RESUMEN
El problema de rutas para vehículos con ventanas de tiempo y programación de la carga no solo requiere el diseño de las rutas que satisfagan las restricciones temporales y de capacidad de los vehículos, sino que también la programación de las salidas de los vehículos desde un terminal dado un tiempo de carga debido a los recursos limitados disponibles para cargar las demandas de los clientes en los vehículos.
No solo se presenta una formulación del problema de diseño de rutas para vehículos con ventanas de tiempo y programación de la carga, sino que también se propone e implementa una metaheurística basada en múltiples sistemas de colonias de hormigas, cada una con una sola función objetivo, organizadas de manera jerárquica. Se incorpora una forma de actualización de tiempo dentro del procedimiento constructivo para actualizar y programar la salida de un vehículo desde el terminal cuando cada hormiga se mueve a un nuevo cliente. Se utiliza propagación de restricciones para determinar un movimiento factible a un nuevo cliente. Como el [VRPTWSL] incorpora la programación de la salida de los vehículos, el algoritmo presentado en este artículo tiene una directa aplicación a problemas reales, de esta manera, el [VRPTWSL] se puede tomar como un importante avance para problemas de diseño de rutas para vehículos.
Palabras clave: Problema de rutas para vehículos, sistema de colonias de hormigas, programación.
ABSTRACT
The vehicle routing problem with time windows and scheduled loading [VRPTWSL] requires not only the design of routes with time windows and capacity constraints, but also a schedule of the departures of vehicles from the depot given a load time due to the limited resources available to load the demand in the vehicles.
A mathematical formulation of the vehicle routing problem with time windows and scheduled loading is presented and a metaheuristics based on Multiple Ant Colony System is proposed and implemented where two ant colonies, each with a single objective function, are organized in a hierarchical way. A time update procedure is incorporated into the ant constructive procedure to update and schedule the departure of a vehicle from the depot when each ant moves to a new customer-node. Constraint programming is used to determine a feasible move to a new customer-node. As [VRPTWSL] incorporates the vehicle departure scheduling, the algorithm presented in this paper has a direct application to real problems, in this way [VRPTWSL] can be taken as an important advance for practical vehicle routing problems.
Keywords: Vehicle routing problem, ant colony system, scheduling.
INTRODUCTION
Throughout the last fifty years, the scientific community has been interested by several vehicle routing mostly because of their inherent complexity. in almost every supply chain problem, some of goods is required between the internal or external components.
The vehicle routing problem with time windows is specified in terms of minimizing the total time and cost for a fleet of vehicles to distribute goods from a depot to customers who need to be visited exactly once within their time windows (time intervals). All routes start and end at the same depot, and the total demand of all customers serviced by a vehicle along a particular route must not exceed its total capacity. Typical objective functions are to minimize the number of vehicles and the total travel lime.
In this paper we propose a variant of the vehicle routing problem with time windows referred to as the Vehicle Routing Problem with Time Windows and Scheduled Loading | VRPTWSL |. In this problem, we have to account for the additional feature related to loading on each vehicle at the depot, the goods delivered to the customers on the associated route. Since the number of loading equipment (loaders, loading platforms, etc.) is limited, this induces a scheduling problem of the vehicles at the loading equipments. Consequently, the loading time and the waiting time before loading have an impact on the vehicle departure times from the depot. Such a problem occurs in the forestry industry where pine plants and eucalyptus trees packed in boxes have to be transported from a plant nursery where the number of loading equipments is limited, to different timber sites.
The [VRPTW] has been widely studied in literature. The first state of art reviews can be found in |9J. f 1 7). 1 1 8) and [24). In 1 5] and [10] the authors mostly focus on exact approaches. Two of the best melaheuristics approaches for [VRPTWI are given in [ 16| and 13].
In this paper we formulate the IVRPTWSL |. The solution procedure implemented considers the special case where only one loading equipment is available (inducing a vehicle scheduling problem at the depoD.This procedure is a hybrid of the Ant Colony System approach with the constraint programming approach to deal with the I VRPTWSLI constraints. We present some numerical results for problems randomly generated according to Solomon [23 1 process modified to account for the loading constraints of the (VRPTWSL].
The rest of the paper is organized as follows. First we include a formal description of the | VRPTWSL). and then, we introduce a mathematical model. After that, we present the hybrid method of the Ant Colony System and Constraint Programming approaches to solve the | VRPTWSL). Later, we include the numerical results. Finally, we summarize the main conclusions and remarks.
