An improved immigration memetic algorithm for solving the heterogeneous fixed fleet vehicle routing problem

Neurocomputing 150 (2015) 58–66

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

An improved immigration memetic algorithm for solving the heterogeneous fixed fleet vehicle routing problem Oliviu Matei a, Petrică C. Pop b,n, Jozsef Laszlo Sas b, Camelia Chira c a

Technical University of Cluj-Napoca, North University Center of Baia Mare, Department of Electrical Engineering, Baia Mare, Romania Technical University of Cluj-Napoca, North University Center of Baia Mare, Department of Mathematics and Computer Science, Baia Mare, Romania c Technical University of Cluj-Napoca, Department of Computer Science, Cluj-Napoca, Romania b

art ic l e i nf o

a b s t r a c t

Article history: Received 30 September 2013 Received in revised form 4 February 2014 Accepted 26 February 2014 Available online 5 October 2014

This paper deals with the heterogeneous fixed fleet vehicle routing problem (HFFVRP) which is a generalization of the classical vehicle routing problem (VRP) in the sense that the fixed fleet of vehicles is assumed to be heterogeneous. The objective of HFFVRP is to find the best fleet composition and the collection of routes such that the total costs are minimized. To address this combinatorial optimization problem, we design and implement a hybrid heuristic model integrating a genetic algorithm, a local search mechanism and an immigration strategy. Several strategies for generating the initial population of the genetic algorithm in relation with six local search heuristics are considered. An important feature of the proposed approach refers to the immigration strategy used to ensure diversification by which the level of evolution for the new immigrant individuals increases along with the evolution of the population. The proposed algorithm is tested on a set of HFFVRP benchmark instances and the preliminary results point out that our approach is an attractive and appropriate method to explore the solution space of this complex problem leading to good solutions within reasonable computational times. & 2014 Elsevier B.V. All rights reserved.

Keywords: Heterogeneous fixed fleet vehicle routing problem Genetic algorithms Immigration techniques Local search Memetic algorithms

1. Introduction Problems associated with determining optimal routes for vehicles from one or several depots to a set of locations/customers are known as Vehicle Routing Problems (VRPs) and have many practical applications in the field of distribution and logistics. A wide body of literature exists on the problem (for an extensive bibliography, see Laporte [9,10] and the book edited by Ball et al. [1]). Given a set of vehicles, a set of locations containing the depot location and the distance between each pair of locations, the VRP consists in finding the minimum cost tour for each vehicle such that all locations are visited and each vehicle returns to the depot. Because of the VRP simplicity, most attractive to many researchers have been the variations of the VRP, built on the basic VRP with extra features such as:


The Capacitated VRP [25] in which each vehicle has finite capacity and each location has a finite demand.


The VRP with Time Windows [19] in which there is a specified


The VRP with Multiple Depots [3] generalizes the idea of a






depot in such a way that there are several depots from which each customer can be served. The Heterogeneous Fixed Fleet VRP [22] in which we have a fleet of heterogeneous (different types) vehicles using the depot as a starting base. The Multi-Commodity VRP [18] in which each location has a demand for different commodities and each vehicle has a set of compartments in which only one commodity can be loaded. The problem then becomes that of deciding which commodities to place in which compartments in order to minimize distance traveled. The Generalized Vehicle Routing Problem (GVRP) [5,15] is the problem of designing optimal delivery or collection routes, subject to capacity restrictions, from a given depot to a number of predefined, mutually exclusive and exhaustive node-sets (clusters) with the property that exactly one node is visited from each cluster.

temporal window of opportunity in which to visit each location.

n

Corresponding author. E-mail address: [email protected] (P.C. Pop).

http://dx.doi.org/10.1016/j.neucom.2014.02.074 0925-2312/& 2014 Elsevier B.V. All rights reserved.

We are concerned in this paper with the heterogeneous fixed fleet vehicle routing problem (HFFVRP) introduced by Taillard [22]. The HFFVRP is an important variant of VRP, since usually the fleets are heterogeneous in most practical situations. Various heuristic and metaheuristic algorithms have been developed for solving the HFFVRP including an algorithm based on Tabu Search, adaptive

