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Asymmetric vehicle routing problem with heterogeneous limited fleet: a case study in a beverage industry

Abstract

This article deals with a case study in a beverage industry concerning the Asymmetric Vehicle Routing Problem with Heterogeneous Limited Fleet (AVRPHLF). The objective of the present paper is to define the vehicle routes so as to reduce distribution costs. The AVRPHLF belongs to the NP-hard class, that is, its resolution through exact methods is an extremely hard task. In practice, problems of this nature are generally treated via heuristics. Among the various solution approaches, we decided to perform an adaptation of a heuristic from the literature that has proved to be efficient, capable of generating high-quality solutions in a reasonable execution time. Computational experiments were conducted in a set of seven instances obtained from the company in question. The results obtained show that there was a considerable reduction in the number of vehicles used and in the total distance traveled with respect to the solutions adopted by the company.

Keywords:
Optimization; Beverage industry; Iterated Local Search; Logistics

Resumo

Este artigo aborda um estudo de caso em uma indústria de bebidas relativo ao Problema de Roteamento de Veículos Assimétrico com Frota Heterogênea Limitada (PRVAFHL). O objetivo é definir as rotas dos veículos de modo a reduzir os custos de distribuição. O PRVAFHL pertence à classe NP-difícil, isto é, sua resolução por meio de métodos exatos é uma tarefa extremamente árdua. Problemas desta natureza são geralmente tratados na prática de forma heurística. Dentre as diversas abordagens existentes, optou-se por realizar uma adaptação de uma heurística da literatura que se mostrou eficiente, sendo capaz de gerar soluções de qualidade elevada em um tempo de execução aceitável. Experimentos computacionais foram realizados em um conjunto de 7 instâncias obtidas junto à empresa em questão. Os resultados obtidos mostram que houve uma redução considerável no número de veículos utilizados e na distância total percorrida em relação às soluções adotadas pela empresa.

Palavras-chave:
Otimização; Indústria de bebidas; Iterated Local Search; Logística

1 Introduction

The Vehicle Routing Problem (VRP) is one of the most well-known and studied logistic distribution problems in the field of combinatorial optimization due to its applicability and importance, especially in the supply chain. In view of its high complexity, solving the VRP is regarded as NP-hard, that is, the difficulty in finding the optimal solution increases exponentially as the number of customers also increases.

In its classic form, the VRP can be defined as follows. Let G = (V,E) be a graph where V = {v0,v1, ...,vn} is the set of vertices and E = {(vi, vj):vi, vjV i<j} is the set of edges. The vertex v0 denotes the depot where a fleet of m identical vehicles with capacity Q are placed, whereas the remaining vertices represent the customers. Each customer vi has a non-negative demand qi. A non-negative travel cost cij, which can be interpreted in terms of distance, time and travel costs, is associated with each edge {vi, vj}. The VRP consists in determining a set of routes (each of them associated with a vehicle), so that each customer is visited exactly once and by a single vehicle, in such a way that the sum of travel costs are minimized and each customer’s demand is met. Each vehicle must start and finish the route at the depot v0 and the sum of the demands of each route must not exceed the vehicle capacity. Such problem is known as Capacited Vehicle Routing Problem. A graphical representation of a VRP solution containing 3 routes is depicted in Figure 1.

Figure 1
VRP solution with 3 vehicles and 56 customers.

When the travel cost from customer i to the customer j is different from the cost in the opposite direction, we have the Asymmetric VRP (AVRP) which, in contrast with the VRP, is defined over a directed graph G = (V, A), where A = {(vi, vj):vi, vjV, ij} is the set of arcs of G.

For the case in which there is a set of K types of vehicles with different Qk capacities, we have the Heterogeneous Fleet VRP (HFVRP). In this variant, two situations can occur: when the number of vehicles of each type is known, the fleet is limited; otherwise, it is unlimited. Furthermore, fixed and/or variable costs associated with each type of vehicle may or may not be considered.

By combining the asymmetric case with a limited fleet of vehicles that have distinct capacities, we have the Asymmetric Heterogeneous Limited Fleet Vehicle Routing Problem (AHLFVRP).