VEHICLE ROUTING PROBLEM WITH TIME WINDOWS AND SCHEDULED LOADING
Consider the | VRPTW] specified as follows:
- n customers must be delivered from a unique depot:
- [a^sub i^, b^sub i^) denotes the lime window to service customer i;
- q^sub i^ denotes the demand of goods of customer i:
- P^sub i^ and h^sub i^ denotes the loadi ng ti me of customer / goods at the depot and the unloading time at the destination, respectively:
- a set V of vehicles having homogeneous capacity Q is available:
- each route completed by a vehicle ? e V starts and ends at the depot:
- a set G of homogeneous loading equipment is avai lable at the depot :
- each vehicle v ∈ V must be loaded by a unique equipment g ∈ G at the depot.
The non negative value t^sub ij^ associated with each are (i,j) denotes the travel time between / and 7. Note that the delivery of goods for each customer / must start during its time window [a^sub i^, b^sub i^]. Hence the vehicle ? delivering the goods to customer ; can arrive before lime or but then it has to wait until this time to make the delivery.
MATHEMATICAL FORMULATION
* n^sub iv^ = the starting time for the delivery of customer / by vehicle v, i ( N, v ( V, If vehicle v does not deliver goods to customer i, then wiv = 0.
The [VRPTWSL] model is summarized as follows:
In the mathematical model, the constraints (2) to (7) are those used by Toth and Vigo 1 251 to formulate die [VRPTWl. and the constraints (8) to (11) are the additional constraints required for the loading feature of the problem.
The objective function ( 1 ) is a linear function formulated as the sum of the total travel times for the selected routes and a total fixed cost increasing with the number of vehicles used. The value of F is a parameter to be adjusted according to priority of the administrator to reduce the number vehicles used with respect to the total travel time.
Constraints (2) impose that each customer has his goods delivered by exactly one vehicle. Constraints (3) require that whenever a vehicle ? leaves the depot (i.e.. vehicle ? is used), it must return to die depot. The flow conservation constraints are formulated in (4). The constraints (5) and (6) ensure that the time window constraints are satisfied. It is worth noticing that in constraints (5) A0, the unloading time at the depot is equal to 0 (i.e., hfì = 0). The constraints (7) are the vehicle capacity constraints.
The other constraints related to the loading feature of the problem can be seen as those of a (VRP) where the loading equipments correspond to the vehicles and the vehicles to the customers. Constraints (8) require that each vehicle is loaded by unique equipment. Constraints (9) indicate that the first and the last vehicle loaded by any equipment are the corresponding dummy vehicles.
The flow conservation constraints (10) guarantee that a unique vehicle / can be the next vehicle to be loaded by any equipment after loading vehicle v. Finally, we use the constraints (11) to determine the time when the loading of each vehicle is completed by some equipment.
MULTI ANT COLONY SYSTEM FOR VRPTWSL
Ant Colony System is a metaheuristics from the family of IACO] algorithms, where a set of artificial Ants share information to build solutions. [ACS] was first proposed by [ 1 1 1 for the Traveling Salesman Problem and in recent years applications for the [VRPTW] have been proposed by [3] and [ 1 6] showing some of the best performances for this kind of problem: in particular [16] proposes a multi ant colony system for the [VRPTW] where the problem is transformed into a [TSP) considering a number of depots equal to the number of vehicles which must be used to serve the customers and in which two colonies in parallel minimize an objective function, sharing the global best solution. We propose an algorithm for the [VRPTWSL 1 based in a modified version of [MACS] presented by [23], where the two colonies run sequentially and we incorporate a time update procedure for the vehicle scheduling at the depot and a constraint programming based procedure to manage the constraints of a | VRPTW].
The aim of this paper is to propose an algorithm for the i VRPTWSLl based in [MACS] when only one load resource g is available at the depot: in that case vehicles scheduling is needed.
Following the previous we have named this procedure as I M ACS- VRPTWSL I (Figure I), where [MACS] is for multiple ant colony system, the first colony ACS-VEl minimizes the number of vehicles used to fulfill the total demand and has priority over the second colony ACSTIME, which minimizes the total amount of time for the minimum number of vehicles found by ACS-VEl. both colonies use independent pheromone trails but share the variable ψ^sup gb^ which save the global best so fai" solution.
Each colony works in a very similar way. where every ant uses a constraint programming based procedure to determine a feasible domain, wiüi probability q^sub 0^ applying the Pseudo Random Proportional Rule exploration or exploitation to choose the next node to be visit and call a Time Update procedure to update the departure and visits time in the previous nodes visited given the new node incorporated to the sub-route.