O. Matei et al. / Neurocomputing 150 (2015) 58–66

memory and column generation described by Taillard [22], an algorithm that extends a number of VRP classical heuristics followed by a local search procedure based on the Steepest Descent Local Search and Tabu Search introduced by Prins [16], a threshold accepting procedure implemented by Tarantilis et al. [23] where a worse solution is only accepted if it is within a given threshold (the same authors [24] later presented another threshold accepting procedure to solve the same problem). Also, a record-to-record travel algorithm was proposed by Li et al. [11] and a multi-start adaptive memory procedure combined with Path Relinking and a modified Tabu Search was developed by Li et al. [12]. More recently, Brandao [2] proposed a Tabu Search algorithm for the HFFVRP which includes additional features such as strategic oscillation, shaking and frequency-based memory while Subramanian et al. [21] described a hybrid algorithm composed by an Iterated Local Search based heuristic and Set Partitioning formulation. In this paper, we present an efficient memetic algorithm for solving the HFFVRP, obtained by combining an immigration-based genetic algorithm with a powerful local search procedure. The genetic algorithm is endorsed by an immigration strategy designed to ensure diversification of the genetic material by inserting new individuals (called immigrants) into the population every generation. The immigrants are not random but generated based on some heuristics allowing the evolution of immigrants along with the evolution of the main population. The initialization of population takes into account several strategies for inserting the route splitters leading to four different approaches tested. The strength of the local search mechanism integrated in the evolutionary process is ensured by six local search heuristics and their diversity. Computational experiments are performed for a set of HFFVRP benchmark instances and the results are presented, analyzed and compared with existing heuristic methods. The experimental results reveal that the solutions provided by the proposed memetic algorithm are of high quality and competent to those existing in the literature. The rest of paper is structured as follows: the definition of the HFFVRP is presented in Section 2; the proposed memetic algorithm is described in Section 3 focusing on the general framework of the genetic algorithm (solution representation, fitness function, genetic search operators and strategies for population initialization), the immigration techniques integrated in the algorithm and the local search procedure; computational experiments and results are discussed in Section 4 and the conclusions of the study are depicted in Section 5.

2. Definition of the problem Formally, the HFFVRP is defined on a directed graph G ¼ ðV; AÞ with V ¼ f0; 1; 2; …; ng as the set of nodes, the set of arcs A ¼ fði; jÞji; jA V; i ajg and a nonnegative distance cij associated with each arc ði; jÞ A A. The set of nodes consists of vertex v¼ 0 which represents the depot and the vertices v ¼ 1; …; n which represent the customers. Each customer has a certain nonnegative amount of demand. There exists a fleet of heterogeneous (different types) vehicles that are using the depot as a starting base. We denote by vt the variable cost per distance unit of a vehicle of type t and by Qt its carrying capacity and we suppose that the number of vehicles of each type is fixed. The HFFVRP consists in finding the minimum cost collection of routes starting and ending at the depot, such that a single vehicle supplies the demand of each customer and the sum of all the demands of any route does not exceed the capacity of the assigned vehicle to it. An illustrative scheme of the HFFVRP and a feasible collection of routes is shown in Fig. 1. The HFFVRP reduces to the classical Capacitated Vehicle Routing Problem (CVRP) when all the vehicles have the same capacity.

59

As a consequence HFFVRP is NP-hard because it includes the Capacitated Vehicle Routing Problem as a special case.

3. An improved immigration memetic algorithm for solving the HFFVRP Memetic algorithms have been introduced by Moscato [14] to denote a family of metaheuristic algorithms that use a populationbased approach with separate individual learning or local improvement procedures for problem search. Therefore, a memetic algorithm is a genetic algorithm (GA) hybridized with a local search procedure to intensify the search. Genetic algorithms are not well suited for fine-tuning structures which are close to optimal solutions. Therefore, incorporating local improvement operators into the recombination step of a GA is essential in order to obtain a competitive GA. We propose an effective heuristic based algorithm for solving the HFFVRP which is an immigration memetic algorithm combining the power of genetic algorithms with that of local search and the immigration techniques. The general scheme of the proposed heuristic is depicted in Fig. 2. As shown in the presented scheme, in order to improve a current generation we use an immigration technique (described in Section 3.2), a local search procedure (described in Section 3.3) and in addition we eliminate the duplicate solutions.

3.1. The genetic algorithm We use a natural, compact and efficient encoding of solutions for HFFVRP similar to that described by [15] in the case of the GVRP. Specifically, 0 represents the depot and each customer is tagged with a non-duplicated natural number from 1 to n. We represent a chromosome by a variable length array so that the gene values correspond to the nodes selected to form the collection of routes which are delimited by 0 representing the depot. The corresponding chromosome representation of the feasible solution of the HFFVRP presented in Fig. 1 is ð6 2 10 1 7 0 11 3 9 4 0 8 5 12Þ; where the values f1; …; 12g represent the customers while the depot denoted by 0 is the route splitter. Route 1 begins at the depot then visits customers 6–2–10–1–7 and returns to the depot. Route 2 starts at the depot and visits the customers 11–3–9–4 and finally, in route 3 are visited the customers 8–5–12.

Fig. 1. An example of a feasible solution of the HFFVRP.

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collection of routes, is given by the sum of the costs of the arcs selected in the routes. The aim is to find the minimum cost collection of routes.

Fig. 2. Generic form of our improved immigration memetic algorithm.

