 50 is best, but the performance is good as well, while 𝑝𝑠=30. From Figures 8–10, the information provided by population is sufficient when 𝑝𝑠 is greater than 20; however, the convergence rate is better when 𝑝𝑠=30 among the three instances. Considering the tradeoff between the stability of algorithm and the rate of convergence, the parameter 𝑝𝑠 takes a compromise values, 𝑝𝑠=30.

4.2. Comparisons of Simulation Results

In order to show the effective of HBA-PR, we carry out a simulation to compare HBA-PR with other state-of-art algorithms, that is, a parallel version of the classical Clarke and Wright Savings (CWS) heuristic and SR-GCWS (Simulation in Routing via the Generalized Clarke and Wright Savings heuristic) proposed by Juan et al. [9], CS-GRASP proposed by Zheng et al. [10]. Results of these simulations are summarized in Table 1, which contain the following information of each instance: vehicle capacity; tightness (demand/capacity); I_BKS is integral best-known solution (BKS) or “optimal” value according to the web http://www.branchandcut.org/; R_BSK is verifies real costs for the best-known solution according to [9]; CWS is the costs associated with the solution given by the parallel version of the CWS heuristic; SR-GCWS is the best solution obtained by SR-GCWS method; HBA-PR is our best solution, where “—” represents no records in the literature.

Table 1 

Comparison of results for Augerat et al. Set A, E, and P instances.

From the simulationresults obtained by testing HBA-PR, it demonstrates that using the proposed HBA-PR to solve the CVRP is effective, and the performance of HBA-PR is prominent. From Table 1, all instances achieved a good quality solution, and solutions of 6 instances are better than the best-known solution (“optimal” value). HBA-PR has matched 10 of the 12 best-known solutions except for P_n51_k10 and A_n39_k6, and the average deviation from the real costs best-known solution is 0.057%. HBA-PR outperforms CWS for the 12 instances. Comparing with CS-GRASP and HBA-PR, HBA-PR outperforms CS-GRASP in terms of A_n33_k5, A_n33_k6, and E_n23_k3; the other instances have same results. The SR-GCWS and HBA-PR have similar results, and the gap is very small.

Furthermore, Figure 11 shows the best solution found so far by using our methodology for the A_n37_k5.vrp file, where the depot (using 1 instead of 0) is at the center. Analogously, Figures 12 and 13 show the best solution found by HBA-PR for the E-n51-k5.vrp file and P_n51_k10.vrp file. The convergence curve for some test problems has been shown in Figure 14. From Figure 14, for instance, A_n37_k5, it converges to an optimal solution after 43 generation, and it expends about 170 generations, for instance, E-n51-k5 and P_n51_k10 when algorithm converges to an optimal solution, which demonstrate that the HBA-PR has a faster convergence rate.

Figure 11 

Optimal routes of A_n37_k5.

Figure 12 

Optimal routes of E_n51_k5.

Figure 13 

Optimal routes of P_n51_k10.

Figure 14 

Convergence curve of three instances.

In general, the proposed HBA-PR can produce good solutions when compared with existing heuristics for solving the CVRP, and the convergence rate of HBA-PR is faster. These results seem to indicate that the hybrid bat algorithm with path relinking is an alternative to solve the capacitated vehicle routing problem.

5. Conclusions

The capacitated vehicle routing problem is important in the fields of Operations Research, which is an NP-hard problem. Bat algorithm is a continuous metaheuristics; it cannot be used to solve the CVRP directly. In this paper, a hybrid bat algorithm with path relinking (HBA-PR) for solving the CVRP has been presented. This methodology, which does not require any particular fine tuning of parameters or configuration process, combines the classical greedy randomized adaptive search procedure (GRASP) with bat algorithm; the path relinking as an intensification strategy to explore local trajectories connecting elite solutions obtained by proposed algorithm; subsequence and single-point local search are effectively integrated into HBA-PR. Results show that our methodology is able to provide fine-quality solutions which can compete with the ones provided by some exact and heuristic approaches. Moreover, because of its simplicity and flexibility, we think that this methodology can easily be adapted to other variants of the vehicle routing problem and even to other combinatorial problems, for example, the vehicle routing problem with time window and traveling salesman problem, which is our further work.

Acknowledgments

This work is supported by National Science Foundation of China under Grant no. 61165015, Key Project of Guangxi Science Foundation under Grant no. 2012GXNSFDA053028, Key Project of Guangxi High School Science Foundation under Grant no. 20121ZD008, the Funded by Open Research Fund Program of Key Lab of Intelligent Perception and Image Understanding of Ministry of Education of China under Grant no. IPIU01201100, and the Innovation Project of Guangxi Graduate Education under Grant no. YCSZ2012063.