A feasible solution is a list of nodes starting from the depot and alternating customer nodes with depots. The last element from the list is the last customer node visited by the vehicle before it goes back to the depot. In this way the number of depots is equal to the number of vehicles. On the other hand the objective function incorporates the distance between the last customer node and the depot.
The first step of the algorithm is to determine an initial solution not necessarily feasible with me nearest neighbor heuristic incorporating the Time Update Procedure after every node selection.
Follow the Nearest Neighbor fNN] heuristic solution with a number of (Vl vehicles. ACS-VEI is calf and tries to find a feasible solution with IV-II vehicles used in ψ^sup gb^. In this way, every time ACS- VEl finds a feasible solution, the process is restarted with one less vehicle until no feasible solution is found. The algorithm determines the minimum number of vehicles to be used and ACS-TIME is activated to improve the total time with the number of vehicles found by ACS-VEI. In ACS-TIME one ant can find a feasible solution with fewer vehicles found by ACS-VEI. In that case a depot node is dropped and the ants will work with the new graph. In this situation ACS-TIME will work with a number of vehicles less than the one found by ACS-VEI.
Time Updating
In mis particular vehicle routing problem we have dropped the assumption that every vehicle starts its service from the depot at time zero because of the limited number of load resources at the depot. Given this we must considérer the loading time of the demand at the depot in the vehicle every time a vehicle adds a new customer node to the sub-route. Thus a lime update procedure must be defined in which dynamically updated total load time at the depot given lhe load of the new customer node incorporated to the sub-route and the delay obtained when the vehicle leaves the depot given this load. This procedure is named Time Updating and it is incorporated to the ant's construction procedure. To understand the Time Updating procedure, three concepts must be defined:
- Delay (Delay): The delay is the time in which the departure of the vehicle from the depot is delayed due to the incorporation of a new customer to the sub-route. This time delay can change the arrival lime of the vehicle in the following customer nodes previously incorporated in the sub-route. The delay at the depots departure is equal to the loading time.
- Delta Time (DeltaJTime): every time a vehicle arrives to a node before its bottom time window (ai) it produces a Delta Time. The Delta Time from every node previously incorporated in the sub-route is equal to the bottom time window less the arrival time of the vehicle to the respective node (a.- Arrival Jtime.); when a vehicle arrives after the bottom time windows Delta Time is zero for the respective node.
- Accumulated Time (Accumulaied_Time): Accumulated Time is the sum of every Delta Times present before a particular node in the sub-route. When accumulated time is bigger than zero for a particular node. Accumulated Time has the ability to diminish part of the Delay in the following nodes; this is because if the accumulated time for a node is big enough then the arrival lime to this node can be delayed enough without affecting the arrival time to the following nodes.
The Time Updating Procedure (Figure 2) includes three stages: Update the accumulated times for every node previously incorporated in the sub-route given the last time update: update the arrival times of the vehicle for every node in the sub-route given the delay produced because of the new node to be visited and the delays that are diminished because the accumulated time of a particular node: finally, update the departure time of the vehicle from every node in the sub-route because the arrival time update to the node.
The Update Time procedure is as follows:
1. Locate the vehicle at the depot with departure time equal to the last vehicle departure; if no previous vehicle or sub-route is present the departure time is zero.
2. Choose a new customer node to be visited and update arrival time to the customer.
3. Calculate Delays at the depot and propagate to the nodes previously incorporated to the sub-route, and update arrival times for each node given the respective Accumulated Times.
4. Update Accumulated Times.
5. Go to stage 2 until there are feasible nodes to be incorporated in the sub-route.
Set of Feasible Nodes
Let N^sup k^^sub i^ set of feasible nodes i to be visited by Ant k. To determine N^sup k^^sub i^ a constraint model is formulated to exploit Constraint Programming propagation properties. If i is a depot. N^sup k^^sub i^ will be the set of nodes not yet visited; if i is a customer N^sup k^^sub i^ is determined according to the constraint model. A constrained variable s^sub i^ is defined, where:
s^sub i^. successor of node i in the sub-route. Where s^sub i^ ∈ [feasible nodes to visit }
At the beginning, the ant is located in a depot. For this reason the domain of possible values for s^sub i^ is the set of all nodes. This domain must be updated as the ant constructs the sub-route. This domain must satisfy the following conditions:
4 No visited nodes by a vehicle in the route must be included.
5 When incorporating s^sub i^ Time windows constraint for the actual sub-route is still feasible after the incorporation, which means Delay is smaller than the difference between time windows b^sub i^ for every node in the sub-route and the sum between arrival time and accumulated time for every node in the sub-route.