3.1.1. Initial population The first step of any GA is to generate a set of possible solutions as the initial population of individuals. Although it seems simple, the convergence, the performance and the ability of the GA are critically affected by the initial generation. In the case of the HFFVRP, we carried out experiments with the initial population constructed as follows: each individual is generated either by selecting a random node at a time or based on a Monte Carlo technique, i.e. meaning that when an individual is created the probability for a node to be selected is proportional with the distance from it to the previous selected node. After this phase, the route splitters are introduced using one the following techniques: 1. Random insertion of the splitters: A vehicle is selected randomly and inserted after the subroute which can be fulfilled with the capacity of the vehicle. In other words, the selection of the vehicle is random, but the insertion itself is deterministic. 2. Deterministic insertion of the splitters: For each vehicle t, the efficiency is computed as vt Et ¼ dist where vt is the variable cost and dist is the traveled distance from the last deposit as long as the demand of the nodes can be fulfilled (as has to return to the deposit) and the most efficient vehicle is inserted. 3. Monte Carlo: For each vehicle, the efficiency is computed as explained in the case of the deterministic insertion of the splitters. A vehicle is selected using the Monte Carlo technique, namely proportionally with their efficiency. The deposit is inserted where the vehicle fulfills the total demand of the subroute. 4. Probabilistic splitter insertion: For each node, the probability of inserting a splitter is computed as the total demand of the subroute (from the last deposit) divided by the capacity of the biggest vehicle. However, from the experiments carried out it turned out that the Monte Carlo method of generating the initial population with respect to the order of the nodes has not brought any improvements. 3.1.2. Fitness function In order for GAs to work effectively, it is necessary to be able to evaluate how “good” a potential solution is relative to other potential solutions. In our case, the fitness value of a feasible solution, i.e.

3.1.3. Crossover Two parents are selected from the population by the binary tournament method, i.e. the individuals are chosen from the population at random. The one with better fitness value is chosen as the first parents. Offspring are produced from two parent solutions using the following 2-point order crossover procedure: it creates offspring which preserve the order and position of symbols in a subsequence of one parent while preserving the relative order of the remaining symbols from the other parent. This recombination operator is implemented by selecting two random cut points which define the boundaries for a series of copying operations. First, the symbols between the cut points are copied from the first parent into the offspring. Then, starting exactly after the second cut-point, the symbols are copied from the second parent into the offspring, omitting any symbols that were copied from the first parent. When the end of the second parent sequence is reached, this process continues with the first symbol of the second parent until all the symbols have been copied into the offspring. The second offspring is produced by swapping round the parents and then using the same procedure. We assume two well-structured parents chosen randomly, with the cutting points between nodes 2 and 3, respectively 5 and 6: P1 ¼ P2 ¼

6 2

0 1

j j

1 6

2 0

0 5

j j

5 4

4 3

3

The sequences between the two cutting-points are copied into the two offspring: O1 ¼ O2 ¼

x x

x x

j j

1 6

2 0

0 5

j j

x x

x x

x

The nodes of the parent P1 are copied into the offspring O2 if O2 does not contain already those nodes. And the same approach is used for the other offspring: O1 ¼ O2 ¼

x x

x x

j j

1 6

2 0

0 5

j j

4 x

3 4

3

The unfilled alleles are completed with the free nodes at random: O1 ¼ O2 ¼

5 1

6 0

j j

1 6

2 0

0 5

j j

4 2

3 4

3

3.1.4. Mutation We have applied a generalized mutation operator, which is able to exchange a sequence (of zero or more nodes) from a subroute with another sequence (of zero or more nodes) from another subroute. This is done based on the following algorithm: Algorithm 1. Mutation. 1: 2: 3: 4: 5: 6: 7:

for all individual A population do for all (allele A individual) AND (allele a splitter) do select randomly r A ½0; 1 LET pm be the probability of mutation if r o pm then select randomly endSeq1 A ½posallele ; possplitter1  select randomly posswap A another subroute

O. Matei et al. / Neurocomputing 150 (2015) 58–66

8: 9: 10: 11: 12: 13: 14:

select randomly endSeq2 A ½posswap ; possplitter2  LET seq1 ¼ falleleposallele ; …; alleleendSeq1 g LET seq2 ¼ falleleposswap ; …; alleleendSeq2 g exchange the two sequences seq1 and seq2 end if end for end for

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Therefore, the probability of mutation pim for the generation i is

8 5% > > > i  20 >

> m  5% > > : pi  20 m

if i r 20 if population stucks for 20 epochs and pim 20 o 30% if population evolved 20 epochs and pim 20 4 5% otherwise:

3.2. Immigration techniques where posallele is the position of the node allele, possplitter1 is the position of the first splitter after posallele, possplitter2 is the first splitter after posswap. We consider this mutation operator very general because it covers several specific cases:


if the length of the two sequences is 1, then we have a classical



and simple swap of two nodes belonging to two different subroutes; if the length of one sequence is zero, then we simply move the other sequence to another subroute; if the length of the two sequences differs, then we have the general case when a sequence of n nodes of a subroute is swapped with other m nodes from another subroute.

