Metaheuristic approaches for the vehicle routing problem with time windows and multiple deliverymen

Abstract

This paper addresses the vehicle routing problem with time windows and multiple deliverymen, a variant of the vehicle routing problem which includes the decision of the crew size of each delivery vehicle, besides the usual scheduling and routing decisions. This problem arises in the distribution of goods in congested urban areas where, due to the relatively long service times, it may be difficult to serve all customers within regular working hours. Given this difficulty, an alternative consists in resorting to additional deliverymen to reduce the service times, which typically leads to extra costs in addition to travel and vehicle usage costs. The objective is to define routes for serving clusters of customers, while minimizing the number of routes, the total number of assigned deliverymen, and the distance traveled. Two metaheuristic approaches based on Iterated Local Search and Large Neighborhood Search are proposed to solve this problem. The performance of the approaches is evaluated using sets of instances from the literature.

Keywords:
Vehicle routing; Multiple deliverymen; Iterated Local Search; Large Neighborhood Search

Resumo

Neste trabalho, aborda-se o problema de roteamento de veículos com janelas de tempo e múltiplos entregadores, uma variante do problema de roteamento de veículos que, além das decisões de programação e roteamento dos veículos, envolve a determinação do tamanho da tripulação de cada veículo de entrega. Esse problema surge na distribuição de bens em centros urbanos congestionados em que, devido aos tempos de serviço relativamente longos, pode ser difícil atender todos os clientes durante o horário de trabalho permitido. Diante dessa dificuldade, uma alternativa consiste em incluir a designação de entregadores adicionais para reduzir os tempos de serviço, o que gera custos adicionais aos custos tradicionais de deslocamento e utilização de veículos. Dessa forma, o objetivo é definir rotas para atender grupos de clientes, minimizando o número de veículos usados, o número de entregadores designados e a distância total percorrida. Para tratar o problema são propostas duas abordagens metaheurísticas baseadas em Busca Local Iterada e Busca em Vizinhança Grande. O desempenho das abordagens propostas é testado utilizando conjuntos de instâncias disponíveis na literatura.

Palavras-chave:
Roteamento de veículos; Múltiplos entregadores; Busca Local Iterada; Busca em Vizinhança Grande

1 Introduction

Transportation processes are involved in multiple ways in production systems, especially in those involving distribution activities. Such processes can have a huge impact on competitiveness and on service levels of industries. For example, transportation processes may represent up to 20% of the final costs of goods produced by a company (Toth & Vigo, 2002). In addition, it is estimated that distribution costs can represent up to 75% of logistics costs of an organization (Bräysy & Gendreau, 2005), making it necessary to make efforts for the improvement of those processes. Amongst the distribution activities, arises the vehicle routing problem (VRP), a challenging problem that is faced daily by many companies dealing with the transportation of goods or people. In practice, the VRP plays an important role in distribution systems and, therefore, solving this problem is a key activity for efficient operations management in the companies.

Recently, it was proposed a new variant of the VRP that treats as a decision variable the number of deliverymen that should be assigned to each delivery route (Pureza et al., 2012; Ferreira & Pureza, 2012). This variant, called as the vehicle routing problem with multiple deliverymen (VRPMD), has applications in real-life transportation, mainly in the distribution of goods in congested urban areas. Examples are soft drinks, dairy and beer companies which must replenish on a regular basis (daily or every few days) small and medium establishments like convenience stores, restaurants, grocery stores, among others. These establishments are typically located in very busy areas, in which it becomes difficult to park the delivery trucks. Thus, the vehicles park in a strategic point of a region having a group of customers and the deliveries are made on foot to these customers by the service workers of the route. By making the deliveries in this way, the service time in the group of customers (from now on, cluster of customers) can be relatively long when compared with travel times, which may difficult serving all customers during regular working hours. In these contexts, the use of additional deliverymen becomes an important feature as it can speed up the delivery of the products, reducing the service time in each cluster. An example of a typical route in the VRPMD is shown in Figure 1, where deliveries are performed in two phases. First, the truck arrives to the parking location of each cluster, then the service workers have to go on foot from the parking location to each customer in the cluster.

Figure 1
A typical route in the VRPMD.

In spite of the theoretical and practical importance of this variant, there are few researches on the VRPMD in the literature, which encourages the development of solution methods for it. In this paper, we propose two metaheuristic approaches for the Vehicle Routing Problem with Time Windows and Multiple Deliverymen, which are based on Iterated Local Search (ILS) and Large Neighborhood Search (LNS). These metaheuristics have been successfully applied to solve different variants of the VRP, e.g., VRP with heterogeneous fleet (Subramanian et al., 2012); VRP with time windows (Pisinger & Ropke, 2007); Dynamic VRP (Hong, 2012); VRP with multiple routes (Azi et al., 2014); VRP with split deliveries (Silva et al., 2015); and VRP with pickup and deliveries (Ropke & Pisinger, 2006). Thus, we believe that ILS and LNS can also be effective for the problem addressed in this research. Using instances from the literature, we compare the performance of the two proposed metaheuristic approaches between themselves, as well as we compare their performances with other methods proposed in the literature.

The remainder of this paper is organized as follows. In Section 2, we describe the problem to be addressed. Section 3 presents the metaheuristic approaches proposed to solve the problem. Next, we show the results of computational experiments in Section 4. Finally, Section 5 highlights general conclusions of this research and the plans for future research.

2 Problem statement

In this article, we address the vehicle routing problem with time windows and multiple deliverymen (VRPTWMD), a variant of the classic VRP with time windows that considers the crew size in the vehicles as a decision, in addition to routing and scheduling decisions. In the VRPTWMD, service times can be very long when compared to travel times because in the problem we must serve clusters of customer instead of serving individual customers. Furthermore, service times depend on the number of deliverymen assigned to the delivery route.