6 The capacity constraint is satisfied.
7 Time windows constraints are satisfied.
If the set of feasible nodes is empty or at least 25% of capacity is used. N^sup k^^sub i^ contains also all non-visited depots.
The constraint programming model was programmed in ILOG/solver 6.0.
ACS-VEI Colony
ACS- VEI searches for a feasible solution by maximizing the number of visited customers, in other words minimize the number of vehicle used.
ACS-VEl (Figure 3) starts its search with one less vehicle than the previous feasible solutions found, that is. a solution with IVI -1 vehicles. During this search ACS-VEl builds unfeasible solutions in which some customers have been not yet visited. The solution with the highest number of customers visited for I VI- 1 is stored in a variable ψ^sup ACS-VEI^, a solution is better than ψ^sup ACS-VEI^ only when the number of visited customers is increased.
In ACS- VEI to maximize the number of visited customer. manages a vector of integers IN. The entry IN^sub j^ stores the number of times a customer j has been not incorporated in the solution. IN is used in the constructive procedure by the Ants for favoring less visited customer.
Finally at the end of each cycle, ACS-VEI increases the pheromone trails τ^sub ij^ in the path stored by the unfeasible solution ψ^sup ACS-VEI^ with the highest number of visited customers so far and the feasible solution ψ^sup gb^ with the lowest number of vehicles computed and the lowest route time calculated so far.
ACS-TIME Colony
ACS- TIME (Figure 4) works very similarly to the traditional I ACS] based colony whose goal is to compute a lour as short as possible. ACS-TIME is activated only once, when ACS-VEI has found the lowest number of vehicles, and its pheromone trails are initialized with the pherecord matrix. In ACS-TIME m ants are activated to construct solutions.
If a feasible solution constructed by an ant is better than the one stored in ψ^sup gb^ then ψ^sup gb^ is replaced with the new solution.
One ant could find a new solution with a number of vehicles Jess than the one found by ACS-TIME, in this case the procedure continues with a number of vehicles equal to the number of vehicles found by ACS-TIME.
The next step is to update the pheromone level as follows:
Solution Model
The solution model for [MACS-VRPTWSL| is very similar to the one proposed by [I6|. where every Ant builds a single tour per time.
A solution is represented as follows:
1 . The depots with all their connections to their customers are duplicated a number of times equal to the number of available vehicles.
2. Distances between copies of the depots are set to a number M. where M is a sufficiently big value.
Here, the routing problem is closer to the traditional |TSPI for the [ACS], where a feasible solution is a path where all the nodes have to be visited exactly once. According to [16| the advantage of such an approach is that the trails in direction to a copy of a depot are less attractive than the case with a single depot, due to the pheromone update.
Constructive Procedure
For [M ACS- VRPTWSLJ the heuristic information η^sub ij^ is calculated as presented in [16] for [VRPTW| problems, where it takes into consideration travel time between nodes t^sub ij^, time windows [a^sub j^, b^sub j^] associated to node j, and die number of times IN^sub j^ the node j has not been inserted into a solution (when the procedure is called by ACSTIME, the variables IN are not used and the value is set to zero).
This is calculated as follows:
∀j ∈ N^sup k^^sub i^
Delivery_Time^sub j^ <- max( Time + 1^sub ij^, a^sub j^)
Delta^sub ij^ <- Delivery_Time^sub j^ - Time
Distance^sub ij^ <- Delta^sub ij^ (b^sub j^ - Time)
Distance^sub ij^ <- max( 1 .0. (Distance^sub ij^ - IN^sub j^))
η^sub ij^ >- 1 .01/Distance^sub ij^
In this way, heuristic information privileges the nodes where its time windows are closer to the current time and nodes less included in a solution. This is relevant for a problem where the vehicles departure has to be updated constantly due to the incorporation of a new customer in the route.
NUMERICAL RESULTS FOR MACS-VRPTWSL
In this section we present some instances for [MACSVRPTWSL], and we show that this approach is effective to solve this kind of problem.
The algorithm was coded in C++ and ILOG/Solver 6.0 and run on an Intel Core Duo T2400 @ 1.83 GHz. 504 Mb RAM.