For instance, applying the mutation operator for the individual ð0 1 2 3 4 5 0 6 7 8 9 0Þ, with posallele ¼ 3, endSeq1 ¼ 5, posswap ¼ 8 and endSeq2 ¼ 9, will result into the following new individual: ð0 1 6 7 5 0 2 3 4 8 9Þ:

3.1.5. Selection Selection is the stage of a GA in which individuals are chosen from a population for later breeding (crossover or mutation). The selection process is deterministic. In our algorithm we have used a (μ þ λ þ λim ) selection procedure, where μ individuals generate an intermediary population of λ offspring, besides which λim individuals are created using the immigration technique described in Section 3.2.

3.1.6. Genetic parameters The genetic parameters are very important for the success of a GA, equally important as the other aspects, such as the representation of the individuals, the initial population and the genetic operators. Based on preliminary computational experiments, we set the following genetic parameters: the population size μ has been set to the five times the number of customers, the size of the intermediary population λ ¼ 5  μ. Regarding the mutation rate, we have adapted the Rechenbergs 1/5 success rule: “The ratio ϕ of successful mutations to all mutations should be 1/5. Increase the variance of the mutation operator if ϕ is greater than 1/5, otherwise decrease it” [13] to the specific needs of our problem. The initial mutation probability pm has been set to 5% and plays an important role in avoiding the local optima. A higher value “throws” the new generation too far from their parents; a smaller probability would be of no evolutionary value. However, every time the population gets stuck into a local optimum for 20 epochs, the mutation probability increases with 5%, until it reaches 30%. If this upper value is reached, the algorithm stops. If the population gets out of the local optimum, which is equivalent with pim o 30%, the probability decreases with 5% and it goes down by 5% every 20 generations while the population evolves, but not below the initial 5%.

It is well known that GAs get stuck in local optima very often. One efficient way of avoiding this problem is maintaining the diversification in population [20]. Among other methods, infusion and migration seem to offer good solutions. Infusion techniques insert new random individuals every generation, as proposed by Grefenstette in [7]. Koza [8] introduced decimation, where a substantial percentage of a population has been replaced by random individuals at regular intervals. Instead of inserting random individuals, Rahnamayan et al. [17] recently proposed the concept of opposition, inserting opposite individuals for diversity maintenance. Parallel GAs preserve diversity by exchanging individuals between sub-populations. Migrants can replace less fit individuals, randomly chosen individuals or the most similar individuals. None of the above-mentioned techniques make use of evolved individuals generated using other methods than GAs. The former one – infusion – uses random individuals, which introduce diversity but, in the same time, draw back the population somehow to the initial state (initial population). The latter one – migration – uses parallel GAs, which means that the subpopulations risk to get trapped in (different) local optima and migrating individuals do not bring enough diversity. Nevertheless, we agree that immigrants bring progress to any population and, moreover, the more different the immigrants are, the more progress and knowledge is brought. That is why we propose an immigration technique in which the immigrants are not random, but rather generated using some heuristics every generation. This way, the level of evolution of the immigrants increases along with the evolution of the population of our proposed GA. The idea of the proposed immigration technique is to bring new individuals (called “immigrants”) to the population each generation. The immigrants are created after applying the recombination and mutation by re-inserting the depots using the same strategy used for generating the initial population. Several questions arise regarding the immigrants:


How much the evolution of the immigrants influence the evolution of the host population?


What is the size of the immigrant population so that the progress is maximal and yet controllable? Fig. 3 gives two snapshots of the evolution of the population in two cases: using a simple GA – the blue line and using immigration techniques – the red line. In Fig. 3, the number of immigrants used is equal with the intermediary population of offspring, i.e. λ ¼ λim . Analyzing the figure, it is obvious that using immigrants improves the evolution significantly. In order to see the effect of the number of immigrants on the evolution of the global population independent trials have been conducted. For the experiments, we considered four instances that are going to be described in the Experimental results section. The obtained results are described in Table 1 for various percentages of immigrants from the total population. The same comparison is depicted in Fig. 4.

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O. Matei et al. / Neurocomputing 150 (2015) 58–66

Fig. 3. A snapshot of the evolution of the population in two cases: a regular GA (blue line), respectively with immigrants (red line). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