This work extends the one of Kramer et al. (2012)Kramer, R. H. F. R., Subramanian, S., & Penna, P. H. V. (2012). Problema de roteamento de veículos assimétrico com frota heterogênea limitada: um estudo de caso em uma indústria de bebidas. In Anais do XVI CLAIO/XLIV SBPO (pp. 2102- 2113). Rio de Janeiro: SOBRAPO., which refers to a case study concerning the AHLFVRP in a beverage industry considered to be the largest bottler of mineral water in Brazil, with various manufacturing units distributed throughout the territory. The objective of the present paper is to define the distribution routes and compare the results obtained with those usually adopted by the company.

As in many combinatorial optimization problems, such as packing, production scheduling, location, work scheduling, etc, VRPs are usually solved using heuristic methods, especially when applied to real-life and large-scale cases. A heuristic can be defined as a set of methods and rules, inspired by intuitive processes that lead to the resolution of problems. In contrast to exact methods, which find the best possible solution at the expense of a high resolution time, heuristic methods do not have this guarantee, but they are capable of generating near-optimal solutions in a relatively short amount of time. A detailed explanation of heuristics and metaheuristics can be found in Souza (2007)Souza, M. J. F. (2007). Inteligência computacional para otimização (Notas de aula). Ouro Preto: Universidade Federal de Ouro Preto., Blum & Roli (2003)Blum, C., & Roli, A. (2003). Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Computing Surveys, 35(3), 268-308. http://dx.doi.org/10.1145/937503.937505.
http://dx.doi.org/10.1145/937503.937505...
and Gendreau & Potvin (2010)Gendreau, M., & Potvin, J.-Y. (2010). Handbook of metaheuristics. 2 ed. Berlin: Springer. 648 p..

With a view of solving the proposed problem, we decided to use the ILS-RVND heuristic (Penna et al., 2013Penna, P. H. V., Subramanian, A., & Ochi, L. S. (2013). An iterated local search heuristic for the heterogenous fleet vehicle routing problem. Journal of Heuristics, 19(2), 201-232. http://dx.doi.org/10.1007/s10732-011-9186-y.
http://dx.doi.org/10.1007/s10732-011-918...
), due to its flexibility, simplicity, robustness and its good computational performance. During data collection, we made use of the Application Programming Interface (API) of Google Maps to acquire the distance and time matrices between customers.

The remainder of the paper is organized as follows. Initially, Section 2 presents a brief description of some related works. Section 3 characterizes the company studied as well as the particularities of the distribution process. Section 4 describes the algorithm developed for the acquisition of the distance and time matrices. Section 5 presents the ILS-RVND algorithm along with the procedures for generating an initial solution and for performing local search and perturbation. Section 6 contains the results obtained and a comparison with those adopted by the company. Finally, Section 7 concludes the work.

2 Related works

The VRP was introduced by Dantzig & Ramser (1959)Dantzig, G. B., & Ramser, J. H. (1959). The truck dispatching problem. Management Science, 6(1), 80-91. http://dx.doi.org/10.1287/mnsc.6.1.80.
http://dx.doi.org/10.1287/mnsc.6.1.80...
in the late 1950s in order to solve a gasoline distribution problem. In this work, the authors presented the first mathematical formulation and algorithmic approach for the problem. Since then, the number of studies and works related to the topic grew significantly. Some authors, such as Toth & Vigo (2002)Toth, P., & Vigo, D. (2002). The vehicle routing problem (SIAM Monographs on Discrete Mathematics and Applications). Philadelphia: SIAM. and Golden et al. (2008)Golden, B., Raghavan, S., & Wasil, E. (2008). The vehicle routing problem: latest advances and new challenges. New York: Springer., list a reasonable number of VRP applications in various branches of activity.

Given the high complexity in solving the problem and depending on the instance characteristics as well as on the difficulty in generating feasible solutions, the use of computational techniques may contribute to a reduction in the transportation costs in the order of 5-20%, as mentioned by Toth & Vigo (2002)Toth, P., & Vigo, D. (2002). The vehicle routing problem (SIAM Monographs on Discrete Mathematics and Applications). Philadelphia: SIAM..

Many of the real-life applications of the VRPs in food industry, solid waste, beverage, dairy and journal distributions can be found in Golden et al. (2002)Golden, B. L., Assad, A. A., & Wasil, E. A. (2002). Routing vehicles in the real world: applications in the solid waste, beverage, food, dairy and newspaper industries. In P. Toth & D. Vigo (Eds.), The vehicle routing problem (SIAM Monographs on Discrete Mathematics and Applications, pp. 245-286). Philadelphia: SIAM.. The authors demonstrate that when operations research techniques are used in distribution logistics, they can contribute to reducing costs and increasing productivity.

