A Literature Review On The Vehicle Routing Problem With Multiple Depepots PDF

Computers & Industrial Engineering 79 (2015) 115129
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Computers & Industrial Engineering

journal homepage: www.elsevier.com/locate/caie
Survey
A literature review on the vehicle routing problem with multiple depots

Jairo R. Montoya-Torres a,, Julin Lpez Franco b, Santiago Nieto Isaza c, Heriberto Felizzola Jimnez d,
Nilson Herazo-Padilla e,f
a
Escuela Internacional de Ciencias Econmicas y Administrativas, Universidad de La Sabana, km 7 autopista norte de Bogot D.C., Chia (Cundinamarca), Colombia
b
Engineering and Consulting SAS, Bogot, D.C., Colombia
c
Departamento de Ingeniera Industrial, Universidad del Norte, Km 5 va Puerto Colombia, Barranquilla (Atlntico), Colombia
d
Departamento de Ingeniera Industrial, Universidad de La Salle, Carrera 2 # 10-70, Bogot, D.C., Colombia
e
Departamento de Ingeniera Industrial, Universidad de la Costa, Calle 58 # 55-66, Barranquilla, Colombia
f
Fundacin Centro de Investigacin en Modelacin Empresarial del Caribe FCIMEC, Carrera 53 # 74-86 Oc.402, Barranquilla, Colombia
a r t i c l e i n f o a b s t r a c t
Article history: In this paper, we present a state-of-the-art survey on the vehicle routing problem with multiple depots
Received 3 August 2012 (MDVRP). Our review considered papers published between 1988 and 2014, in which several variants of
Received in revised form 28 October 2014 the model are studied: time windows, split delivery, heterogeneous eet, periodic deliveries, and pickup
Accepted 31 October 2014
and delivery. The review also classies the approaches according to the single or multiple objectives that
Available online 10 November 2014
are optimized. Some lines for further research are presented as well.
2014 Elsevier Ltd. All rights reserved.
Keywords:
Vehicle routing
Multiple depots
Exact algorithms
Heuristics
Survey
1. Introduction Gutirrez-Franco, & Halabi, 2009; Ozrat & Ozkarahan, 2010;

Thangiah & Salhi, 2001). Solving the VRP is vital in the design of
Physical distribution is one of the key functions in logistics distribution systems in supply chain management.
systems, involving the ow of products from manufacturing plants
or distribution centers through the transportation network to
consumers. It is a very costly function, especially for the distribu- 1.1. VRP versus MDVRP
tion industries. The Operational Research literature has addressed
this problem by calling it as the vehicle routing problem (VRP). The Formally, the classical vehicle routing problem (VRP) is repre-
VRP is a generic name referring to a class of combinatorial optimi- sented by a directed graph G(E,V), where V = {0,1, . . ., n} represents
zation problems in which customers are to be served by a number the set of nodes and E is the set of arcs. The depot is noted to be
of vehicles. The vehicles leave the depot, serve customers in the node j = 0, and clients are nodes j = 1, 2, . . ., n, each one with
network and return to the depot after completion of their routes. demand dj > 0. Each arc represents a route from node i to node j.
Each customer is described by a certain demand. This problem The weight of each arc Cij > 0 corresponds to the cost (time or even
was rstly proposed in the literature by Dantzig and Ramser distance) of going from node i to node j. If Cij = Cji then we are
(1959). After then, considerable number of variants has been facing the symmetric VRP, otherwise the problem is asymmetric.
considered: hard, soft and fuzzy service time windows, maximum From the complexity point of view, the classical VRP is known to
route length, pickup and delivery, backhauls, etc. (Cordeau, NP-hard since it generalizes the Travelling Salesman Problem
Gendreau, Hertz, Laporte, & Sormany, 2005; Juan, Fauln, (TSP) and the Bin Packing Problem (BPP) which are both
Adelanteado, Grasman, & Montoya Torres, 2009; Lpez-Castro & well-known NP-hard problems (Garey & Johnson, 1979). A review
Montoya-Torres, 2011; Montoya-Torres, Alfonso-Lizarazo, of mathematical formulations for the classical VRP can be found in
the work of Laporte (1992).
Corresponding author. In the literature, lots of surveys have been presented analyzing
E-mail addresses: jairo.montoya@unisabana.edu.co (J.R. Montoya-Torres), julian.
published works on either the classical version (Bodin, 1975; Bodin
lopez@eic-sas.com (J. Lpez Franco), nietos@uninorte.edu.co (S. Nieto Isaza), & Golden, 1981; Desrochers, Lenstra, & Savelsbergh, 1990;
healfelizzola@unisalle.edu.co (H. Felizzola Jimnez). Eksioglu, Volkan, & Reisman, 2009; Laporte, 1992; Liong, Wan,
http://dx.doi.org/10.1016/j.cie.2014.10.029
0360-8352/ 2014 Elsevier Ltd. All rights reserved.
116 J.R. Montoya-Torres et al. / Computers & Industrial Engineering 79 (2015) 115129
Khairuddin, & Mourad, 2008; Mafoli, 2002) or its different vari-