In practical terms, the problem involves two stages: first, the customers must be clustered around parking locations; then, routes must be designed to visit the defined clusters. Vehicle capacity, time windows and available deliverymen constraints must be satisfied while the total cost is minimized (vehicle usage, deliverymen assignment and traveled distance costs). Nevertheless, given the complexity of the complete problem, clustering and routing stages are addressed separately (Senarclens de Grancy & Reimann, 2014). Therefore, regarding the VRPTWMD, it is assumed that the clustering stage is performed in advance, thus each cluster has its predefined parking location, cumulated demand and service time, which includes the transportation of the goods from the parking location to the customers of the cluster.

After describing the context above, the VRPTWMD can be formally stated as follows. Given a homogeneous fleet of vehicles located in a central depot, each one with capacity Q, it must be used to visit n clusters, aiming to serve its demands $d_i,i=\mathrm{1,}\dots ,n$. The objective is to define a set of minimum cost routes, satisfying the following constraints: each cluster must be visited exactly once and within its time window $\left[w_i^a,w_i^b\right]$, i.e., the vehicle must serve the cluster before the time instant $w_i^b$ and must wait until the time instant $w_i^a$ to start serving the cluster, if the vehicle arrives before this time. The service time in cluster i for a route with l deliverymen is known in advance and denoted by$s_{il}$. The travel time between clusters i and j is given by $t_{ij}$. The vehicles are required to come back to the depot at the end of their trips.

The cost of a solution S is defined by Equation 1, where V denotes the number of vehicles used, E denotes the total number of deliverymen assigned and D denotes the total traveled distance. Constants $p_1$, $p_2$ and $p_3$ denote the weights of each objective, which are used to impose priority over them.

$c\left(S\right)=p_{1}V+p_{2}E+p_{3}D$ (1)

In order to show how the use of additional deliverymen may improve the service levels, consider the next illustrative example with three clusters, a vehicle with a capacity large enough to serve all clusters and the data of Table 1. In the table, $d_i,s_{i1},s_{i2},w_i^a,w_i^b$ denote the demand, the service times with one and two deliverymen and the time window of cluster i, respectively. Cluster $i=0$ represents the depot. Solutions for the problem in Table 1, considering routes with one and two deliverymen are presented in Figure 2. In the figure, $w_i$ represents the service start time in cluster i. Note that in the route with two deliverymen (left), all clusters can be served within the planning horizon (latest arrival time at the depot). On the other hand, the route with one deliveryman (right) cannot serve cluster 3 because, departing from cluster 2, it is not possible to arrive in cluster 3 before its latest arrival time ($w_3^b$).

Table 1
Data for a VRP with multiple deliverymen.

$i$	$d_i$	$s_{i1}$	$s_{i2}$	$w_i^a$	$w_i^b$	$t_{ij}$
						0	1	2	3
0	0	0	0	0	50	0	10	12	9
1	8	6	3	8	15	10	0	5	12
2	6	8	4	18	26	12	5	0	6
3	7	6	3	25	30	9	12	6	0

Figure 2
(left) Solution with two deliverymen in the route; (right) Solution with one deliveryman in the route.

3 Metaheuristic approaches

In this section, we describe the metaheuristic approaches developed to solve the VRPTWMD. The first approach is based on the metaheuristic ILS and it is presented in Section 3.2. The second approach is based on the metaheuristic LNS and its description is shown in Section 3.3. Both approaches use the same constructive heuristic, which is described in Section 3.1.

3.1 Constructive heuristic

To generate an initial solution for the metaheuristic approaches, we developed a constructive heuristic similar to the used by Senarclens de Grancy & Reimann (2014), which is based on the classic Solomon's insertion heuristic I1 (Solomon, 1987). In our implementation, routes are constructed sequentially, starting with the farthest unrouted cluster and using the maximum possible number of deliverymen on the vehicle. Next, clusters are inserted in the current route minimizing a weighted sum of additional time and distance when the cluster is inserted into the route. When no more clusters can be inserted into the route, a new one is initialized and the process is repeated until a solution serving all clusters is reached.

3.2 ILS-based metaheuristic approach

The first developed approach is based on ILS (Lourenço et al., 2003), a metaheuristic that applies a local search repeatedly to a set of solutions obtained by perturbing previously visited local optimal solutions. An ILS algorithm uses four basic components: (i) an initial solution; (ii) a local search procedure; (iii) a perturbation mechanism; and (iv) an acceptance criterion. For further information on ILS algorithms, see Lourenço et al. (2010).

In addition to the usual components of an ILS metaheuristic, the proposed approach uses two additional heuristics to enhance its performance, namely, deliverymen reduction heuristic and route reduction heuristic, which are specifically designed for the VRPTWMD. The overall structure of this approach is shown in Figure 3. First, an initial solution is generated with the constructive heuristic (line 2). This solution is improved through the local search heuristic and defined as the best initial solution in each iteration of the approach (line 5). The main loop of the algorithm is given on lines 7 to 23 and aims at improving the current best solution using the local search procedure (line 9) combined with the deliverymen reduction heuristic (line 10) and the perturbation mechanism (line 8). The acceptance criterion defines that the perturbation is performed on the incumbent solution of the current iteration of the approach ($S^{+}$). The main loop comprises two phases, each one of them terminates when the algorithm reaches MaxIterILS consecutive perturbations without improvements (lines 13-15 and 16-18, respectively). Then, the best global solution is updated (lines 24-26) and a new main loop is started, in case that the overall stopping criterion has not been reached.

Figure 3
ILS-based metaheuristic approach.

The two phases of the approach are needed to consider different parts of the objective function of the VRPTWMD. The first phase focuses on reducing the number of vehicles used in the solutions. For this, the perturbation is performed with the route reduction heuristic of Section 3.2.3. The second phase of the approach focuses on reducing the traveled distance and therefore it uses the perturbation procedure of Section 3.2.2. Note that in the first phase the route reduction heuristic not always obtain a different solution. In these cases, the perturbation procedure of the second phase is applied. Finally, also note that the reduction of the number of deliverymen is directly addressed in the approach by using the deliverymen reduction heuristic.