Gonçalves et al. (2005)Gonçalves, S. M., & Steiner, M. T. A., & Zamboni, V. S. (2005). Uma metodologia para o problema de roteamento de veículos - um estudo de caso para a distribuição de água mineral. In Anais do XXXVII SBPO (pp. 1413-1423). Gramado: SOBRAPO. studied the mineral water distribution problem in the city of Itu. To solve it, the authors defined customers’ clusters after determining medians and then performed the routing process using the savings heuristics of Clarke & Wright (1964)Clarke, G., & Wright, J. W. (1964). Scheduling of vehicles from a central depot to a number of delivery points. Operations Research, 12(4), 568-581. http://dx.doi.org/10.1287/opre.12.4.568.
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together with the nearest insertion approach and the Tabu Search metaheuristic.

The practical importance of HFVRP can be verified by the large variety of studies found in the literature. Tarantilis & Kiranoudis (2001)Tarantilis, C., & Kiranoudis, C. (2001). A meta-heuristic algorithm for the efficient distribution of perishable foods. Journal of Food Engineering, 50(1), 1-9. http://dx.doi.org/10.1016/S0260-8774(00)00187-4.
http://dx.doi.org/10.1016/S0260-8774(00)...
, Prins (2002)Prins, C. (2002). Efficient heuristics for the heterogeneous fleet multitrip VRP with application to a large-scale real case. Journal of Mathematical Modelling and Algorithms, 1(2), 135-150. http://dx.doi.org/10.1023/A:1016516326823.
http://dx.doi.org/10.1023/A:101651632682...
and Tarantilis & Kiranoudis (2007)Tarantilis, C., & Kiranoudis, C. (2007). A flexible adaptive memory-based algorithm for real-life transportation operations: Two case studies from dairy and construction sector. European Journal of Operational Research, 179(3), 806-822. http://dx.doi.org/10.1016/j.ejor.2005.03.059.
http://dx.doi.org/10.1016/j.ejor.2005.03...
presented limited fleet applications in dairy, furniture, food and construction industries. Belfiore & Yoshizaki (2009)Belfiore, P., & Yoshizaki, H. T. Y. (2009). Scatter search for a real-life heterogeneous fleet vehicle routing problem with time windows and split deliveries in Brazil. European Journal of Operational Research, 199(3), 750-758. http://dx.doi.org/10.1016/j.ejor.2008.08.003.
http://dx.doi.org/10.1016/j.ejor.2008.08...
described an unlimited fleet application in a Brazilian retailer, in which each store had time windows and certain vehicle sizes constraints, and they could be served by more than one vehicle. Campos et al. (2006)Campos, G. G., Yoshizaki, H. T. Y., & Belfiore, P. P. (2006). Algoritmos genéticos e computação paralela para problemas de roteirização de veículos com janelas de tempo e entregas fracionadas. Gestão & Produção, 13(2), 271-281. http://dx.doi.org/10.1590/S0104-530X2006000200009.
http://dx.doi.org/10.1590/S0104-530X2006...
solved a HFVRP with time windows constraints of one of Brazil’s largest retailers by means of genetic algorithms and parallel computing. The approach employed has proved capable of reducing the costs of the solutions adopted by the company by about 13%.

Marmion et al. (2010)Marmion, M.-E., Humeau, J., Jourdan, L., & Dhaenens, C. (2010). Comparison of neighborhoods for the HFF-AVRP. In International Conference on Computer Systems and Applications (AICCSA) (pp. 1-7). Hammamet: IEEE. http://dx.doi.org/10.1109/AICCSA.2010.5586979.
http://dx.doi.org/10.1109/AICCSA.2010.55...
proposed the first asymmetric version of HFVRP by developing four algorithms whose main objective was to compare the performance of two classic operators: swap and reinsertion.