ants: the capacitated VRP (Baldacci, Toth, & Vigo, 2010; Cordeau,
Laporte, Savelsbergh, & Vigo, 2007; Gendreau, Laporte, & Potvin,
2002; Laporte & Nobert, 1987; Laporte & Semet, 2002; Toth &
Vigo, 2002), the VRP with heterogeneous eet of vehicles
(Baldacci, Battarra, & Vigo, 2008; Baldacci, Toth, & Vigo, 2007;
Baldacci et al., 2010), VRP with time windows (VRPTW), pickup
and deliveries and periodic VRP (Solomon & Desrosiers, 1988),
dynamic VRP (DVRP) (Psaraftis, 1995), Periodic VRP (PVRP)
(Mourgaya & Vanderbeck, 2006), VRP with multiple trips (VPRMT)
(S
en & Blbl, 2008), Split Delivery vehicle routing problem
(SDVRP) (Archetti & Speranza, 2008). All of these works consider
only one depot. Fig. 1 presents a hierarchy of VRP variants. One
of these variants considers a well-known (Crevier, Cordeau, &
Laporte, 2007) more realistic situation in which the distributions
of goods is done from several depots to nal clients. This particular
distribution network can be solved as multiple individual single
depot VRPs, if and only if clients are evidently clustered around
each depot; otherwise a multi-depot-based approach has to be
used where clients are to be served from any of the depots using
the available eet of vehicles. In this paper, we consider the variant Fig. 2. Comparison between VRP versus MDVRP.
of the vehicle routing problem known as Multiple Depots Vehicle
Routing Problem (MDVRP) in which more than one depot is consid-
ered (see Fig. 2). The reader must note that most exact algorithms demand of each route does not exceed the vehicle capacity, and
for solving the classical VRP model are difcult to be adapted for (iv) the total cost of the distribution is minimized. According to
solving the MDVRP. Kulkarni and Bhave (1985), a mathematical model of the MDVRP
According to Renaud, Laporte, and Boctor (1996), the MDVRP requires the denition of binary decision variables xijk to be equal
can be formally described as follows. Let G = (V,E) be a graph, to 1 if the pair of nodes i and j are in the route of vehicle k, and
where V is the set of nodes and E is the set of arcs or edges 0 otherwise. Auxiliary variables yi are required in order to avoid
connecting each pair of nodes. The set V is further partitioned into subtour elimination. According to this last reference, the model is
two subsets: Vc = {v1, v2, . . . , vN} which is the set of customers to be as follows:
served; and Vd = {vN+1, vN+2, . . . , vM} which is the set of depots. Each
customer vi 2 Vc has a nonnegative demand di. Each arc belong to
X NM
NM XXK
the set E has associated a cost, distance or travel time cij. There Minimize cij xijk 1
are a total of K vehicles, each one with capacity Pk. The problem i1 j1 k1
consists on determining a set of vehicle routes in such a way that: Subject to :
(i) each vehicle route starts and ends at the same depot, (ii) each
XX
NM K
customer is serviced exactly once by a vehicle, (iii) the total xijk 1 j 1; .. .; N 2
i1 k1
XX
NM K
xijk 1 i 1;. .. ;N 3
j1 k1
periodic Multiple X
N M X
N M
k 1;. .. ;K
VRP depots xihk xhjk 0 4
i1 j1
h 1; .. .; N M
constraints
distance or time
X
NM X
NM
capacity
constraints time windows Qi xijk 6 Pk k 1; .. .; K 5

i1 j1
X NM
NM X
cij xijk 6 T k k 1; .. .; K 6
i1 j1
capacity & time of
X
NM X
N
with loading
& unloading
distance constraints
backhaul xijk 6 1 k 1; .. .; K 7
iN1 j1
X
NM X
N
xijk 6 1 k 1; .. .; K 8
jN1 i1
yi yj M Nxijk 6 N M 1 For 1 6 i j 6 N and 1 6 k 6 K 9