3.3. Local improvement procedure Classical GAs are not aggressive enough for some combinatorial optimization problems. A possibility to obtain more competitive heuristics is to combine the GAs with local search procedures. For each solution belonging to the current generation we use a local improvement procedure that runs several local search heuristics sequentially. Once an improvement move is found, it is immediately executed. In our algorithm, we used 6 local search heuristics divided into two classes depending on the following aspect: do they operate on a single route or do they consider more than one route simultaneously. The local search heuristics from the first class are obtained by moving one or more customers from one position in the route to another position in the same route and are called intra-route neighborhoods. We considered in our algorithm three such neighborhoods: Two-opt neighborhood, Three-opt neighborhood and Or-opt neighborhood. The moves defined within the intra-route neighborhoods are used in order to reduce the overall distance. The other class, called inter-route neighborhoods work with two routes. They are used in order to reduce the overall distance and in some cases they can reduce as well the number of vehicles. We considered in our algorithm three such neighborhoods: 1–0 Exchange neighborhood, 1–1 Exchange neighborhood and the Relocate neighborhood. 3.3.1. Two-opt neighborhood The Two-opt algorithm was first introduced in the case of the traveling salesman problem (TSP) and it was later extended to other related network optimization problems. In the case of the HFFVRP, in a Two-opt neighborhood two arcs belonging to a single route are replaced by two other arcs in order to improve the total cost of the route. The size of the Two-opt neighborhood is quadratic (w.r.t. the number of customers) and there is only one proper move type. Fig. 5 illustrates this process. 3.3.2. Three-opt neighborhood The Three-opt neighborhood was also introduced in the case of the TSP. The Three-opt neighborhood extends the Two-opt neighborhood and involves deleting three arcs in a route and reconnecting the three remaining paths in all other possible ways. Each of

Table 1 The influence of the number of immigrants on the evolution of the global population. Instance

10%

25%

50%

100%

200%

500%

13 15 18 20

3283.48 3164.46 3788.32 4975.44

3102.34 3102.67 3743.58 4893.4

3185.09 3068.53 3752.03 4834.17

3196.12 3066.86 3784.49 4910.37

3201.44 3122.37 3874.44 5035.74

3374.47 3097.45 3901.38 5083.55

Fig. 4. The influence of the number of immigrants on the evolution of the global population.

the reconnecting methods is then evaluated in order to find the optimum one. The size of the Three-opt neighborhood is cubic and there are three proper move types. Fig. 6 illustrates this process and contains an example of a proper move. 3.3.3. Or-opt neighborhood Designed for TSP, the Or-opt heuristic attempts to improve the current tour by first moving a chain of one, two or three consecutive customers in a different location (and possibly reversing it) until no further improvement can be obtained. The size of the Or-opt neighborhood is quadratic with the condition that the length of the sequence is bounded. Fig. 7 illustrates this process with an example of a proper move. 3.3.4. 1–0 Exchange neighborhood Given a pair of routes corresponding to a current solution of the HFFVRP, the 1–0 exchange neighborhood simply moves a vertex

O. Matei et al. / Neurocomputing 150 (2015) 58–66

from one route to the other, by replacing three arcs. Fig. 8 illustrates the 1–0 exchange neighborhood method. 3.3.5. 1–1 Exchange neighborhood Given a pair of routes corresponding to a current solution of the HFFVRP, the 1–1 exchange neighborhood swaps the positions of a vertex pair belonging to two different routes, by removing four arcs and creating four new ones. Fig. 9 illustrates this process. 3.3.6. Relocate neighborhood Given a pair of routes corresponding to a current solution of the HFFVRP, the relocate neighborhood simply moves a sequence of 2, 3 or 4 global arcs from one route to another one. Fig. 10 illustrates this process. 3.3.7. Proposed local improvement procedure Our improvement local search procedure is a Variable Neighborhood Descent (VND) algorithm, which systematically changes all the described neighborhoods in the following order: 1–0 exchange neighborhood, 1–1 exchange neighborhood, relocate neighborhood, two-opt neighborhood, Or-opt neighborhood and three-opt neighborhood. We observe that the neighborhoods are applied in an increasing order of their sizes. We point out three important aspects concerning our local improvement procedure:

63

1. Within our algorithm the infeasible solutions that are arising by applying inter-route neighborhood structures are handled as follows: the route that exceeds the vehicle capacity is split at the node (customer) that causes the violation of capacity restrictions and the rest of customers in that route are distributed among the other routes at random as long as all restrictions are met. 2. Applying the local search procedure to all the individuals of a current population will lead to highly time consuming procedure. Therefore, a subset of individuals is selected in each generation with a specified probability (in our case 10%) and then the VND procedure is applied to each of them separately. In the case that better individuals are found they are introduced in the current population. 3. The procedure is applied only once even for the identical solutions carried over to the next generation(s). Otherwise, it would be a waste of resources and an increase of the computation time.

4. Computational experiments and results In order to assess the performance of the proposed memetic algorithm for solving the HFFVRP, we conducted experiments on two well-known sets of instances: one reported by Taillard [22] and the second one introduced by Li et al. [11].

Fig. 5. Example showing a two-opt exchange move.

Fig. 6. Example showing a three-opt exchange move.

Fig. 7. Example showing a Or-opt exchange move.

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O. Matei et al. / Neurocomputing 150 (2015) 58–66

Fig. 8. Example showing a 1–0 Exchange move.

Fig. 9. Example showing a 1–1 Exchange move.

Fig. 10. Example showing a Relocate move.