In addition to the methodology used to solve the routing problem, the way in which the information is collected is a crucial factor to ensure the quality of the solutions. Given that the metric used to assess the VRP solutions are mostly based on the total distance traveled by the vehicles, the precise definition of the location and distance between customers is an essential prerequisite to increase the reliability of the quality of the solutions. In view of this, Galvão et al. (1997)Galvão, R. D., Barros Neto, J. F., Ferreira Filho, V. J. M., & Henriques, H. B. S. (1997). Roteamento de veículos com base em sistemas de informação geográfica. Gestão & Produção, 4(2), 159-174. investigated the integration of routing models and Geographic Information Systems (GIS) applied to a simulated soft drinks distribution problem in Rio de Janeiro. Tarantilis & Kiranoudis (2002)Tarantilis, C., & Kiranoudis, C. (2002). Using a spatial decision support system for solving the vehicle routing problem. Information & Management, 39(5), 359-375. http://dx.doi.org/10.1016/S0378-7206(01)00103-3.
http://dx.doi.org/10.1016/S0378-7206(01)...
proposed a decision support system (DSS) which combines GIS with heuristic methods to solve VRPs in Athens.

Due to the aforementioned practical importance of VRPs, there are various softwares available in the market capable of solving them. Baker (2002)Baker, E. K. (2002). Evolution of microcomputer-based vehiclerouting software: case studies in the United States. In P. Toth & D. Vigo (Eds.), The vehicle routing problem (SIAM Monographs on Discrete Mathematics and Applications, pp. 353-361). Philadelphia: SIAM. compared different softwares focusing his attention on the optimization algorithms used. More recent studies regarding prices, computational performance, algorithms used and solvable variants, among other aspects, can be found in Partyka & Hall (2010)Partyka, J., & Hall, R. (2010). On the road to connectivity. ORMS Today, 37(1), 42-49..

3 Description of the company and its distribution process

The manufacturing plant under analysis is located in the state of Paraíba, Brazil, and has thousands of customers, ranging from small retailers to large supermarket chains, spread over the Brazillian Northeast region. In addition to the bottling of mineral water, the company produces soft drinks, juices and energy drinks. The company is responsible for the distribution of the products to the customers and in order to do so, it relies on the aid of a routing software.

The fleet available to perform the distribution is composed of 37 vehicles, where 36 of them are outsourced and they have distinct capacities; whereas the remaining one is owned by the company itself and it has a capacity of 25 tons. This fleet is responsible for delivering customer's orders from all over the state of Paraiba as well as from some other states of the Northeast region. Table 1 presents the characteristics of the vehicles.

Table 1
Vehicle Fleet.

The data associated with the customers orders were restricted to region of the Greater João Pessoa (GJP), which consists of 5 cities: João Pessoa, Cabedelo, Santa Rita, Bayeux and Lucena. On average, 300 customers are served per day in this area.

The sales orders are delivered daily to the distribution sector once they are collected by the retailers. Since nearly all vehicles are outsourced and their payment is directly proportional to both the distance traveled and the use of the vehicles, it is in the company’s interest to reduce them.

4 Obtaining the distances and travel times between customers

The data used to solve the problem was provided by the company studied. However, while collecting such data, it was not possible to obtain the distance and travel time matrices between customers. This information belongs to the companz that developed the routing software, which was not contacted due to financial issues. As a result, we decided to use Google Maps to obtain the referred data.

Google Maps is a Google service that offers a powerful and friendly mapping technology. Among many of its features, it allows users to find addresses and obtain routes between two or more locations, giving a description of the path as well as the distance and time necessary to perform it.

The Google Maps JavaScript API is a free service and available to any free access site. However, some functions are restricted to a certain amount and frequency of requests. For instance, when demanding a route from one place to another, we have a requisition. At the time that the research was conducted, this service was limited to 15000 daily requisitions. Once this limit is reached, the user's Internet Protocol (IP) address is temporarily blocked for a period of 24 hours.

Assuming that 300 customers, on average, place orders daily to the company, the distance and travel time matrices will have 90.000 elements each, respectively. Therefore, in order to completely fill the matrices, the number of requisitions would exceed the limits established by Google, in case they were requested point by point. Nevertheless, instead of sending a route request providing the location of 2 customers at a time, we developed an algorithm that provides a path (restricted to 25 points) as requisition. Hence, when taking a path that contains 25 points (customers), it is possible to obtain 24 distances and 24 times per requisition, making it feasible to obtain the matrices without exceeding the service limits. For instance, when taking a path with 5 points, A-B-C-D-E, it is possible to obtain 4 distances (d) and 4 times (t) between customers: dAB, dBC, dCD, dDE and tAB, tBC, tCD, tDE.

The Algorithm 1 shows the method developed to obtain the distance and time matrices between customers. The geographical coordinates of the customers were provided by the company.

Algorithm 1
Obtaining the distance and time matrices between customers.