depot is source
& destination
xijk 2 f0; 1g 8i; j;k 10

In this formulation, Constraints (2) and (3) ensure that each
customer is served by one and only one vehicle. Route continuity
is represented by Constraints (4). The sets of constraints (5) and
(6) are the vehicle capacity and total route cost constraints. Vehicle
availability is veried by Constraints (7) and (8) and subtour elim-
ination is provided by Constraints (9). In this formulation, it is
Fig. 1. Different variants of the VRP (adapted from Weise et al., 2010). assumed that total demand at each node is either less than or at
J.R. Montoya-Torres et al. / Computers & Industrial Engineering 79 (2015) 115129 117
the most equal to the capacity of each vehicle. Similar to single short-listed in this review (see Tables A1 and A2 in Appendix).
depot VRP, the subtour elimination constraint (9) can be rewritten The short-listed publications are then examined in more detail.
in a more compact form as: The analysis of methodological issues and problem variants are
presented in detail in Section 4 of the paper.
X
V
yi yj M N xijk 6 M N 1 for i 6 ij 6 M N 1
k1 1.4. Paper organization
11
The rest of this paper is organized as follows. Section 2 presents
the review of papers studying the single-objective problem, while
1.2. Motivation for a review of research papers on MDVRP those focused on multiple objectives are reviewed in Section 3. In
both sections, papers are briey commented and most important
The MDVRP is more challenging and sophisticated than the sin- facts are highlighted. The literature is quantied and measured in
gle-depot VRP. The variant with multiple depots appears rst in Section 4. The paper ends in Section 5 by presenting some conclud-
the literature on the works of Kulkarni and Bhave (1985), and ing remarks and discussing some research opportunities.
Laporte, Nobert, and Taillefer (1988) and Carpaneto, Dellamico,
Fischetti, and Toth (1989). Since then, considerable amount of
2. Single objective MDVRP
research has been published (see Table 1) in the form of journal
paper, conference paper, research/technical report, thesis or book.
This section presents a literature review of papers considering
To the best of our knowledge, despite the great amount of research
the single objective case of the MDVRP. For the total of papers con-
papers published, there is no rigorous literature survey exclusively
sidered in this review, 88.44% considers only one optimization
devoted to the vehicle routing problem with multiple depots. A
objective. The list of reviewed papers is presented in Table A1 in
short overview of academic works was proposed by Liu, Jiang,
the Appendix. This section is divided into three main parts, each
Liu, and Liu (2011), but only presenting the most representative
one corresponding to the type of solution procedure employed:
research papers. From the 58 cited references in their paper, only
exact method, heuristic or meta-heuristic. After the rst works
23 of them explicitly refer to the MDVRP. Besides, these authors
were published in the decade of 1980, more than one hundred of
focus on the problem denition, solution methods (dividing them
papers have studied the classical version of the MDVRP and its
into exact algorithms, heuristics and meta-heuristics) and mention
variants, some of them inspired from real-life applications.
some problem variants. In fact, no actual systematic review was
presented in that paper. Our intention now is to present a rigorous
review of scientic literature, by presenting a taxonomic classica- 2.1. Exact methods
tion of those works. Most of the published works focus on the sin-
gle objective problem, while a few consider the multi-objective In the decade of the 1970s, some works already mentioned
case. In this paper, we intend to provide an analysis of both single some problems related to distribution of goods with multiple
and multiple objective problems. depots. However, to the best of our knowledge, the rst paper pre-
senting formal models or procedures to nd optimal solution for
the multi-depot vehicle routing problem are those of Laporte,
1.3. Review methodology
Nobert, and Arpin (1894) who formulated the symmetric MDVRP
as integer linear programs with three constraints. These authors
This paper presents a review of relevant literature on the
then proposed a branch-and-bound algorithm using a LP relaxa-
vehicle routing problem with multiple depots, with both single
tion. The works of Kulkarni and Bhave (1985), Laporte et al.
and multiple objective functions. An ambitious search was con-
(1988) and Carpaneto, Dellamico, Fischetti, and Toth (1989) can
ducted using the library databases covering most of the major jour-
also be considered as part of the pioneer works on exact methods
nals. Some conference papers are also included in this review. In
for the MDVRP. The mathematical formulation proposed by
addition, the websites of leading research groups and the principal
Kulkarni and Bhave (1985) was later revised by Laporte (1989).
authors of major publications were also searched for further infor-
More recently, Baldacci and Mingozzi (2009) proposed mathemat-
mation about their research projects (Ph.D. projects and sponsored
ical formulations for solving several classes of vehicle routing
projects) and publications. We intentionally excluded working
problems including the MDVRP, while Nieto Isaza, Lpez Franco,
papers, theses and research reports not available online on the
and Herazo Padilla (2012) presented an integer linear program
Internet because they are very difcult to obtain.
for solving the heterogeneous eet MDVRP with time windows.
The initial collection of references was screened rst for their
Dondo, Mendez, and Cerd (2003) proposed a mixed-integer linear
relevance and their signicance for the purpose of this review. In
programming (MILP) model to minimize routing cost in the
order to control the length of this paper, only some representative
HFMDVRP, in which heterogeneous eet of vehicles are available.
publications were selected to be explained in detail within the text
The variant with pickup and deliveries and heterogeneous eet
of this manuscript, which are authored by leading researchers or
was modeled by Dondo, Mndez, and Cerd (2008) using MILP
groups. These selected authors and research groups have, in fact,
model: this model is able to solve only small sized instances, and
published a long list of research papers and reports in the eld. A
hence a local search improvement algorithm was then proposed
collection of a total of 147 representative publications are
by the authors for medium to large sized instances. This approach
Table 1 was later employed by Dondo and Cerd (2009) to solve the
Number and types of publications on MDVRP. HFMDVRPTW. The work of Kek, Cheu, and Meng (2008) proposes
Type of publication Total a mixed-integer linear programming model and a branch-and-
Journal 125
bound procedure for the MDVRP with xed eet and pickup and
Conference 28 delivery. The objective function is the minimization of the total
Thesis 7 cost of routes. Cornillier, Boctor, and Renaud (2012) presented a
Technical report 4 MILP model for the problem in which heterogeneous eet of
Book/book chapter 9
vehicles is available and with maximization of total net revenue
Total 173
as objective function, while maximum and minimum demands
constraints are given. Seth, Klabjan, and Ferreira (2013) studied an experimental results showed that the one-stage algorithm outper-
application case from the printed circuit board production process forms the other one.
modeled as a MDVRP with mobile depots. They presented an exact The HFMDVRP, in which heterogeneous eet of vehicles is
algorithm based on network-ow formulation and proposed a iter- available have captured the attention of researchers since the work
ated tour partitioning heuristic. Branch-and-cut algorithms were presented by Salhi and Sari (1997). Irnich (2000) proposed a set
proposed by Benavent and Martnez (2013) and Braekers, Caris, covering heuristic coupled with column generation and branch-
and Jenssens (2014). Former authors focused also on studying and-price algorithm for cost minimization for the heterogeneous
the polyhedral structure of the problem which allowed the exten- eet and pickup and delivery MDVRP. Dondo and Cerd (2007)
sion of their procedure to the location-routing problem (LRP), proposed a MILP model as well as a three-stage heuristic. Before,
while latter authors considered a dial-a-ride problem with multi- a preprocessing stage for node clustering is performed and a more
ple depots. The LRP with multiple depots was also studied by compact cluster-based MILP problem formulation is developed.
Contardo, Cordeau, and Gendron (2014) who proposed a cut-and- Many other papers have been appeared in literature on or before
column generation procedure for the capacitated case. Other 2007 and solution approaches have primarily been focused on
congurations of the MDVRP have been studied through exact meta-heuristic algorithms. Hence, this will be discussed more in
algorithms. For instance, Contardo and Martinelli (2014) studied detail in the next subsection.
the capacitated MDVRP with route length constraints; Ray, Concerning the periodic MDVRP, few works appear in literature
Soeanu, Berger, and Debbabi (in press) modeled the version with with heuristic algorithm as solution approach. We have identied
split deliveries; Li, Li, and Pardalos (in press) proposed an integer only the works of Hadjiconstantinou and Baldacci (1998), Vianna,
programming model for the MDVRP with shared depots and time Ochi, and Drummond (1999), Yang and Chu (2000), Maischberger
windows; and Adelzadeh, Asi, and Koosha (in press) proposed a and Cordeau (2011), and Maya, Srensen, and Goos (2012). The
mathematical model and a fuzzy-based heuristic for the particular problem with time windows was studied by Chiu, Lee, and Chang
case of the MDVRP with fuzzy time windows and heterogeneous (2006) who presented a two-phase heuristic method. In contrast
eet of vehicles. with other works in literature, these authors considered the wait-
ing time as objective function. Results indicate that the waiting
time has a signicant impact on the total distribution time and
2.2. Heuristics the number of vehicles used when solving test problems with nar-
row time windows. The authors also considered a real-life case
Because the NP-hardness of the MDVRP, several heuristic study of a logistics company in Taiwan.