For the computational experiments, we performed 30 independent runs of the proposed memetic algorithm for each problem instance. Results are analyzed based on the best and average solutions obtained over the 30 runs. The machine used for our experiments is equipped with processor Intel Core 2 Duo, 2.4 GHz and 2GB RAM. The algorithm was developed in Java, JDK 1.6. The first set of instances introduced by Taillard [22] contains 8 problems labeled as 13, 14,…, 20. In total, there are six types of vehicles and the number of customers range from 50 to 100, all randomly located over a square in the Euclidean plane. These instances have fixed fleet, capacity restrictions, no route length constraints and no service times at the nodes. Table 2 reports the experimental results obtained with the proposed memetic algorithm in the four cases considered for the route splitters. The first column in the table provides the name of the problem, the second column gives the number of customers and in the next columns we report the best solutions obtained using the described techniques for route splitters (described in Section 3.1.1) and the corresponding computational time out of 30 runs. Analyzing the results presented in Table 2, we can observe that the best results were obtained using the technique 2 (see Section 3.1.1) for introducing the route splitters. Therefore, in our improved immigration memetic based heuristic algorithm we considered the deterministic insertion of the splitters. Table 3 presents the results obtained using the proposed memetic algorithm in comparison to the best-found solutions by the heuristic column generation method (HCG) [22], the backtracking adaptive threshold accepting metaheuristic (BATA) [23], the record-to-record travel algorithm (HRTR) [11] and the multistart adaptive memory programming (MAMP) [12]. Table 3 gives the following information:

the name of the problem, the number of the customers, the best solution and the corresponding computational time obtained by the each of the compared methods (i.e. HCG [22], BATA [23], HRTR [11], MAMP [12] and our immigration memetic algorithm in the last column). The results written in bold represent cases for which the solution found is equal to the best known solution. Analyzing the results presented in Table 3, we observe that the solutions provided by the proposed algorithm are comparable from a quality perspective with the best solutions existing in the literature, namely our immigration memetic algorithm was able to find all the best known solutions except one for problem 15. Regarding the computational times, it is difficult to make a fair comparison between algorithms, because they have been evaluated on different computers and they are implemented in different languages. The running time of our algorithm is proportional with the number of generations. Our proposed heuristic algorithm seems to be slower than MAPN, comparable with HRTR and faster than HCG and BATA. Table 4 presents some more information about the performance of our immigration memetic algorithm: the average solutions and standard deviations (best solutions and the CPU time are the same given in Table 3). These additional data were not reported in the previous table because the other authors used different ways to present their results and in addition they did not provided them, reporting only the best solutions. As shown in Table 4, the proposed immigration memetic algorithm is able to trigger good solutions in each of the 30 runs considered, leading to average solutions which are close to the best solutions obtained. This result is also supported by the low standard deviations value reported in Table 3. The second set of instances used in our computational experiments was introduced by Li et al. [11] and consists of 5 problems

O. Matei et al. / Neurocomputing 150 (2015) 58–66

65

Table 2 The experimental results for choosing the population generation approach. n

Problem

13 14 15 16 17 18 19 20

Technique 1

50 50 50 50 75 75 100 100

Technique 2

Technique 3

Technique 4

Best solution

CPU

Best solution

CPU

Best solution

CPU

Best solution

CPU

3199.96 10 112.51 3067.94 3345.52 2111.67 3776.35 10 425.64 4858.56

321 346 372 473 445 903 783 890

3185.09 10 107.53 3066.86 3278.96 2076.96 3743.58 10 417.51 4834.17

246 377 301 224 412 498 518 634

3197.14 10 111.39 3065.29 3354.05 2111.79 3782.13 10 420.34 4834.17

288 310 329 324 436 486 692 706

3199.96 10 111.20 3067.94 3348.19 2111.01 3782.13 10 423.83 4856.35

285 404 350 287 422 515 676 709

Table 3 The experimental results obtained using our improved immigration memetic algorithm in comparison to best-found solutions of different algorithms. Pb.