3.2.1 Local search

The local search procedure plays the role of the intensification tool in ILS. In our approach, the local search procedure is a variable neighborhood descent heuristic (Mladenovic & Hansen, 1997) with random neighborhood ordering (RVND). This heuristic applies a set of neighborhood structures (local search operators) to progressively improve the solution. First, a set of neighborhood structures $V=\left\{v^1,\dots ,v^n\right\}$, is initialized. While $V$ is not empty, a neighborhood structure $v^i$ is chosen at random and applied to the solution. In case of improvement, $V$ is reestablished to its initial form (containing all the neighborhood structures). Otherwise, $v^i$ is deleted of the set. Infeasible solutions are forbidden and the first improvement strategy is adopted. Moreover, to reinforce the RVND heuristic, the route reduction heuristic is applied when one neighborhood structure improves the solution. The set of used neighborhood structures contains the following movements:

• Inter-routes neighborhood structures:

  • o

    Shift(k,0): move k adjacent clusters from route $r_1$ to route $r_2,k=\left\{\mathrm{1,2,3}\right\}$ (see Figure 4).

    

    Figure 4
    Shift(2,0) movement.
    

  • o

    Swap(1,1): exchange cluster $c_1$ of route $r_1$ with cluster $c_2$ of route $r_2$ (see Figure 5).

    

    Figure 5
    Swap(1,1) movement.
    

  • o

    Swap(2,1): exchange two adjacent clusters $c_1$ and $c_2$ of route $r_1$ with cluster $c_3$ of route $r_2$.

  • o

    Swap(2,2): exchange two adjacent clusters $c_1$ and $c_2$ of route $r_1$ with two adjacent clusters $c_3$ and $c_4$ of route $r_2$.

• Intra-routes neighborhood structures:

  • o

    Or-opt-1: move one cluster from its current position to another one in the same route (see Figure 6).

    

    Figure 6
    Or-opt-1 movement.
    

  • o

    2-opt: two nonadjacent arcs $\left(i,i^{+}\right)$ and $\left(j,j^{+}\right)$ are removed and another two arcs $\left(i,j\right)$ and $\left(i^{+},j^{+}\right)$are added in such a way that a new route is generated (see Figure 7).

    

    Figure 7
    2-opt movement.
    

3.2.2 Removal and insertion heuristic

The perturbation mechanism is responsible for the diversification in the ILS since it changes the current local optimal solution. In our ILS approach, one perturbation mechanism consists of one operation of removal and relocation of a set of clusters, based on the procedure of Melechovsky (2012). For a given route, up to nP cluster nodes are randomly selected and removed from the route. Each removed cluster is then tested for a feasible insertion into the remaining routes of the solution. If such feasible insertion exists, the cluster is relocated to its new position. If some of the clusters cannot be inserted, a new single route with the maximum possible crew is created for serving them.

3.2.3 Route reduction heuristic

Given that reducing the number of routes can also reduce the number of allocated deliverymen, the route reduction heuristic of Senarclens de Grancy & Reimann (2014) was extended and used in the ILS approach. For a given solution, one route at a time, the heuristic attempts to relocate all clusters of the route inserting them into their best possible position in other routes. If any cluster cannot be reallocated, the crews of the receiving routes are temporarily increased by one unit (if its crew size is less than the maximum number of deliverymen) to increase the slack of these routes and hence creating opportunities for receiving unrouted clusters. If the reallocations become feasible after increasing these crews, the temporary crews of the receiving routes are maintained. Remember that this heuristic is used as both perturbation mechanism of the approach and improvement heuristic inside the local search phase.

3.2.4 Deliverymen reduction heuristic

Routes in a solution can have more deliverymen than necessary, because of a possible slack in the construction phase or local search phase. To improve this, we present a heuristic to reduce the number of deliverymen in a given solution. Let $S=\left\{r_1,r_2,\dots ,r_n\right\}$ be a feasible solution, composed by routes $\left(r_1,r_2,\dots ,r_n\right)$. Let $cre{w}_i$ be the number of deliverymen of route $r_i,\forall r_i\in S$.One route at a time, its crew is decreased by one unit (if $cre{w}_i>1$). If the route becomes infeasible (in terms of time windows), the first violated cluster of the route is removed in order to increase the slack of the route from the removal position. This process is performed until restoring the feasibility of the route. Then, the heuristic tries to insert the removed clusters into their best possible position in other routes and, if some clusters cannot be inserted, a new single route is created visiting only these clusters. The resulting solution is denoted by $S^{\prime }$ and if it uses the same number of routes of the original solution $S$, then $S^{\prime }$ replaces $S$.

3.3 LNS-based metaheuristic approach

The second metaheuristic approach proposed in this paper is based on the LNS metaheuristic (Shaw, 1998). LNS tries to overcome the difficulties faced by many local search algorithms, which only generate little changes in the solutions. Therefore, for these local searches it is often very difficult to escape from local optimal solutions and explore promising areas of the solution space when dealing with tightly constrained problems. In LNS, a solution is gradually improved by alternately destroying and repairing it. A detailed description of LNS algorithms is presented by Pisinger & Ropke (2010).

In the proposed approach, the LNS guides the search based on destroy and repair operators, in addition to the same improvement heuristics used in the ILS-based approach (RVND, route reduction heuristic and deliverymen reduction heuristic). The structure of the metaheuristic approach is shown in Figure 8. First, an initial solution is generated with the constructive heuristic (line 2). An outer loop defines the initial solution as incumbent (line 5) and apply the LNS operations until the stopping criteria is reached. In the main loop of the approach (lines 7-16), the first step is to apply the destroy and repair operators (line 8), which are described in Sections 3.3.1 and 3.3.2. These operators are chosen at random from a set of available operators, which will be described below. Then, the route reduction, deliverymen reduction and RVND heuristics are applied (lines 9-11). Only improved solutions are accepted (lines 12-14) and the main loop finishes when the algorithm reaches MaxIterLNS iterations. After that, the best global solution is updated (lines 17-19) and the outer loop is repeated in case that the overall stopping criterion has not been satisfied.