As far as we know there are not instances in the literature for [VRPTWSL]. in this case [MACS-VRPTWSL] has been tested in a subset of 17 instances presented by |23] for the [VRPTW]. The instances we have chosen are problems with 25 nodes and belong to the class of C1 and C2 problems, where C is for Clustered Customers. whose Time Windows where generated based on a known solution. The set of type I problems has narrow time windows and small vehicle capacity, and set of type 2 has large time windows and large vehicle capacity. The service time and load time of every customer demand for every instance is 0.5 demand time unities. Results in Table 1 are obtained with the following parameters:
- β = 2
- m = 10
- q^sub 0^ = 0.9
- ρ = 0.1
- maximum number of iterations = 659
- maximum number of iterations without improvement = 10
- maximum computational time = 250 seconds
It is observable that for the state transition rule presented in equation (17) and ( 18) the importance level has been taken for the pheromone levels first and second on the heuristic information.
CONCLUSIONS
This paper introduces a mathematical formulation for problem [VRPTWSL| based in the model presented by Toth and Vigo [25 1 for the IVRPTW]. in which the assumption that every vehicle departs from the depot at time zero is eliminated. This model incorporates new variables and sequencing constraints due to the limited number of resources, as well a multi-objective function.
An algorithm approach based in multi-ant colony system is designed and implemented, in which two objective functions are minimized: the number of vehicles and total tour length.
A Time Update and constrained procedures are proposed to be implemented in the Ants constructive procedure, to perforin the update of a new customer incorporated into the sub-route and the schedule that this action implies the load resources and to determine the set of feasible nodes to be visited by the ant, respectively.
Even if it not possible to judge the results presented for [MACS-VRPTWSL], this paper shows that fMACSVRPTWSL) is effective in the resolution of this kind of problem, and is a good first step for further for the problems of vehicle routing problem with time windows and vehicle departure scheduling. Possible developments for [VRPTWSL] can incorporate a local search procedure or a more dominant use of constraint programming for the time update procedure. Also, one can formulate an algorithm that for a first step avoids the elimination of the time zero departure assumption and a second step implements a local search procedure with constraint programming to make feasible solution in a similar way presented by [22].
Also, further analysis can take the case of multiple load resources to perform the demand load in the As [VRPTWSL] incorporates the vehicle departure scheduling, the algorithm presented in this paper has a direct application to real problems, in this way I VRPTWSL] can be taken as an important advance for practical vehicle routing problems.
ACKNOWLEDGEMENT
The first author has been partially supported by 205.097.009- 1 .0 of the University of Concepción. Chile. The second and fourth authors have been partially supported by DIN 1 0/2008 of Universidad Católica de la Santísima Concepción. Chile.
REFERENCES
[1] E. Aarts and J. K. Lenstra. "Local Search in Combinatorial Optimization". Wiley Interscience Series in Discrete Mathematical and Optimization. UK. 1997.
[2] J. Bramel and D. Simchi-Levi. "On the effectiveness of set covering formulations for the vehicle routing problem with time windows". Operations Research. Vol.45, pp. 295-301. 1997.
[3] O. Bräysy and W. Dullaert. "A Fast Evolutionary Metaheuristic for the Vehicle Routing Problem with Time Windows". International Journal on Artificial Intelligence Tools. Vol. 12 N° 2. pp. 153-172, 2003.
[4] A. Colorni. M. Dorigo and V. Maniezzo. "Distributed optimization by ant colonies". In: Varela F.J. and Bourgine P. (Eds.). Proceedings of the First European Conference on Artificial Life. Elsevier publishing, pp. 134 - 142. Paris. France. 1991.
[5] J. Cordeau. G. Desaulniers. J. Desrosiers, M.M. Solomon and F. Soumis. "Vip with time windows". In: Toth P. and Vigo D. (Eds.). The vehicle routing problem. Society for Industrial and Applied Mathematics monograph on discrete mathematics and applications, pp. 157-193. 2001.
[6] C. de Jong. G. Kant and A. Van Vlient. "On finding minimal route durations in the vehicle routing problem with multiple time windows". Technical report. Manuscript. Department of Computer Science. Utrecht University. Netherlands. 1996.
[7] J. L. Deneubourg, S. Aron. S. Goss and J. M. Pasteéis. "The self-organizing exploratory pattern of the argentine ant". Journal of Insect Behavior. Vol. 3. pp. 159-168. 1990.