13 14 15 16 17 18 19 20

n

50 50 50 50 75 75 100 100

HCG

BATA

HRTR

MAMP

Memetic

Best

CPU

Best

CPU

Best

CPU

Best

CPU

Best

CPU

3198.05 10 115.64 3066.86 3354.05 2111.79 3800.16 10 417.51 4859.77

473 575 335 350 2245 2876 5833 3402

3199.96 10 111.39 3065.29 3345.52 2111.01 3776.35 10 423.83 4856.35

843 387 368 341 363 971 428 1156

3197.14 10 107.53 3065.29 3344.94 2076.96 3743.58 10 420.34 4834.17

358 141 166 188 216 366 404 447

3185.09 10 107.53 3065.29 3278.96 2076.96 3743.58 10 420.34 4834.17

110 34 46 99 148 119 287 200

3185.09 10 107.53 3066.86 3278.96 2076.96 3743.58 10 417.51 4834.17

246 377 301 224 412 498 518 634

labeled as Hi, i ¼ 1; 2; …; 5. These instances were generated for HFFVRP through an adaptation of the existing five large-scale VRPs with customers ranging from 200 to 360 provided by Golden et al. [6]. Each instance has a geometric symmetry and the customers are located in concentric circles around the depot. The fleet is composed of six vehicle types and the demand of each customer node may take only two different values. Table 5 summarizes the comparative results between the proposed immigration memetic algorithm, the tabu search algorithm (TS) described by Brandao [2] and the record-to-record travel algorithm provided by Li et al. [11]. To conduct their respective experiments, Brandao [2] used first a Pentium II at 400 MHz, 128 MB of RAM and then for the TSA Intel Pentium M at 1.4 GHz, 512 MB of RAM while Li et al. [11] used an Athlon at 1 GHz, 256 MB of RAM. In order to compare the speeds of the different computers measured in millions of floating-point operations per second (Mflop/s), we used the Dongarras [4] tables and we have obtained: Pentium II¼80 Mflop/s, Athlon¼450 Mflop/s and Pentium M¼250 Mflop/s. The speed of the computer used in our experiments is 1408 Mflop/s. The results provided in Table 5 show that our hybrid algorithm was able to find three best known solutions out of five existing instances. The solutions reported for instances H1, H3 and H4 correspond to the best known HFFVRP solutions while the solution obtained for instance H2 is indeed very close to the best solution (10 210.34 reported by the proposed algorithm vs. 10 208.32 best known solution). Regarding instance H5, we suspect – as Brandao did in [2] – that the variable cost per unit of distance traveled for each type of vehicle is not the value written by Li et al. [11] but the same as the corresponding one for H4. The motivation for this statement is that we obtained the same solution as Brandao's best solution (i.e. 23 166.56) for instance H5. We can observe that the computing times of our hybrid algorithm are better than those of the TSA, but higher than those of Li et al. algorithm. Furthermore, the average solutions over 30 runs reported by the proposed memetic algorithm are very close to the best solutions obtained, which supports a stable efficient performance of the introduced method. This result is further endorsed by the values of standard deviations reported in Table 5.

Table 4 Performance of our improved immigration memetic algorithm. Problem n

Immigration memetic algorithm Best solution

13 14 15 16 17 18 19 20

50 3185.09 50 10 107.53 50 3066.86 50 3278.96 75 2076.96 75 3743.58 100 10 417.51 100 4834.17

Average solution

Standard deviation

Mean CPU

3201.45 10 112.36 3072.53 3352.27 2103.54 3764.82 10 420.77 4852.26

16.36 4.83 2.83 36.65 13.29 10.62 1.63 18.09

246 377 301 224 412 498 518 634

Overall, the proposed immigration memetic algorithm found ten best known solutions out of thirteen instances. The obtained results are within 0.9% of the best known solutions on average, proving that our hybrid algorithm is competitive to all of the known heuristics published to date.

5. Conclusions and future work In this paper, we developed an improved immigration memetic algorithm for solving the heterogeneous fixed fleet vehicle routing problem (HFFVRP). The proposed hybrid heuristic integrates a number of original features: we combine a genetic algorithm with a powerful local search procedure and an immigration approach, the initial population of the GA is constructed using four different approaches and the search is further guided by generalized mutation operator. We used an efficient way of avoiding getting trapped in local optima by maintaining the diversification of the population using an immigration technique. Our local search procedure consists of six local search heuristics, their diversity and power being an important factor for solving successfully the HFFVRP.

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Table 5 The experimental results obtained using our improved immigration memetic algorithm in comparison to Brandao' algorithm, TS and Li et al. algorithm. Instance

H1 H2 H3 H4 H5

n

200 240 280 320 360

Brandao

Li et al.

TSA

Best sol.

Best sol.

CPU

Best sol.

CPU

Best sol.

Avg. sol.

Std. dev.

CPU

12 050.08 10 208.32 16 223.39 17 458.65 23 166.56

12 067.65 10 234.40 16 231.80 17 576.10 21 850.40

688 995 1438 2256 3277

12 050.08 10 226.17 16 230.21 17 458.65 23 220.72

1395 3650 2822 8734 13 321

12 050.08 10 210.34 16 223.39 17 458.65 23 166.56

12 073.4 10 212.42 16 228.31 17 463.2 23 230.76

9.31305 0.57729 1.81254 0.45976 26.38

1372 2038 1637 3263 8493

The experimental results show that our algorithm is robust and is comparable with all known heuristic approaches to the problem in terms of solution quality. The use of immigrant individuals to ensure diversification coupled with the use of local improvement methods to intensify search is a powerful approach able to lead to competitive results. Future work focuses on the investigation of different immigration strategies integrated with local search procedures and their mutual influences. In the future, we also plan to assess the generality and scalability of the proposed hybrid heuristic by testing it on more instances in extensive computational experiments.