Figure 8
ILS-based metaheuristic approach.

3.3.1 Destroy operators

The approach uses four different destroy operators. Each one of them takes as input a complete solution and returns a partial solution from which q clusters were removed. The used operators are:

• Random removal: this operator selects q clusters at random and removes them from the solution. As pointed out by Pisinger & Ropke (2007), this operator clearly has the effect of diversifying the search.

• Worst removal: this operator tries to remove clusters that are very expensive, or that somehow increase the cost of the current solution. Let $i$ be a cluster, $i^{-}$ its predecessor and $i^{+}$ its successor in the route. The cost $c_i$ of cluster $i$ is computed according to Equation 2.

  $c_i=d_{i^{-},i}+d_{i,i^{+}}-d_{i^{-},i^{+}}$(2)

where $d_{ij}$ is the distance between clusters i and j. Next, the removal operator repeatedly chooses a new cluster i that has the largest cost, until q clusters have been removed. The removal operator has a randomization component, which is controlled by the parameter p as follows. Let L be the number of clusters in the solution. When a new cluster must be removed, a random number y is chosen from the interval (0,1] and we calculate $k=\lceil y^{p}L\rceil$. Next, the cluster with the kth largest cost is removed and L is updated. This procedure is repeated until q clusters are removed. Note that if p is large, more expensive clusters are more likely to be selected, while less expensive clusters may be chosen for smaller values of p. This component was incorporated to avoid situations where the same clusters are removed over and over again, as in (Pisinger & Ropke, 2007).

• Related removal: the purpose of the related removal operator is to remove clusters that are related in some sense and therefore it is expected that may be easy to interchange. The relatedness measure between two clusters i and j was computed as the distance between them, removing the clusters as follows. The first selected cluster is chosen at random. Then, other clusters are selected, but they must be closely related to a previously selected cluster. This procedure is applied until q clusters are marked as selected. Then, these clusters are removed. Similar to the worst removal operator, the selection process contains a randomized component, which is controlled by the parameter p.

• Time-oriented removal: this operator is a variant of the related removal operator. In this operator, the relatedness measure is based on the service start time on clusters. Hence, this operator tries to remove clusters that are served approximately at the same time, as it is expected that can may be easy interchangeable. The operator works as follows. First, a cluster r is chosen at random and its 2q closest clusters are marked as potential clusters. The relatedness measure between clusters r and i is given by Equation 3.

  ${\Delta }_{ri}=\left|w_r-w_i\right|$(3)

where $w_r$ and $w_i$ are the service start times at clusters r and i, respectively. Among the potential clusters, the operator selects the $q-1$ clusters that are the most related to r. These clusters are removed together with r. This operator also has a randomization component, similar to the worst removal and related removal operators.

3.3.2 Repair operators

After applying the destroy operator, the partial solution must be repaired in order to render it feasible again. Each repair operator takes a partial solution as input and returns a complete feasible solution. The used repair operators are described below.

• Greedy insertion: this operator tries to insert the clusters in the cheapest possible position. Formally, it can be stated as follows. Let $\Delta f_{ir}$ denote the change in the objective function incurred when inserting cluster i in the cheapest position in route r. If cluster i cannot be inserted in route r, then $\Delta f_{ir}=\infty$. Following a greedy criterion, Equation 4 is applied.

  $\left(i^{\prime },r^{\prime }\right)=\arg \underset{i,r}{\min }\Delta f_{ir}$(4)

and cluster $i^{\prime }$ is inserted into the best position in route $r^{\prime }$. This operation is performed until all clusters have been reinserted into the solution or no more insertions are feasible. In the latter case, new routes are created (with the maximum possible crew) to serve those clusters.

• Regret insertion: this operator tries to overcome the difficulties of the greedy insertion operator as it often postpones the insertion of difficult clusters to the last iterations, when its insertion becomes more constrained. This operator tries to incorporate an anticipation component when selecting the next cluster to be inserted, as follows. Let $\Delta f_i^q$ denote the change in the objective function when cluster i is inserted into its best position in the qth cheapest route. Then, in each call of the operator, we choose cluster $i^{\prime }$ in accordance with Equation 5

  $i^{\prime }=\arg \underset{i}{\max }\Delta f_i^2-\Delta f_i^1$(5)

and the cluster is inserted into the route with the lowest insertion cost. In other words, the operator maximizes the difference of cost of inserting the cluster i in its second best route and its best route, meaning that groups with fewer feasible insertion positions tend to be inserted first. This process is repeated until no more clusters can be inserted. As in the greedy insertion operator, if any cluster cannot be inserted new routes are created to visit those clusters.

4 Computational experiments

In this section we present the results of the computational experiments using the metaheuristic approaches proposed in Section 3. We also compare the proposed approaches against other metaheuristic proposed in the literature. Furthermore, a statistical analysis is conducted to calibrate the parameters of the metaheuristic approaches. All algorithms were implemented in C++.

4.1 Benchmark instances

In all the experiments we use the benchmark instances proposed by Pureza et al. (2012), which are based on the well-known benchmark instances proposed by Solomon (1987) for the VRP with time windows. The instance set is composed of 56 instances involving 100 clusters. They are divided into six classes based on planning horizon length, vehicle capacity, width of the time windows and distribution of the customers, namely: R1, R2, C1, C2, RC1 and RC2. Classes R1, C1 and RC1 (R2, C2 and RC2) contain instances with short (long) planning horizon, narrow (loose) time windows and vehicles with small (large) capacity. On the other hand, classes R1/R2, C1/C2 and RC1/RC2 have randomly distributed, grouped and a mix of randomly distributed and grouped clusters, respectively. Note that the characteristics of classes R2, C2 and RC2 allow more clusters to be served per route than in routes of classes R1, C1 and RC1.