[8] M. Desrochers. J. Desrosiers, M.M. Solomon. "A new optimization algorithm for the vehicle routing problem with time windows". Journal of Operations Research. Vol. 40. pp. 342-354. 1992.
[9] M. Desrochers and F. Soumis. "A generalized permanent labeling algorithm for the shortest path problem with time windows". Society for Industrial and Applied Mathematics, pp. 157-193. 1988.
[10] J. Desrosiers. Y. Dumas, M.M. Solomon and F. Soumis. 'Time constrained routing and scheduling". In: Ball M.O., Magnanti T.L., Monma CL. and Nemhauser G.L. (Eds), Network Routing. Handbooks in Operations Research and Management Science. Vol. 8, pp. 35-139. 1995.
[11] M. Dorigo and L.M. Gambardella. "Ant colonies for the traveling salesman problem". BioSystems. Vol. 43 N^sup o^ 2, pp. 73-81 . 1997a.
[12] M. Dorigo and L.M. Gambardella. "Ant colony system: A cooperative learning approach to the traveling salesman problem". IEEE Transactions on Evolutionary Computation. Vol. 1 N^sup o^ I, pp. 53-66. 1997b.
[13] M. Dorigo and T. Stuzle. "Ant Colony Optimization". MTT Press, Massachussets Institute of Technology. Cambridge. 2004.
[14] D. Favaretto. E. Moretti and P. Pellegrini. "Ant colony system fora vrp with multiple time windows and multiple visits". Accepted for publication in Journal of Interdisciplinary Mathematics. 2006.
[15] M. Flood. "The traveling salesman problem". Operations Research. Vol. 4, pp. 61-75. 1956.
[16] L.M. Gambardella, E. Taillard and G. Agazzi. "A multiple ant colony system for vehicle routing problem with time windows". Technical Report IDSIA-06-99. IDSIA. Lugano, Switzerland. 1999.
[17] B. Golden and A. Assad. "Vehicle routing with time-window constraints". American Journal of Mathematical and Management Sciences. Vol. 6 N° 3-4, pp. 251-260. 1986.
[18] B.L. Golden, E.A. Wasil. J.P. Kelly and LM. Chao. "Metaheuristics in vehicle routing". In: CrainicT.G.. Lapote C. (Eds). Fleet Management and Logistics, pp. 33-56. Kiuwer, Boston, MA. 1998.
[19] N. Kohl, J. Desrosiers, O.B.G. Madsen. M.M. Solomon and F. Soumis. "2-path cuts for the vehicle routing problem with time windows". Transportation Science. Vol. 33, pp. 101-116, 1999.
[20] G. Laporte. "The traveling salesman problem: an overview of exact and approximate algorithms". European Journal of Operational Research. Vol. 59, pp. 231-247, 1992.
[21] M. Montemanni, L.M. Gambardella, A.E. Rizzoli. A.V Donati. "Ant colony system for a dynamic vehicle routing problem". Journal of Combinatorial Optimization. Vol. 10. pp. 327-343. 2005.
[22] P. Shaw. "Using constraint programming and local search methods to solve vehicle routing". Lecture Notes in Computer Science. Vol. 1520. pp. 417431. 1998.
[23] M.M. Solomon. "Algorithms for the vehicle routing and scheduling problem with time windows constraints". Operations Research. Vol. 35, pp. 254-365, 1987.
[24] M.M. Solomon and J. Desrosiers. "Time window constrained routing and scheduling problems". Transportation science. Vol. 22 N° I, pp. 1-13. 1988.
[25] P. Toth and D. Vigo. "The Vehicle Routing Problem". Society for Industrial and Applied Mathematic. Siam Monographs on Discrete Mathematics and Applications. 2001.
Pablo Ortega1
Cristian Oliva2
Jacques Ferland3
Manuel Cepeda2
Recibido 28 de febrero de 2008. aceptado 10 de septiembre de 2009
Received: February 28. 200H Accepted: September /0. 2009
1 Departamento de Ingeniería Industrial. Facultad de Ingeniería. Universidad de Concepción. Concepción. Chile. E-mail: [email protected]
2 Departamento de Ingeniería Industrial. Facultad de Ingeniería. Universidad Católica de la Santísima Concepción. Concepción. Chile. Corresponding author. E-mail: [email protected]; [email protected]
3 Département d'informatique et de recherche opérationnel le. Faculté des arts el des sciences. Université de Montreal. Francia. E-mail: [email protected]
Copyright Universidad de Tarapacá Sep-Dec 2009