Acknowledgments This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PNII-RU-TE-2011-3-0113. The authors are grateful to the anonymous referees for reading the manuscript very carefully and providing constructive comments which helped to improve substantially the paper. References [1] M. Ball, T. Magnanti, C. Monma, G. Nemhauser, Network Routing, Handbooks in Operations Research and Management Science, Vol. 8, Elsevier, Amsterdam, 1995. [2] J. Brandao, A tabu search algorithm for the heterogeneous fleet vehicle routing problem, Comput. Oper. Res. 38 (2011) 140–151. [3] B. Crevier, J.-F. Cordeau, G. Laporte, The multi-depot vehicle routing problem with inter-depot routes, Eur. J. Oper. Res. 176 (2) (2007) 756–773. [4] J. Dongarra, Performance of Various Computers using Standard Linear Equations Software, Report CS-89-85, University of Tennessee, 2006. [5] G. Ghiani, G. Improta, An efficient transformation of the generalized vehicle routing problem, Eur. J. Oper. Res. 122 (1) (2000) 11–17. [6] B. Golden, E. Wasil, J. Kelly, I.-M. Chao, The impact of metaheuristics on solving the vehicle routing problem: algorithms, problem sets, and computational results, in: T. Crainic, G. Laporte (Eds.), Fleet Management and Logistics, Kluwer, Boston, MA, 1998, pp. 33–56. [7] J.J. Grefenstette, Genetic algorithms for changing environments, in: Proceedings of Parallel Problem Solving from Nature, Elsevier, Amsterdam, 1992, pp. 137–144. [8] J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT Press, Cambridge, MA, 1992. [9] G. Laporte, What you should know about the vehicle routing problem, Nav. Res. Logist. 54 (2007) 811–819. [10] G. Laporte, I. Osman, Routing problems: a bibliography, Ann. Oper. Res. 61 (1995) 227–262. [11] F. Li, B. Golden, E. Wasil, A record-to-record travel algorithm for solving the heterogeneous fleet vehicle routing problem, Comput. Oper. Res. 34 (2007) 2734–2742. [12] X. Li, P. Tian, Y. Aneja, An adaptive memory programming metaheuristic for the heterogeneous fleet vehicle routing problem, Transp. Res. Part E: Logist. Transp. Rev. 46 (2010) 1111–1127. [13] Z. Michalewicz, Genetic Algorithms þ Data Structures¼ Evolution Programs, Springer-Verlag, Berlin, Heidelberg, 1996. [14] P. Moscato, On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms, Caltech Concurrent Computation Program Report, 826, 1989. [15] P.C. Pop, O. Matei, C. Pop Sitar, An improved hybrid algorithm for solving the generalized vehicle routing problem, Neurocomputing 109 (2013) 76–83. [16] C. Prins, Efficient heuristics for the heterogeneous fleet multi-trip vrp with application to a large-scale real case, J. Math. Modell. Algorithms 1 (2002) 135–150.

Immigration memetic

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Oliviu Matei obtained his Ph.D. in the field of practical applications of evolutionary computing in solving combinatorial optimization problems from Technical University Cluj-Napoca. Before that, he studied Artificial Intelligence at Vrije Universiteit, Amsterdam. Currently, he is a member of the Department of Electrical Engineering at North University Center of Baia Mare, Technical University of ClujNapoca, Romania. His research interests focus on evolutionary and natural computing methods and their application to real-world optimization problems.

Petrică C. Pop is a professor within the Department of Mathematics and Computer Science at North University Center of Baia Mare, Technical University of Cluj-Napoca, Romania. He published more than 80 scientific papers, majority of them being published in Theoretical Computer Science, Neurocomputing, Applied Mathematical Modelling, European Journal of Operational Research, Annals of Operations Research, Advances in Intelligent Systems and Computing, Lecture Notes in Computer Science. His scientific contributions have a very good visibility with more than 155 citations in ISI journals and international journals. He is a member of the Editorial Board of 5 international journals. His research interests include metaheuristics, nature-inspired computing and combinatorial optimization problems.

Jozsef Laszlo Sas received B.S. degree in computer engineering from the North University Center at Baia Mare, Technical University of Cluj Napoca, Romania. Currently, he is a Master student in Informatics and Software Engineering at the same university and a junior software developer in Baia Mare.

Camelia Chira received the M.Sc. (2002) and Ph.D. (2005) degrees from Galway-Mayo Institute of Technology, Ireland, in the area of agent-based systems for distributed collaborative design environments. From 2006 to 2013, Camelia was a researcher at Babes-Bolyai University, Romania, working on several research projects both as a member and principal investigator. She is now with Technical University of Cluj-Napoca, Department of Computer Science, Romania. Her main research interests include nature-inspired computing, complex systems and networks, multi-agent systems and bioinformatics. She published more than 70 papers in international conferences and ISI journals.