5 The ILS-RVND algorithm

The proposed algorithm consists of and adaptation of the ILS-RVND heuristic presented in Penna et al. (2013)Penna, P. H. V., Subramanian, A., & Ochi, L. S. (2013). An iterated local search heuristic for the heterogenous fleet vehicle routing problem. Journal of Heuristics, 19(2), 201-232. http://dx.doi.org/10.1007/s10732-011-9186-y.
http://dx.doi.org/10.1007/s10732-011-918...
. ILS-RVND is based on the Iterated Local Search (ILS) metaheuristic (Lourenço et al., 2002Lourenço, H. R., Martin, O. C., & Stützle, T. (2002). Iterated local search. In F. Glover & G. A. Kochenberger (Eds.), Handbook of metaheuristics (pp. 321-353). Norwell: Kluwer Academic Publishers.) which makes use of a Randomized Variable Neighborhood Descent procedure (RVND) (Hansen et al., 2010Hansen, P., Mladenović, N., & Pérez, J. A. M. (2010). Variable neighborhood search: methods and applications. Annals of Operations Research, 175(1), 367-407. http://dx.doi.org/10.1007/s10479-009-0657-6.
http://dx.doi.org/10.1007/s10479-009-065...
) in the Local Search phase. In order to contribute for the diversification of the search space, ILS-RVND employs more than one type of procedure for generating initial solutions, as well as for performing local search and perturbation. The original ILS-RVND algorithm was adapted to take into account the particularities of the case study, such as the route duration constraints and the asymmetrical distances between customers.

The pseudocode of the ILS-RVND heuristic is illustrated in Algorithm 2. The input parameters of the algorithm are MaxIter, MaxIterILS and v, where the latter represents the total number of vehicles available. MaxIter represents the number of restarts of the algorithm (lines 4-21) and an initial solution is generated at each restart. After the generation of the initial solution, the main loop of the ILS algorithm is executed MaxIterILS times (lines 8-16), as an attempt to improve the initial solution by means of local search (line 9) and perturbation (line 14) procedures. In case the best solution found after the ILS loop (MaxIterILS times) is better than the initial solution generated, this becomes the current solution and the algorithm is restarted from line 3.

Algorithm 2
ILS-RVND.

5.1 Generation of the initial solution

The procedure for generating initial solutions makes use of two insertion criteria, more precisely, the Modified Cheapest Feasible Insertion Criterion (MCFIC) and the Nearest Feasible Insertion Criterion (NFIC). In addition, two insertion strategies were applied: the Sequential Insertion Strategy (SIS) and the Parallel Insertion Strategy (PIS).

The pseudocode of the procedure for generating an initial solution is represented in Algorithm 3. Initially, the vehicles are sorted in descending order, with respect to their capacity, in such a way that each vehicle serves a route with a single customer, chosen according to the largest demand (lines 4-8). Next, an insertion criterion and strategy are randomly selected (lines 9-10). An initial solution is then built, using the criterion and the strategy selected (lines 11-15). If an infeasible solution is generated, the procedure is restarted from line 3. In the case of the AHLFVRP, an infeasible solution is obtained when the available vehicles are not capable of serving all customers or the duration of one or more routes exceed the maximum limit. If it is not possible to produce a feasible initial solution after a certain number of consecutive attempts, an extra vehicle is added in order to facilitate the process of generating a complete initial solution (this vehicle cannot be used in the final solution).

Algorithm 3
Generation of the Initial Solution.

MCFIC consists of inserting, in a given route, a customer k belonging to the Candidate List (CL) that has the cheapest insertion cost denoted by Expression (1), in which the first part computes the insertion cost of customer k between two adjacent customers i and j, while the second one is a mechanism used to avoid the late insertion of customers located far from the depot, representing a reduction of the insertion cost as a function of the distance from the depot (back and forth).The parameter γ can assume any value in the following set: {0.00, 0.05, 0.10, ..., 1.65, 1.70} (Subramanian et al., 2010Subramanian, A., Drummond, L. M. A., Bentes, C., Ochi, L. S., & Farias, R. (2010). A parallel heuristic for the vehicle routing problem with simultaneous pickup and delivery. Computers & Operations Research, 37(11), 1899-1911. http://dx.doi.org/10.1016/j.cor.2009.10.011.
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).

min { ( c i k + c k j c i j ) γ ( c 0 k + c k 0 ) | k C L } (1)

In NFIC, one should include in the route a customer kCL with the shortest distance with respect to a given customer i already inserted, as represented in Expression (2). In this case, k is always inserted after i.

min { c i k | k C L } (2)

In SIS, only one route is considered for insertion at each iteration. If MCFIC is the insertion criterion, the γ value is then chosen at random from the set already specified. While the CL is not empty and there is at least one customer kCL that can be added to the current solution without violating any constraint, a customer is inserted in each route according to the insertion criterion selected.