algorithms have been proposed in the literature. This section sum- Tsirimpas, Tatarakis, Minis, and Kyriakidis (2007) considered
marizes some of the most relevant works concerning different vari- the case of a single vehicle with limited capacity, multiple-prod-
ants of the problem. The rst works were published in the 1990s, ucts and multiple depot returns. Another characteristic of their
in order to solve the capacitated version. Min, Current, and problem is that the sequence of visits to customer is predened.
Schilling (1992) studied the version of the MDVRP with backhaul- They developed a suitable dynamic programming algorithm for
ing and proposed a heuristic procedure based on problem the determination of the optimal routing policy. For the MDSDVRP
decomposition. Hadjiconstantinou and Baldacci (1998) considered which consists on the combination of the MDVRP and the Split
a real-life problem taken from a company supplying maintenance Delivery VRP (SDVRP). The work of Gulczynski, Golden, and
services. Their problem consists on determining the boundaries Wasil (2011) developed an integer programming-based heuristic.
of the geographic areas served by each depot, the list of customers The objective of this study was to determine the reduction in trav-
visited each day and the routes followed by the gangs. The objec- eled distance that can be achieved by allowing split deliveries
tive is to provide improved customer service at minimum operat- among vehicles based at the same depot and vehicles based at dif-
ing cost subject to constraints on frequency of visits, service time ferent depots. The multi-depot capacitated vehicle routing prob-
requirements, customer preferences for visiting on particular days lem with split delivery (MDCVRPSD) is studied by Liu, Jiang,
and other routing constraints. This situation was solved using the Fung, Chen, and Liu (2010). A mathematical model is proposed
periodic variant: MDPVRP for which a ve-level heuristic was pro- based on a graph model. The objective function is the minimization
posed: rst and second levels solve the problem of determining the of movements of empty vehicle. A greedy algorithm is proposed as
service areas and service days (the periodic VRP); third level solves well, in order to solve large-scale instances. More recent works on
the VRP for each day; fourth level solves a TSP for each route, and the application of dedicated heuristics include the work of Vahed,
fth level seeks the optimization of routes. Crainic, Gendreau, and Rei (in press) for the case of a MDVRP with
Salhi and Sari (1997) proposed the so-called multi-level com- the objective of determining the optimal vehicle eet size, and the
posite heuristic. This heuristic found as good solutions as those work of Afshar-Nadja and Afshar-Nadja (in press) for the study
known at that time in the literature but using only 5 to 10% of their of the time-dependent MDVRP with heterogeneous vehicles and
computing time. The heuristic was also tested on the problem with time windows.
heterogeneous eet. Salhi and Nagy (1999) proposed an insertion-
based heuristic in order to minimize routing cost. Later, these 2.3. Meta-heuristics
authors (Nagy & Salhi, 2005) also studied the problem with pickup
and deliveries (MDVRPPD). Their approach avoids the concept of As for other NP-hard combinatorial optimization problems,
insertion and proposes a method that treats pickups and deliveries meta-heuristic procedures have been employed by several
in an integrated manner. The procedure rst nds a solution to the researchers for efciently solving the single-objective MDVRP.
VRP, then it modies this solution to make it feasible for the VRPPD The rst meta-heuristic was proposed in the work of Renaud,
and it nally ensures that it is also feasible for the multi-depot Laporte et al. (1996) who study the MDVRP with the constraints
case. Jin, Guo, Wang, and Lim (2004) modeled the MDVRP as a of vehicle capacities and maximum duration of routes (e.g. the
binary programming problem. Two solving methodologies were time of a route cannot exceed the maximum working time of the
presented. The rst one is a two-stage approach that decomposes vehicle). The objective to be optimized is the total operational cost.
and solves the problem into two independent subproblems: These authors proposed a Tabu Search algorithm for which the ini-
assignment and routing. The second proposed approach treats both tial solution is built using the Improved Petal heuristic of Renaud,
assignment and routing problems in an integrated manner. Their Boctor, and Laporte (1996b). Experiments were carried out using
classical instances of Christodes and Eilon (1969) and Gillett and reverse logistics problem in which the aim is to collect cores from
Johnson (1976). Later, Cordeau, Gendreau, and Laporte (1997) an enterprises dealers. The problem is called the selective MDVRP
proposed a Tabu Search algorithm with the initial solution being with price. In addition to the Tabu Search procedure, the authors
generated randomly for the MDVRP that can also be used to solve also formulated two mixed-integer linear programming (MILP)
the periodic VRP (PVRP), while Tzn and Burke (1999) proposed a models.
Tabu Search procedure for minimizing the total cost of the routing. Other meta-heuristics, such as GRASP, are presented in the
Cordeau, Laporte, and Mercier (2001) also proposed a TS procedure works of Villegas, Prins, Prodhon, Medaglia, and Velasco (2010)
with the objective of minimizing the number of vehicles. An and Maya et al. (2012), respectively minimizing route cost and
approximation to real industrial practice was studied by distance. zyurt and Aksen (2007) solved the problem of depot
Parthanadee and Logendran (2006). In their problem, depots location and vehicle routing using a hybrid approach based on
operate independently and solely within their own territories. Lagrangian relaxation (LR) and Tabu Search (TS). These procedures
The distributors may hence satisfy customers requests by deliver- improve the best solutions found for the set of instances proposed
ing products from other depots that hold more supplies. They by Tzn and Burke (1999). A case study taken from waste collec-
proposed a mixed-integer linear programming model for the tion system involving 202 localities in the city of Viseu, Portugal, is
multi-product, multi-depot periodic distribution problem and pre- presented by Matos and Oliveira (2004). An Ant Colony Optimiza-
sented three Tabu Search heuristics with different long-term mem- tion (ACO) algorithm is proposed and compared with other proce-
ory applications. Results revealed that interdependent operations dures from the literature.
provide signicant savings in costs over independent operations The great amount of heuristics algorithms proposed for the
among depots, especially for large-size problems. More recently, problem variant with heterogeneous eet (HFMDVRP) has been
Escobar, Linfati, Toth, and Baldoquin (in press) evaluated a Granu- focused on the design of meta-heuristics algorithms. We can
lar Tabu Search algorithm to minimize the total sum of vehicle highlight the works of Jeon, Leep, and Shim (2007), who proposed
traveled distances. a hybrid genetic algorithm that minimizes the total distance
The rst genetic algorithms were proposed by Filipec, Skrlec, traveled, and that of Flisberg, Lidn, and Rnnqvist (2009) who
and Krajcar (1997) for the problem of minimizing total travel dis- considered a Tabu Search procedure. Simulated Annealing (SA)
tance, by Salhi, Thangiah, and Rahman (1998) and by Skok, Skrlec, has been employed as well. Wu et al. (2002) coupled SA with Tabu
and Krajcar (2000), Skok, Skrlec, and Krajcar (2001). An evolution- Search to solve the heterogeneous eet case of the integrated loca-
ary algorithm coupled with local search heuristic was proposed by tion-routing problem. In their problem, location of depots, routes
Vianna et al. (1999) in order to minimize the total cost. Thangiah of vehicles and client assignment problems are solved simulta-
and Salhi (2001) proposed the use of genetic algorithms to dene neously. The multi-depot heterogeneous vehicle routing problem
clusters of clients and then routes are found by solving a traveling with time windows (MDHVRPTW) was studied by Dondo and
salesman problem (TSP) using and insertion heuristic. This Cerd (2009), who proposed a MILP and a Local Search Improve-
approach is called Genetic Clustering (GenClust). Solutions are ment Algorithm that explores the neighborhood in order to nd
nally optimized using the post-optimization procedure of Salhi the lowest cost feasible solution.
and Sari (1997). Recently, Ycenur and Demirel (2011a) proposed Other research papers have also been very interested in the
geometric shape based genetic clustering algorithm for the classi- analysis of the problem with time windows (MDVRPTW). This
cal MDVRP. The procedure is compared with the nearest neighbor variant is studied in 25% of the single-objective focused papers
algorithm. Their experiments showed that their algorithm provides considered in this review. The rst meta-heuristics reported in lit-
a better clustering performance in terms of the distance of each erature was the Tabu Search procedure of Cordeau et al. (2001) in
customer to each depot in clusters, in a considerably less computa- which the objective function is the minimization of the number of
tional time. vehicles. Polacek, Hartl, Doerner, and Reimann (2005) proposed a
In the survey by Gendreau, Potvin, Brumlaysy, Hasle, and Variable Neighborhood Search (VNS) algorithm for the MDVRP
Lkketangen (2008) focused on the application of meta-heuristics with time windows and with xed distribution of vehicles. This
for solving various variants of the VRP, a short revision of the problem was also studied by Lim and Wang (2005) with the char-
multi-depot problem is presented. The equivalence between the acteristic of having exactly one vehicle at each depot. Jin, Mitsuo,
MDVRP and the PVRP is also analyzed. Among the meta-heuristics and Hiroshi (2005), Yang (2008), Ghoseiri and Ghannadpour
presented therein, we can highlight the use of Genetic Algorithms (2010) and Samanta and Jha (2011) proposed Genetic Algorithms,
(Filipec, Skrlec, & Krajcar, 2000), Simulated Annealing (Lim & Zhu, Pisinger and Ropke (2007) presented an Adaptive Large Neighbor-
2006) of the case of xed vehicle eet, and Tabu Search (Angelelli hood Search (ALNS) procedure with minimization of routing cost.