In the original Solomon instances there is no differentiation of the service times according to the number of deliverymen. Thus, Pureza et al. (2012) proposed to modify them to represent the delivery time of the accumulated demand of the customers on clusters, defined in Equation 6.

$s_{il}=\frac{\min \left\{rs*q_i,T-\max \left\{w_i^a,d_{0i}\right\}-d_{i0}\right\}}{l},i=\mathrm{1,}\dots ,n,l=\mathrm{1,}\mathrm{2,}3.$ (6)

where $q_i$ is the demand of cluster i, rs is the service rate, which in our experiments was defined with value 2, T is the latest arrival time at the depot and $d_{0i}=d_{i0}$ is the distance between the depot and cluster i, whereas the second term in the min operation in the equation guarantees the instance feasibility.

4.2 Parameter tuning

The metaheuristic approaches were calibrated using a Design of Experiments (Montgomery, 2012), as in Naderi et al. (2010). To do so, we carried out a full factorial experiment testing the parameters of the approaches in the levels shown in Table 2. The tested levels were determined through preliminary experiments. All levels combination resulted in nine different configurations for each algorithm. We used 20 instances (chosen at random from the instance set) to calibrate the algorithms. All 20 instances were solved five times by each configuration of the algorithms, resulting in 900 observations per approach. A time limit of five minutes was imposed for each single execution of the algorithms, which were run on a PC Dell Precision T7600 CPU E5-6280 2.70 GHz and 192 GB of RAM, using a single core. To measure the performance, we used the relative gap between the cost of the solution found by the algorithm ($A{\lg }_{sol}$) and the cost of the best solution found by all the configurations ($Mi{n}_{sol}$), as stated in Equation 7.

Table 2
Factors and levels of the design of experiments.

Algorithm	Parameter/factor	Levels
ILS	MaxIterILS	100; 150; 200
	nP	1; 3; 5
LNS	MaxIterLNS	500; 1000; 1500
	q	Up to 0.2n; 0.3n; 0.4n

$gap=\frac{A{\lg }_{sol}-Mi{n}_{sol}}{Mi{n}_{sol}}$ (7)

The results were analyzed using the analysis of variance technique, checking the three hypotheses of this analysis (normality, homoscedasticity and independence of residuals) using the appropriate techniques. No evidence was found to question the validity of the experiment. Figures 9-10 show the mean plots and Tukey intervals with 95% confidence for the levels of the parameters of the metaheuristic approaches. In these plots, overlapping between confidence intervals indicates that there is no statistically significant difference between the means.

Figure 9
Means plots and Tukey intervals for the parameters of the ILS.

Figure 10
Means plots and Tukey intervals for the parameters of the LNS.

The plots in Figures 9 and 10 indicate that there is no statistically significant difference between the performances of the methods, considering all levels of the parameters. This result is explained by the fact that the metaheuristic approaches depend heavily on the additional heuristics (route reduction heuristic and deliverymen reduction heuristic), which have no parameters, thus they reduce the sensitivity of the metaheuristic approaches regarding to its parameters. The dependency of the approaches on the additional heuristics is a result of the specific characteristics of the VRPTWMD.

Since there is no significant difference in the performance of the approaches for different levels of the parameters, values for them were defined based on the best average results of the calibration tests, as follows: MaxIterILS = 200, nP = 3, MaxIterLNS = 1000 and q = $rand\left(\mathrm{0,1}n;\mathrm{0,2}n\right)$. In addition, following the proposal of Ropke & Pisinger (2006), the value of the randomization component of the removal operators in the LNS was defined as $p=3$. This value is large enough to allow removing clusters with large costs. Still, this value allows the removal operators to have an adequate performance, avoiding cases where the same clusters are removed repeatedly.

4.3 Experiments with the metaheuristic approaches

This section shows the results of the computational experiments after the tuning phase performed in the last section. For these experiments, the weights of the objective function are defined with the same values as used by Pureza et al. (2012), namely: $p_1=\mathrm{1,}{p}_2=0.1$ and $p_3=0.0001$. They prioritize the minimization of the number of vehicles used in the solution, followed by the number of deliverymen and then the traveled distance. The experiments were run in a PC Intel Core i7 3.40 GHz with 16 GB RAM, using a single core, with a time limit of 600 seconds and running five times each algorithm.

The results of the metaheuristic approaches are compared with the results reported by Pureza et al. (2012), regarding a Tabu Search approach (TS-PMR) and an Ant Colony Optimization algorithm (ACO-PMR), which were both run in a PC Intel Core2 2.40 GHz with 2 GB RAM. Also, we use the results reported by Senarclens de Grancy & Reimann (2014), which were obtained by an ACO algorithm (ACO-SR), in a PC Intel E8400. In the following tables, labels Cost, Veh, Dist, Del and Time denote the total cost of the solutions, the number of vehicles used, the distance traveled, the number of deliverymen assigned and the running time (in seconds), respectively. The best results (in terms of cost) for each instance/class are highlighted in boldface. Costs are presented with two decimal places only and ties are broken by selecting the solution with the shortest distance.

First, we compare the overall performance of the approaches in all the instance classes. In this sense, it is only possible to compare our approaches with the approaches of Pureza et al. (2012), since Senarclens de Grancy & Reimann (2014) only report the results for class R1. Table 3 shows the best results obtained by the metaheuristic approaches, grouped for each instance class. Note that the proposed approaches always find solutions with costs that are better than or equal to those found by TS-PMR and ACO-PMR in all instance classes. Moreover, comparing in detail the performance of the two proposed approaches, we obtain that ILS finds the best solution in 9 out of 12 instances in class R1, in 5 out of 8 in class RC1, in 9 out of 11 of class R2 and in 4 out of 8 of class RC2. In classes C1 and C2, both approaches find the same solutions for each instance. As a result, ILS is superior to LNS regarding the number of best solutions found. On the other hand, in terms of average results ILS is better than LNS only in one instance class, whereas LNS is superior in three classes and tie in the two remaining classes. This result indicates that when LNS surpass ILS in a single instance, the difference is large enough to allow dominating in terms of average results.