As opposed to the previous case, PIS takes all routes into consideration when computing the insertion costs. Therefore, one evaluates the insertion of customer k in every possible position of all routes. The best possible insertion, that is, the one associated with the smallest cost and that does not violate the constraints of the problem is then performed (lines 7-12).

5.2 Local search

One of the main characteristics of the ILS-RVND metaheuristic is the ability to perform a randomized ordering of the neighborhood structures (RVND), in contrast to the usual VND approaches that employ a deterministic order, thus avoiding both the use of parameter tuning and the premature convergence to poor-quality local optimal solutions.

The pseudocode of the RVND procedure is shown in Algorithm 4. Initially, a inter-route neighborhood structure is selected randomly from a Neighborhood List NL (line 4). Next, the best solution is determined from the set of moves of the neighborhood selected and, in case the solution found is better than the current solution, this becomes the current solution and then, an intra-route local search is performed (lines 6-9). If the inter-route neighborhood search leads to an improvement, the NL is repopulated with all neighborhoods (line 9). In case the solution found with the use of inter-route neighborhood is not better than the current solution, such neighborhood is removed from NL (line 12) and the same procedures (from line 4) are repeated until the list becomes empty.

Algorithm 4
RVND.

Similarly to the inter-route, the intra-route local search algorithm consists of randomly choosing a neighborhood contained in a list NL2, determining the best solution by applying moves of the neighborhood selected and, in case it is better than the current solution, it becomes the new current solution. In case of no improvement, the neighborhood is removed from NL2 and the same procedures are repeated until the list becomes empty. The NL2 list is always restarted after the inter-route neighborhood search leads to an improvement.

In both inter-route and intra-route local search procedures, only moves that do not violate the vehicle capacity and the route duration constraints are accepted. The algorithm verifies if the move is feasible or not, always after ascertaining the improvement in the current solution. Hence, it is possible to reduce the runtime of the algorithm.

The verification of the vehicle capacity constraint can be performed in a constant time by comparing the sum of the customers’ demands with the vehicle capacity of a given route. As for performing the verification of the route duration constraint in constant time, one needs to keep track of the duration of each route during the whole local search procedure. Some auxiliary structures are employed by the ILS-RVND algorithm in order to yield an improvement of the computational performance. A detailed description of these structures can be found in Penna et al. (2013)Penna, P. H. V., Subramanian, A., & Ochi, L. S. (2013). An iterated local search heuristic for the heterogenous fleet vehicle routing problem. Journal of Heuristics, 19(2), 201-232. http://dx.doi.org/10.1007/s10732-011-9186-y.
http://dx.doi.org/10.1007/s10732-011-918...
.

5.2.1 Inter-route neighborhood structures

In order to solve the AHLFVRP, six inter-route neighborhood structures were used. Five of them are based on the λ-interchanges scheme (Osman, 1993Osman, I. H. (1993). Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem. Annals of Operations Research, 41(4), 421-451. http://dx.doi.org/10.1007/BF02023004.
http://dx.doi.org/10.1007/BF02023004...
), which consists of exchanging up to λ customers between two routes; while the remaining one is based on the Cross-exchange operator (Taillard et al., 1997Taillard, E., Badeau, P., Gendreau, M., Guertin, F., & Potvin, J.-Y. (1997). A tabu search heuristic for the vehicle routing problem with soft time windows. Transportation Science, 31(2), 170-186. http://dx.doi.org/10.1287/trsc.31.2.170.
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), which consists of exchanging two segments of different routes. To limit the number of possibilities, we considered λ=2. A description of each structure is presented as follows.

  1. 1

    Shift(1,0) – One customer is moved from a route to another one.

  2. 2

    Shift(2,0) – Two customers k1 and k2 are moved from one route to another one. In this case, both arcs (k1, k2) and (k2, k1) are considered.

  3. 3

    Swap(1,1) – Interchange between one customer of a route and a customer from another route.