and Speranza, 2002). Other works proposing meta-heuristics can Ting and Chen (2009) presented a hybrid algorithm that combines
be found in (Chao, Golden, & Wasil, 1993 and Chen, Takama, & multiple ant colony systems (ACS) and Simulated Annealing (SA).
Hirota, 2000). Zarandi, Hemmati, and Davari (2011) presented a SA procedure
The most studied variant of the problem has been the capaci- to minimize routing cost, while Wang, Zhang, and Wang (2011)
tated MDVRP. Among the meta-heuristics proposed in literature, coupled SA with a modied Variable Neighborhood Search algo-
we can highlight the Simulated Annealing algorithms of Wu, rithm, and a clustering algorithm is used to allocate customers in
Low, and Bai (2002) and Lim and Zhu (2006), the Variable Neigh- the initial solution construction phase. A branch-and-cut-and-
borhood Search procedure proposed by Polacek, Hartl, Doerner, price algorithm for the multi-depot heterogeneous vehicle routing
and Reimann (2005), Polacek, Benkner, Doerner, and Hartl problem with time windows (MDHVRPTW) was recently proposed
(2008), Tabu Search algorithms from Lim and Wang (2005), Aras, by Bettinelli, Ceselli, and Righini (2011). Computational experi-
Aksen, and Tekin (2011) and Maischberger and Cordeau (2011). ments showed that this procedure is competitive in comparison
Genetic Algorithms has been proposed as well for this problem with local search heuristics.
variant, as illustrated in the works of Bae, Hwang, Cho, and Goan The variants with split delivery (MDVRPSD) or with pickup &
(2007), Vidal, Crainic, Gendreau, Lahrichi, and Rei (2010). All of delivery (MDVRPPD) have been considered by very few authors
these works seek for the minimization of total route distance or in the scientic literature. The work of Wasner and Zapfel (2004)
cost, except the work of Aras et al. (2011) in which the objective presents an application to postal, parcel and piece goods service
is the maximization of vehicle utilization rate. It is to note here that provider in Austria. The model employed is the MDVRPPD (MDVRP
the work of Aras et al. (2011) is inspired by a particular case of with pickup and deliveries) with the objective of determining the
number and location of depots and hubs. Also, the client assign- combines the exploration breadth of population-based evolution-
ment problem is addressed. These authors develop a local search ary search, the aggressive improvement capabilities of neighbor-
heuristic. As a real-life problem is solved, additional features are hood search based procedures and advanced population diversity
included in the algorithm in order to take into account the topog- management strategies. These authors improved the best-known
raphy of the country (which is characterized by mountains) by solutions and even obtained optimal values for these three problem
considering maximum route length. The decision support system cases. Recent years on MDVRP research have witnessed the use of
allowed the solution of large-sized instances with various millions variable neighborhood search (VNS) algorithm for the resolution
of variables and constraints. The paper of Pisinger and Ropke of various variants of the problem (Kuo & Wang, 2012; Salhi,
(2007) studied the MDVRPPD, together with the variants of time Imran, & Wassan, 2014; Xu & Jiang, 2014; Xu, Wang, & Yang,
windows and vehicle capacity constraint. These authors proposed 2012). A realistic application of MDVRP found in vessel routing
an Adaptive Large Neighborhood Search procedure in order to was studied by Hirsch, Schroeder, Maggiar, and Dolinskaya
minimize total routing cost. Flisberg, Lidn, and Rnnqvist (2009) (2014). These authors proposed the implementation of various heu-
also considered heterogeneous eet of vehicles and time windows ristics, including GRASP (Greedy Randomized Adaptive Search Pro-
constraints, in addition to pickups and split deliveries: their case- cedure). More recently, the trend has been focused on the use of
study is taken from a forest company in Sweden. Schmid, hybrid meta-heuristics algorithms. Liu and Yu (2013) presented a
Doerner, Hartl, and Salazar-Gonzlez (2010) studied a realistic case hybridized genetic algorithm ant colony optimization procedure
inspired from companies in the concrete industry, and presented a to minimize the maximum travel distance of a vehicle in a system
mixed integer linear program (MILP) and a Variable Neighborhood with heterogeneous eet of vehicles. Liu (2013) proposed to couple
Search (VNS) procedure to minimize routing cost for the variant the genetic algorithm with bee colony optimization and simulated
with split deliveries. Mirabi, Fatemi Ghomi, and Jolai (2010) annealing to solve the classical MDVRP. Rahimi-Vahed, Crainic,
addressed the problem of minimizing the delivery time of vehicles. Gendreau, and Rei (2013) employed path relinking for the case of
They compared three hybrid heuristics, each one combining capacitated MDVRP with route duration constraint. Vidal, Crainic,
elements from both constructive heuristic search and improve- Gendreau, and Prins (2014) proposed a hybrid genetic algorithm
ment techniques. The improvement techniques are deterministic, with iterated local search and dynamic programming was pre-
stochastic and simulated annealing (SA) methods. sented for the classical MDVRP with unconstrained vehicle eet.
Crevier et al. (2007) considered a MDVRP in which there are Subramanian, Uchoa, and Ochi (2013) proposed a matheuristic pro-
intermediate depots along vehicles routes where they may be cedure for the cases of the MDVRP and MDRVP with mixed pick up
replenished. This problem was inspired from a real-life application and deliveries. Their algorithm is based on iterated local search and
at the city of Montreal, Canada. A heuristic combining adaptive exploits set partitioning models at certain stages of the procedure
memory, tabu search and integer programming was proposed. to obtain competitive solutions. Sitek, Wikarek, and Grzybowska
The model allows the assignment of vehicles to routes that may (2014) presented a multi-agent system coupled with a mixed-inte-
begin and nish at the same depot or that connect two depots to ger linear programming (MILP) model and Constraint Programming
increase the capacity of vehicles to deliver goods. Zhen and (CP) for the multi-echelon capacitated vehicle routing problem.
Zhang (2009) considered a similar problem and proposed a heuris-
tic combining the adaptive memory principle, a Tabu Search
method for the solution of subproblems, and integer programming. 3. The MDVRP with multiple objectives
Another variant of the MDVRP appears in the work of Zarandi et al.
(2011). These authors studied the fuzzy version of the Capacitated An important characteristic of real-life logistics problems found
Location-Routing Problem (CLRP) with multiple depots in which in enterprises is that decision-makers, very often, have to simulta-
the location of depots have to be dened as well as the routes of neously deal with multiple objectives. These objectives are some-
vehicles. Fuzzy travel times between nodes and time window to times contradictory (e.g., minimizing number of vehicles and
meet the demand of each customer are considered. A simulation- maximizing service level). In the literature, there are very few
embedded Simulated Annealing (SA) procedure was proposed. papers on the MDVRP that consider multiple objectives (MOM-
The framework was tested using standard data sets. DVRP): about 11.56% of the papers reviewed here. This percentage
A good manner of improving the performance of meta-heuris- corresponds to a total of 17 papers reviewed here (see Table A2 in
tics is to generate good initial solutions. Ho, Ho, Ji, and Lau the Appendix).
(2008) proposed the use of the well-known Clarke & Wright The work of Lin and Kwok (2005) studies a realistic particular
Savings (C&WS) algorithm (Clarke & Wright, 1964) to generate case of the MDVRP, named as location-routing problem (LRP) with
initial solutions, as commonly used for other vehicle routing prob- multiple uses of vehicles. In this problem, decisions on location of
lems (Juan et al., 2009). Once the solution is generated, the Nearest depots, vehicle routing and assignment of routes to vehicles are
Neighbor (NN) heuristic is employed to improve such solution. In considered simultaneously. Tabu search and simulated annealing
comparison with the random generation of initial solutions, their procedures are designed and tested on both random-generated
experiments showed that this hybrid C&WS + NN approach pro- and real data. The objectives are the minimization of total opera-
duces better results regarding total delivery time. Li and Liu tional cost and the balance on vehicle load. Both simultaneous
(2008) considered the multi-depot open vehicle routing problem and sequential procedures for the assignment of routes to vehicles
with replenishment during the execution of routes. They proposed are tested. Results show that the simultaneous versions have
a model and an Ant Colony Optimization resolution procedure. advantage over the sequential versions in problems where routes
Other application of the Ant Colony Optimization paradigm can are capacity-constrained, but not in the time dimension. The
be found in the works of Wang (2013) and Narasimha, simultaneous versions are also more effective in generating non-
Kivelevitch, Sharma, and Kumar (2013). These last authors studied dominated solutions than the sequential versions.
the MDVRP with minimization of the longest travel distance of a A real-life transportation problem consisting on moving empty
vehicle. or laden containers is studied by Tan, Chew, and Lee (2006). They
Vidal et al. (2010, 2011) proposed a general framework to solve a called the problem as the truck and trailer vehicle routing problem
family of vehicle routing problems, including the multi-depot VRP, (TTVRP), but in fact it corresponds to a variant of the MDVRP: the
the periodic VRP and the multi-depot periodic VRP with capacitated solution to the TTVRP consists of nding a complete routing
vehicles and constrained route duration. Their meta-heuristic schedule for serving the jobs with minimum routing distance
160
140
120
100
Number of
80 papers
60 52 51
40
22
20 13
2 4 3
0
1980-1985 1986-1990 1991-1995 1996-2000 2001-2005 2006-2010 2011-2014
Fig. 3. Distribution of published papers per year for the MDVRP.
Fig. 4. Distribution of objective functions.
well-known algorithms non-dominated sorting genetic algorithms