Table 3
Best results (grouped) of the metaheuristic approaches.

Method		Instance class
		R1	RC1	C1	R2	RC2	C2
TS-PMR (best out of 5)	Cost	15.70	16.64	11.08	3.75	4.45	3.36
	Veh	12.33	13.00	10.00	2.90	3.40	3.00
	Dist	1258.00	1527.90	830.70	1034.00	1230.40	597.20
	Del	32.42	34.90	10.00	7.50	9.30	3.00
	Time	640.10	677.10	265.10	425.40	419.10	246.80
ACO- PMR (best out of 5)	Cost	15.77	16.70	11.08	3.86	4.58	3.36
	Veh	12.50	13.00	10.00	3.10	3.60	3.00
	Dist	1261.50	1480.10	833.60	1064.20	1296.00	609.30
	Del	31.40	35.50	10.00	6.50	8.50	3.00
	Time	575.80	508.60	375.20	600.60	462.00	243.30
ILS (best out of 5)	Cost	15.34	16.59	11.08	3.63	4.31	3.36
	Veh	12.17	13.00	10.00	2.91	3.38	3.00
	Dist	1271.71	1482.46	827.64	993.17	1186.61	587.51
	Del	30.50	34.38	10.00	6.18	8.13	3.00
	Time	600.75	601.25	600.22	604.09	603.25	601.50
LNS (best out of 5)	Cost	15.32	16.50	11.08	3.64	4.29	3.36
	Veh	12.08	12.88	10.00	2.91	3.38	3.00
	Dist	1271.64	1492.29	827.64	998.00	1197.49	587.51
	Del	31.08	34.75	10.00	6.27	8.00	3.00
	Time	602.42	603.25	600.67	610.09	609.50	602.75

Table 4 presents the average results (grouped for each instance class), considering the five runs of each instance. Similar to results of Table 3, it can be seen that the ILS and LNS approaches dominated ACO-PMR and TS-PMR in all instance classes.

Table 4
Average results (grouped) of the metaheuristic approaches.

Method		Instance class
		R1	RC1	C1	R2	RC2	C2
TS-PMR (average of 5)	Cost	16.20	17.12	11.08	3.89	4.50	3.37
	Veh	12.80	13.40	10.00	3.00	3.40	3.00
	Dist	1272.40	1511.40	847.70	1046.80	1251.30	653.50
	Del	32.70	35.70	10.00	7.90	9.70	3.00
	Time	658.20	686.50	245.70	393.00	400.50	261.90
ACO- PMR (average of 5)	Cost	15.90	17.08	11.08	3.89	4.69	3.36
	Veh	12.60	13.40	10.00	3.10	3.70	3.00
	Dist	1263.20	1496.30	838.80	1070.30	1307.90	623.70
	Del	31.70	35.30	10.00	6.80	8.60	3.00
	Time	508.80	472.80	389.10	527.50	455.60	263.80
ILS (average of 5)	Cost	15.45	16.68	11.08	3.76	4.35	3.36
	Veh	12.25	13.10	10.00	3.05	3.43	3.00
	Dist	1260.58	1478.90	827.64	988.71	1203.95	587.51
	Del	30.73	34.35	10.00	6.07	8.03	3.00
	Time	601.03	600.98	600.44	606.27	604.20	603.29
LNS (average of 5)	Cost	15.51	16.60	11.08	3.73	4.31	3.36
	Veh	12.30	12.98	10.00	3.02	3.38	3.00
	Dist	1267.65	1488.90	827.64	997.41	1217.48	587.51
	Del	30.87	34.80	10.00	6.15	8.10	3.00
	Time	602.45	602.65	601.89	613.05	608.95	606.42

The results obtained for the instance class R1 are now described in detail, as their characteristics better reflect the importance of the service times, since the routes in these instances are in general shorter. This class was also addressed in detail in previous researches related to the VRPTWMD. Table 5 shows the best solutions found by the ILS and LNS metaheuristic approaches, as well as the best solutions reported by Pureza et al. (2012) and Senarclens de Grancy & Reimann (2014). In the latter, two metaheuristic algorithms were proposed, namely, ACO and GRASP algorithms. We compare only with ACO, as it had the best performance in class R1. The results show that ILS and LNS can find the best solution in 6 out of 12 instances of the class. Comparing with TS-PMR, ILS finds solutions using 1.35% and 5.91% less vehicles and deliverymen respectively, whereas LNS finds solutions using 2.03% and 4.11% less vehicles and deliverymen respectively. On the other hand, comparing with ACO-SR, ILS finds solutions using the same number of vehicles and 0.81% less deliverymen, while LNS finds solutions using 0.68% less vehicles and 1.08% more deliverymen. Recall that for instance R101, whose optimal solution is known, ILS and LNS find solutions with the optimal number of vehicles and deliverymen, while with travel distances 2.17% and 2.03% longer than in the optimal solution, respectively.

Table 5
Best solutions for the instance class R1.