  4. 4

    Swap(2,1) – Interchange between two adjacent customers k1 and k2 from a route and one customer k3, from another route. In this case, both arcs (k1,k2) and (k2,k1) are considered.

  5. 5

    Swap(2,2) – Interchange between two adjacent customers k1 and k2 from one route and two adjacent customers k3 and k4, from another route. In this situation, all the four possible combinations of the interchanging arcs formed by the two pairs of customers are considered.

  6. 6

    Cross – Interchange between a segment of a route and a segment of another route. A segment of route is determined by the set of customers located along the path between two customers. For the proposed algorithm, only the segments of route ending at the depot are considered.

A graphic illustration of the inter-route neighborhoods implemented is depicted in Figure 2, in which σ represents the segment of route containing one or more customers and ⊕ represents the concatenation operator between two segments of route.

Figure 2
Inter-route Neighborhoods.

5.2.2 Intra-route neighbourhood structures

This set of neighborhoods consists of rellocating customers of a same route. In this case, it is not necessary to verify the vehicles’ capacities constraints, as long as the previous solution is feasible, the exchange of positions between customers in a same route will not affect its total demand. However, since the exchange of positions between customers will modify the path traveled and, consequently, the time of the duration of the route, verifying the total duration time of the route becomes necessary. Four intra-route neighborhood structures were adopted and their description is as follows.

  1. 1

    Reinsertion – One customer is removed and inserted in another position of the route.

  2. 2

    Or-Opt2 – Two adjacent customers are removed and inserted in another position of the route.

  3. 3

    Or-Opt3 – Three adjacent customers are removed and inserted in another position of the route.

  4. 4

    Exchange – Permutation between two non-adjacent customers.

A graphic illustration of the intra-route neighborhoods implemented can be observed in Figure 3.

Figure 3
Intra-route Neighborhoods.

5.3 Perturbation mechanism

According to Lourenço et al. (2002)Lourenço, H. R., Martin, O. C., & Stützle, T. (2002). Iterated local search. In F. Glover & G. A. Kochenberger (Eds.), Handbook of metaheuristics (pp. 321-353). Norwell: Kluwer Academic Publishers., the perturbation mechanisms are considered to be key aspects for the high performance of the ILS metaheuristic, as well as the local search.

In contrast to the local search procedures, the perturbation mechanisms are not intended to improve the solution. They are, however, useful in performing modifications in the current solution, as an attempt to escape from local optima.

As in the previous procedures, two perturbations were used, and they are randomly selected by the algorithm. Such procedures are described as follows:

  1. 1

    Multiple-Swap(1,1) – Multiple Swap(1,1) moves are executed between different pairs or routes selected at random.

  2. 2

    Multiple-Shift(1,1) – Multiple Shift(1,1) moves are executed randomly.

6 Computational results

In this section, we present a description of the instances (daily orders) concerning the fleet of vehicles and the demand. Next, we compare the results achieved by ILS-RVND with those adopted by the company in question. In this section, we present a description of the instances concerning the fleet of vehicles and the demand. We then compare the results achieved by ILS-RVND with those adopted by the company.

The proposed algorithm was coded in C++ and executed in an intel Core i5-2410M 2.3 GHz with 4 GB of RAM memory running Ubuntu Linux 10.10.

After conducting some preliminary experiments, we considered the value of parameter MaxIter to be 10 in all instances, while we set the MaxIterILS parameter, according to the characteristics of the instances, more precisely, as the number of customers plus the number of vehicles available. Each instance was executed 10 times.

In spite of the 12-hour limit on the route duration adopted by the company’s software, the instances were solved by ILS-RVND considering an 8-hour limit, so as to be within the limits established by Brazilian labor legislation.

6.1 Instances

The quantity and the types of vehicles considered in the instances were the same adopted by the company when serving only the customers located in GJP (according to the routes generated by the software). The tests were performed on a set of 7 instances, as shown in Table 2.

Table 2
Instances.

Regarding the solutions generated by the company’s software, it was verified that occasionally some vehicles were overloaded. Such practice is required in cases where the customer’s demand exceeds the vehicle’s capacity.

The KSP4, KSP5 E KSP6 instances refer to the orders of a single day. Such division was done due to the existence of customers’ orders located in a region outside GJP, so that one of the routes generated by the company’s software have customers from GJP as well as customers from another region. The KSP4 instance has all the customers. In the KSP5 instance, we removed both the customers located outside GJP and the vehicle used in this route. As for the KSP6 instance, we disregarded all the customers of this route, including the vehicle (based on the company’s solution). Another reason for dividing this instance was to try to solve the AHLFVRP in such a way that it would be suitable for all GJP customers without requiring a vehicle responsible for serving customers from another region. The solutions found by ILS-RVND for the instance KSP5 could not be compared because the company did not have a solution available for this situation.