2 (NSGA2), strength Pareto evolutionary algorithm 2 (SPEA2) with
and without fuzzy logic and the micro-genetic algorithm (MIC-
ROGA) with and without fuzzy logic. Experiments showed that
the proposed FL-NSGA2 procedure outperformed the other
procedures. This technique was also used by Lau, Chan, Tsui, and
Pang (2010) to solve the problem in which the cost due to the total
traveling distance and the cost due to the total traveling time are
minimized. In their work, several search methods, branch-and-
bound, standard GA (i.e., without the guide of fuzzy logic),
simulated annealing, and tabu search procedure are adopted to
Fig. 5. Distribution of employed solution techniques. compare with FLGA in randomly generated data sets. Results of
their experiments show that FLGA outperforms the other methods.
Ombuki-Berman and Hanshar (2009) and Weise, Podlich, and
and number of trucks, subject to time windows and availability of Gorldt (2010) also proposed a genetic algorithm. The rst authors
trailers. These authors solved the multi-objective case using a considered the objectives of minimizing the total cost and the
hybrid multi-objective evolutionary algorithm (HMOEA) with number of vehicles, while the latter authors considered the total
specialized genetic operators, variable-length representation and distance, the idle capacity of vehicles and the number of
local search heuristic. Lau, Tang, Ho, and Chan (2009) considered externalized deliveries. A Simulated Annealing (SA) procedure
the multi-objective problem in which the travel time is not a con- was presented by Hasanpour, Mosadegh-Khah, and Tavakoli
straint but an objective function to be optimized in the model Moghadam (2009) for minimizing transportation costs and
together with the total traveled distance. The proposed solution maximizing probability of delivery to customers.
procedure is a hybrid meta-heuristic named fuzzy logic guided Weise, Podlich, and Gorldt (2010) presented the use of
non-dominated sorting genetic algorithm (FL-NSGA2). The proce- evolutionary computation for a real-life problem inspired from a
dure uses fuzzy logic to dynamically adjust the probabilities of joint enterprise-academia research project. Results of the imple-
mutation and crossover. The algorithm is compared with the mentation are compared against the traditional routing structure
Table A1
122
Reviewed papers about the single-objective MDVRP.
Year Reference Constraints Solution method