Method	Instance
		R101	R102	R103	R104	R105	R106	R107	R108	R109	R110	R111	R112	Avg	Sum
	Cost	23.67	21.05	16.33	12.91	17.84	15.23	13.01	12.80	15.43	14.12	13.11	12.90	15.70	188.41
TS-PMR	Veh	19	17	13	10	14	12	10	10	12	11	10	10	12.33	148
(best out	Dist	1740.00	1520.00	1285.00	1057.00	1446.00	1323.00	1112.00	967.00	1296.00	1217.00	1137.00	996.00	1258.00	15096.00
of 5)	Del	45	39	32	28	37	31	29	27	33	30	30	28	32.42	389
	Time	645	655	959	692	463	492	473	953	428	620	616	686	640.17	7682
	Cost	23.67	20.95	15.94	12.70	17.64	15.13	12.81	11.70	15.12	13.92	13.01	11.80	15.37	184.40
ACO-SR	Veh	19	17	13	10	14	12	10	9	12	11	10	9	12.17	146
(best out	Dist	1725.46	1533.40	1371.63	1045.68	1412.52	1301.34	1108.92	967.18	1229.72	1154.95	1134.16	996.32	1248.44	14981.28
of 5)	Del	45	38	28	26	35	30	27	26	30	28	29	27	30.75	369
	Time	960	960	960	960	960	960	960	960	960	960	960	960	960.00	11520
	Cost	23.68	20.95	15.84	12.71	17.64	15.04	12.81	11.70	15.03	13.92	13.02	11.80	15.34	184.13
ILS	Veh	19	17	13	10	14	12	10	9	12	11	10	9	12.17	146
(best out	Dist	1757.40	1530.92	1410.73	1062.58	1412.52	1352.09	1140.79	965.31	1291.62	1187.49	1161.96	987.14	1271.71	15260.57
of 5)	Del	45	38	27	26	35	29	27	26	29	28	29	27	30.50	366
	Time	600	601	601	601	600	600	603	601	600	600	602	600	600.75	7209
	Cost	23.68	20.95	15.94	12.71	17.64	15.03	12.91	11.70	14.43	13.92	13.11	11.80	15.32	183.83
LNS	Veh	19	17	13	10	14	12	10	9	11	11	10	9	12.08	145
(best out	Dist	1755.04	1531.36	1369.12	1072.41	1412.52	1328.21	1139.96	1011.37	1281.84	1219.12	1149.03	989.70	1271.64	15259.70
of 5)	Del	45	38	28	26	35	29	28	26	33	28	30	27	31.08	373
	Time	600	603	603	603	600	601	600	601	605	603	607	603	602.42	7229

Finally, to show the robustness of the proposed approaches and, considering that both algorithm have randomized components, Table 6 presents the average values and standard deviations (in parentheses) of the cost of the solutions, the numbers of vehicles, the distances, the number of deliverymen and the running times (in seconds), considering the five runs of the algorithms for instance class R1. Note that the standard deviations are relatively small when compared to the average values.

Table 6
Average solutions (standard deviation) of the approaches for the instance class R1.

Instance	Average results (standard deviation) ILS					Average results (standard deviation) LNS
	Cost	Veh	Dist	Del	Time	Cost	Veh	Dist	Del	Time
R101	23.75(0.04)	19.00(0.00)	1737.37(11.99)	45.80(0.45)	600.20(0.45)	23.73(0.05)	19.00(0.00)	1739.67(14.54)	45.60(0.55)	600.60(0.89)
R102	20.95(0.00)	17.00(0.00)	1531.27(0.19)	38.00(0.00)	600.80(0.84)	20.95(0.00)	17.00(0.00)	1531.66(0.34)	38.00(0.00)	601.40(1.14)
R103	15.94(0.07)	13.00(0.00)	1375.83(23.83)	28.00(0.71)	601.60(1.95)	15.94(0.00)	13.00(0.00)	1380.77(13.08)	28.00(0.00)	602.60(1.14)
R104	12.73(0.04)	10.00(0.00)	1063.26(13.97)	26.20(0.45)	602.20(1.30)	12.79(0.04)	10.00(0.00)	1061.47(6.51)	26.80(0.45)	605.60(2.51)
R105	17.64(0.00)	14.00(0.00)	1412.52(0.00)	35.00(0.00)	600.00 (0.00)	17.64(0.00)	14.00(0.00)	1413.21(1.02)	35.00(0.00)	600.40(0.55)
R106	15.11(0.04)	12.00(0.00)	1307.63(25.17)	29.80(0.45)	601.40(1.14)	15.07(0.05)	12.00(0.00)	1334.82(43.79)	29.40(0.55)	600.40(0.55)
R107	12.95(0.11)	10.00(0.00)	1127.93(15.21)	28.40(1.14)	601.20(1.10)	12.97(0.05)	10.00(0.00)	1137.38(21.06)	28.60(0.55)	600.80(1.79)
R108	11.72(0.04)	9.00(0.00)	975.94(9.06)	26.20(0.45)	601.40(0.89)	11.94(0.37)	9.20(0.45)	989.26(16.40)	26.40(0.89)	602.60(2.07)
R109	15.09(0.05)	12.00(0.00)	1261.06(35.47)	29.60(0.55)	600.40(0.55)	14.99(0.31)	11.80(0.45)	1278.42(6.91)	30.60(1.34)	602.80(1.48)
R110	13.94(0.04)	11.00(0.00)	1184.00(26.59)	28.20(0.45)	601.00 (1.00)	13.96(0.05)	11.00(0.00)	1203.71(34.82)	28.40(0.55)	602.40(2.30)
R111	13.61(0.34)	10.80(0.45)	1138.48(25.28)	27.00 (1.22)	601.40(1.34)	13.65(0.30)	10.80(0.45)	1135.86(9.86)	27.40(1.52)	604.40(2.70)
R112	11.96(0.36)	9.20(0.45)	1011.68(15.77)	26.60(0.89)	600.80(1.10)	12.52(0.40)	9.80(0.45)	1005.54(15.27)	26.20(0.45)	605.40(4.51)

5 Conclusions and perspectives

In this paper, we have addressed the vehicle routing problem with time windows and multiple deliverymen using two metaheuristic approaches, which are based on Iterated Local Search and Large Neighborhood Search. Both approaches use the same constructive heuristic and also a set of additional heuristics, used to enhance the performance of the approaches. The additional heuristics were developed to address specific features of the VRPTWMD. Using six instance classes from the literature, we have compared the performance of the approaches between themselves and against existing metaheuristics. Computational experiments showed that the developed metaheuristic approaches do not dominate one another in all the instance classes, and that both are capable of producing good solutions for the instances when compared to the other approaches from the literature. Perspectives for future research include the combination of these approaches with exact methods, such as column generation or general-purpose integer programming solvers, in order to derive hybrid methods for the VRPTWMD. Also, the current problem can also be extended to incorporate practical features as heterogeneous fleet, multiple depots or split deliveries.