6.2 Comparison of results

The comparison of results was performed with respect to both the total distance traveled by the vehicles and the number of vehicles used.

The best solutions found by the ILS-RVND heuristic are shown in Table 3, as well as the averages and the standard deviations concerning the 10 executions. The results are expressed in terms of the distance traveled and the execution time of the algorithm.

Table 3
ILS-RVND Statistics.

A considerable reduction in the total distance traveled was achieved for all instances, as observed in Table 4. Such reductions can be regarded as reductions in variable costs since both fuel costs and vehicle depreciation increase with the distance traveled.

Table 4
Comparison of results in terms of distance traveled.

When comparing in terms of vehicles, the use of the ILS-RVND heuristic contributed to the reduction of the number of vehicles in 3 out of 6 instances, as pointed out in Table 5. Since all vehicles used by the company are outsourced, reducing the number of vehicles leads to a reduction in the company’s fixed costs.

Table 5
Comparison of results in terms of the vehicles used.

The best solutions achieved by ILS-RVND for the instances presented were illustrated with the aid of Google Maps, as can be observed in Figure 4. Each polygon represents a route and each vertex of the polygon represents a customer (or the depot).

Figure 4
Best solutions.

6.3 Convergence Graphs

With a view of evaluating the ILS-RVND performance and comparing it with the results obtained by the company’s routing software, we have generated convergence graphs for each instance in which the results found by the company were available.

The convergence graphs presented in Figure 5 represent the time spent by the ILS-RVND algorithm to find a better or equal solution to the one achieved by the company. To this end, the ILS-RVND heuristic was executed 100 times for each instance, but this time using a different stopping criterion, more precisely, the algorithm stops once it finds a solution better or equal to the one reported by the company’s software. For each execution, we stored the computational time required until the stopping criterion is met, so that it is possible to infer the probability of equaling or enhancing the company’s results for a given processing time.

Figure 5
Convergence Graphs.

From Figure 5, it can be observed that the maximum time required by the ILS-RVND algorithm to obtain the same results or to outperform the routing software was approximately 6 seconds (for an instance of 341 customers). In general, such results were achieved in less than one second, which emphasizes the high performance of the ILS-RVND algorithm.

7 Conclusions

This article dealt with an Asymmetric Heterogeneous Limited Fleet Vehicle Routing Problem (AHLFVRP) with route duration constraints in the context of a large-scale beverage industry. Due to the high computational complexity of the problem, the use of exact methods becomes prohibitive, especially when dealing with 300-customer instances, which is the average size considered in this work. Hence, it was verified that the use of heuristics revealed to be an adequate strategy to achieve the objective of this paper, that is, to generate good solutions for the problem in a reasonably short execution time.

As expected, the ILS-RVND heuristic proposed by Penna et al. (2013)Penna, P. H. V., Subramanian, A., & Ochi, L. S. (2013). An iterated local search heuristic for the heterogenous fleet vehicle routing problem. Journal of Heuristics, 19(2), 201-232. http://dx.doi.org/10.1007/s10732-011-9186-y.
http://dx.doi.org/10.1007/s10732-011-918...
proved to be very efficient, obtaining high quality solutions. The algorithm was tested on a set of 7 instances associated with the daily orders of the company, in which an average improvement of approximately 14% was achieved, when compared with the practical solutions adopted by the company in terms of distance traveled. There was also a reduction in the number of vehicles used to serve all customers. Such results can be regarded as possible reductions in fixed and variable costs, since the vehicles are outsourced and the company not only pays for their daily usage but also for their operation and maintenance costs.

  • Financial support: Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq (Process 471158/2012-7).

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Publication Dates

  • Publication in this collection
    04 Sept 2015
  • Date of issue
    Jan-Mar 2016

History

  • Received
    05 May 2014
  • Accepted
    20 Nov 2014
Universidade Federal de São Carlos Departamento de Engenharia de Produção , Caixa Postal 676 , 13.565-905 São Carlos SP Brazil, Tel.: +55 16 3351 8471 - São Carlos - SP - Brazil
E-mail: gp@dep.ufscar.br