Time Windows Heterogeneous eet Capacitated Periodic Pick up & Delievery Split delivery Other Exact Heuristic Meta-heuristic
% of papers 26% 21% 38% 12% 11% 5% 26% 31% 45% 57%
1984 Laporte et al. (1894) X
1985 Kulkarni and Bhave (1985) X X
1987 Laporte and Nobert (1987) X
1988 Laporte et al. (1988) X X
1989 Laporte (1989) X
Carpaneto, Dellamico, Fischetti, and Toth (1989) X X
1992 Min et al. (1992) X X X
Wilger and Maurer (1992) X
1993 Chao et al. (1993) X
1996 Renaud, Laporte et al. (1996) X X X
J.R. Montoya-Torres et al. / Computers & Industrial Engineering 79 (2015) 115129

1997 Filipec et al. (1997) X X
Salhi and Sari (1997) X X
Cordeau et al. (1997) X X
1998 Salhi et al. (1998) X
Hadjiconstantinou and Baldacci (1998) X X
1999 Vianna et al. (1999) X X X
Tzn and Burke (1999) X X
Salhi and Nagy (1999) X X
2000 Irnich (2000) X X X X
Filipec et al. (2000) X X X
Skok, Skrlec, and Krajcar (2000) X X X
Yang and Chu (2000) X X
2001 Thangiah and Salhi (2001) X X
Skok, Skrlec, and Krajcar (2001) X X X
Cordeau et al. (2001) X X X
Chan, Carter, and Burnes (2001) X X
2002 Wu et al. (2002) X X X
Angelelli and Speranza (2002) X X X
Zhang, Jiang, and Tang (2002) X X X
Giosa, Tansini, and Viera (2002) X X
2003 Dondo et al. (2003) X X X
Kazaz and Altinkemer (2003) X X X
2004 Matos and Oliveira (2004) X X
Wasner and Zapfel (2004) X X
Jin et al. (2004) X
2005 Mingozzi (2005) X X
Nagy and Salhi (2005) X X
Polacek et al. (2005) X X X
Lim and Wang (2005) X X X
Baltz, Dubhashi, Tansini, Srivastav, and Werth (2005) X
Songyan, Akio, and Bai (2005) X
Jin et al. (2005) X X
Songyan and Akio (2005) X
2006 Parthanadee and Logendran (2006) X X
Yang, Cui, and Cheng (2006) X X
Chiu et al. (2006) X X
Lim and Zhu (2006) X X X X
2007 Dondo and Cerd (2007) X X X X
Jeon et al. (2007) X X
Crevier et al. (2007) X X
Bae, Hwang, Cho, and Goan (2007) X X X
Pisinger and Ropke (2007) X X X X
zyurt and Aksen (2007) X X
Carlsson, Ge, Subramaniam, Wu, and Ye (2007) X
Hu, Chen, and Guo (2007) X X
Wang, Gao, Cui, and Chen (2007) X
Lou (2007) X X X
Tsirimpas et al. (2007) X X X
2008 Ho et al. (2008) X X
Kek et al. (2008) X X X
Dondo et al. (2008) X X X X X
Polacek, Benkner, Doerner, and Hartl (2008) X X X
Dai, Chen, Pan, and Hu (2008) X X
Li and Liu (2008) X
Chen, Guo, and Fan (2008) X X
Yang (2008) X X X
Chen, Sheng-Zhang, and Shi (2008) X X X
Goela and Gruhn (2008) X X X X X X X
2009 Flisberg et al. (2009) X X X X X X
J.R. Montoya-Torres et al. / Computers & Industrial Engineering 79 (2015) 115129