Acknowledgements

This research was supported by CAPES-DS, FAPESP under project number 2014/00939-8 and CNPq under project number 482664/2013-4.

• Financial support: CAPES, FAPESP, CNPq.

Referências

• Azi, N., Gendreau, M., & Potvin, J. Y. (2014). An adaptive large neighborhood search for a vehicle routing problem with multiple routes. Computers & Operations Research, 41(1), 167-173. http://dx.doi.org/10.1016/j.cor.2013.08.016
  » http://dx.doi.org/10.1016/j.cor.2013.08.016
• Bräysy, O., & Gendreau, M. (2005). Vehicle Routing Problem with Time Windows, Part I: Route Construction and Local Search Algorithms. Transportation Science, 39(1), 104-118. http://dx.doi.org/10.1287/trsc.1030.0056
  » http://dx.doi.org/10.1287/trsc.1030.0056
• Ferreira, V., & Pureza, V. (2012). Some experiments with a savings heuristic and a tabu search approach for the vehicle routing problem with multiple deliverymen. Pesquisa Operacional, 32(2), 443-463. http://dx.doi.org/10.1590/S0101-74382012005000016
  » http://dx.doi.org/10.1590/S0101-74382012005000016
• Hong, L. (2012). An improved LNS algorithm for real-time vehicle routing problem with time windows. Computers & Operations Research, 39(2), 151-163. http://dx.doi.org/10.1016/j.cor.2011.03.006
  » http://dx.doi.org/10.1016/j.cor.2011.03.006
• Lourenço, H., Martin, O., & Stützle, T. (2010). Iterated local search: framework and applications. In M. Gendreau & J.-Y. Potvin (Eds.), Handbook of metaheuristics (pp. 363-397). Boston, MA: Springer US.
• Lourenço, H. R., Martin, O. C., & Stutzle, T. (2003). Iterated local search. In F. Glover & G. A. Kochenberger (Eds.), Handbook of metaheuristics (pp. 321-353). Boston: Springer US.
• Melechovsky, J. (2012). Evolutionary local search algorithm to solve the multi-compartment vehicle routing problem with time windows. In Proceedings of the XXX International Conference Mathematical Methods in Economics (pp. 564-568). Karviná: Silesian University.
  » XXX
• Mladenovic, N., & Hansen, P. (1997). Variable neighborhood search. Computers & Operations Research, 24(11), 1097-1100. http://dx.doi.org/10.1016/S0305-0548(97)00031-2
  » http://dx.doi.org/10.1016/S0305-0548(97)00031-2
• Montgomery, D. C. (2012). Design and analysis of experiments (Vol. 2). Hoboken: Wiley.
• Naderi, B., Ruiz, R., & Zandieh, M. (2010). Algorithms for a realistic variant of flowshop scheduling. Computers & Operations Research, 37(2), 236-246. http://dx.doi.org/10.1016/j.cor.2009.04.017
  » http://dx.doi.org/10.1016/j.cor.2009.04.017
• Pisinger, D., & Ropke, S. (2007). A general heuristic for vehicle routing problems. Computers & Operations Research, 34(8), 2403-2435. http://dx.doi.org/10.1016/j.cor.2005.09.012
  » http://dx.doi.org/10.1016/j.cor.2005.09.012
• Pisinger, D., & Ropke, S. (2010). Large neighborhood search. In M. Gendreau & J.-Y. Potvin. Handbook of metaheuristics (Vol. 146, pp. 1-22). Boston: Springer US.
• Pureza, V., Morabito, R., & Reimann, M. (2012). Vehicle routing with multiple deliverymen: Modeling and heuristic approaches for the VRPTW. European Journal of Operational Research, 218(3), 636-647. http://dx.doi.org/10.1016/j.ejor.2011.12.005
  » http://dx.doi.org/10.1016/j.ejor.2011.12.005
• Ropke, S., & Pisinger, D. (2006). An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows. Transportation Science, 40(4), 455-472. http://dx.doi.org/10.1287/trsc.1050.0135
  » http://dx.doi.org/10.1287/trsc.1050.0135
• Senarclens de Grancy, G., & Reimann, M. (2014). Vehicle routing problems with time windows and multiple service workers: a systematic comparison between ACO and GRASP. Central European Journal of Operations Research, 24(1): 29-48.
• Shaw, P. (1998). Using constraint programming and local search methods to solve vehicle routing problems. In M. Maher & J.-F. Puget. Principles and practice of constraint programming - CP98 (pp. 417-431). Berlin: Springer Berlin.
• Silva, M. M., Subramanian, A., & Ochi, L. S. (2015). An iterated local search heuristic for the split delivery vehicle routing problem. Computers & Operations Research, 53, 234-249. http://dx.doi.org/10.1016/j.cor.2014.08.005
  » http://dx.doi.org/10.1016/j.cor.2014.08.005
• Solomon, M. (1987). Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations Research, 35(2), 254-265. http://dx.doi.org/10.1287/opre.35.2.254
  » http://dx.doi.org/10.1287/opre.35.2.254
• Subramanian, A., Penna, P. H. V., Uchoa, E., & Ochi, L. S. (2012). A hybrid algorithm for the Heterogeneous Fleet Vehicle Routing Problem. European Journal of Operational Research, 221(2), 285-295. http://dx.doi.org/10.1016/j.ejor.2012.03.016
  » http://dx.doi.org/10.1016/j.ejor.2012.03.016
• Toth, P., & Vigo, D. (2002). The vehicle routing problem: the vehicle routing problem. Canada: Society for Industrial and Applied Mathematics. http://dx.doi.org/10.1137/1.9780898718515
  » http://dx.doi.org/10.1137/1.9780898718515

Publication Dates

• Publication in this collection
  14 June 2016
• Date of issue
  Apr-Jun 2016

History

• Received
  23 June 2015
• Accepted
  28 Dec 2015