Dondo and Cerd (2009) X X X X
Baldacci and Mingozzi (2009) X
Ting and Chen (2009) X X
Zhen and Zhang (2009) X X X
2010 Schmid et al. (2010) X X X X
Mirabi et al. (2010) X X
Liu et al. (2010) X X X
Vidal et al. (2010) X X X X
Ma and Yuan (2010) X
Sombunthama and Kachitvichyanukulb (2010) X X X
Sepehri and Kargari (2010) X X
Gajpal and Abad (2010) X X
Villegas et al. (2010) X X X X
Garaix, Artigues, Feillet, and Josselin (2010) X X X X
Ghoseiri and Ghannadpour (2010) X X X X
Kansou and Yassine (2010) X X
2011 Gulczynski et al. (2011) X X X
Bettinelli et al. (2011) X X X X X
Ycenur and Demirel (2011a) X
Aras et al. (2011) X X
Zarandi et al. (2011) X X X
Yang, Zhou, Cui, and He (2011) X X X
Wang et al. (2011) X X X X
Samanta and Jha (2011) X X X
Ycenur and Demirel (2011b) X X
Yu, Yang, and Xie (2011) X
Lei, Xing, Wu, and Wen (2011) X
Fard and Setak (2011) X X X
Zhang, Tang, and Fung (2011) X X
Surekha and Sumathi (2011) X X
Maischberger and Cordeau (2011) X X X X X X
2012 Cornillier et al. (2012) X X X X X
Kuo and Wang (2012) X X X
Lpez Franco and Nieto Isaza (2012) X X
Maya et al. (2012) X X X X
Nieto Isaza et al. (2012) X X X
Xu et al. (2012) X X X X
2013 Ben Alaia, Dridi, Bouchriha, and Borne (2013) X X X
Benavent and Martnez (2013) X
Liu (2013) X X X
Liu and Yu (2013) X X X X
Venkata et al. (2013) X X X
123
(continued on next page)
employed by the enterprises associated with the research. The

Meta-heuristic
multi-objective MDVRP with time windows and split delivery is

studied by Dharmapriya and Siyambalapitiya (2010). The objec-
tives to be optimized were dened to be the total transportation
cost, the total distance traveled, full use of vehicle capacity and
X
X
X
X
X
X
X
X
X
load balancing. The problem is solved using Tabu Search, Simu-
lated Annealing and a Greedy algorithm. Tavakkoli-Moghaddam,
Heuristic
Solution method
Makui, and Mazloomi (2010) studied the multi-objective problem

in which depot location and routes of vehicles have to be dened
X
X
X
X
X
X
simultaneously. This problem is known in the literature as the
Exact
Multi-Depot Location-Routing Problem. Traditionally, this problem

X
X
X
X
X is solved sequentially: rst, the location of depots is addressed and

then the routing of vehicles is approached. These authors proposed
Other
a scatter search algorithm that seeks to maximize total demand

X
X
X
X
X
X
X
X
X
served and to minimize the total operational cost (cost of opening
depots and variable delivery costs). Computational experiments
Split delivery
showed that the proposed multi-objective scatter search (MOSS)

algorithm outperformed an Elite Tabu Search (ETS) procedure.
Scatter Search operates on a set of solutions, the reference set, by
X
combining these solutions to create new ones. Then, a subset is

Pick up & Delievery
non-randomly selected from this reference set and selected

solutions are combined to get starting solutions to run an improve-
ment mechanism. The reader interested in more details of the
multi-objective of Scatter Search can refer to the works of
Beausoleil (2001), Beausoleil (2004, Beausoleil (2006). On the other
hand the ETS algorithm considered by Tavakkoli-Moghaddam et al.
X
X
(2010) rst determines the minimum number of required facilities

Periodic
and then evaluates all combinations of facilities by incorporating

an elite tabu search procedure at the routing phase.
X
X
X
Jiang and Ding (2010) minimized the distribution cost, the

customer dissatisfaction and the changes of routes in a disruption
Capacitated
measurement model and an immune algorithm. The procedure is

tested using a simple example. Ghoseiri and Ghannadpour (2010)
considered the problem of simultaneously optimizing total eet
X
X
X
X
X
X
X
X
X
X
X
X
size and total distance deviation by using a genetic algorithm

Heterogeneous eet
coupled with goal programming. Finally, Venkatasubbaiah,

Acharyulu, and Chandra Mouli (2011) proposed the use of a Fuzzy
Goal Programming Method to solve the multi-objective problem. A
fuzzy max-min operator is also proposed to improve the effective-
ness of the procedure. The algorithm is tested on simple transpor-
X
X
X
X
X
X
tation problems from literature, and compared with previous

works, while Li and Liu (2011) proposed a genetic algorithm so
Time Windows
as to minimize the number of vehicles and the total travel distance.

Constraints
Adelzadeh et al. (in press) proposed a mathematical model and a

fuzzy-based heuristic for the particular case of the MDVRP with
fuzzy time windows and heterogeneous eet of vehicles. Mart-
X
X
X
X
nez-Salazar et al. (2014) studied the location-routing problem

(LRP) with transportation considerations in order to minimize
transport cost and load balancing. The bi-objective problem is for-
malized using mathematical programming and then solved with
Afshar-Nadja and Afshar-Nadja (in press)
both a Scatter Tabu Search procedure and the well-known NSGA-

II (Non-dominated Sorting Genetic Algorithm II). Mirabi (2014)
proposed a hybrid meta-heuristic algorithm coupling simulated
annealing and electromagnetism to minimize travel distance and
Contardo and Martinelli (2014)
customer waiting time for service. Finally, Rodrigues Pereira

Rahimi-Vahed et al. (2013)
Adelzadeh et al. (in press)

Subramanian et al. (2012)
Ramos, Gomes, and Barbosa-Pvoa (2014) simultaneously evalu-

Escobar et al. (in press)
Vahed et al. (in press)

Contardo et al. (2014)
ated travel distance and level of carbon emissions in a multi-depot

Braekers et al. (2014)
Luo and Chen (2014)
Xu and Jiang (2014)

Ray et al. (in press)
Hirsch et al. (2014)
vehicle routing problem found in waste collection supply chain.

Vidal et al. (2014)
Li, et al. (in press)
Sitek et al. (2014)

Salhi et al. (2014)
Seth et al. (2013)
Wang (2013)
Reference
Table A1 (continued)
4. Analysis of literature
As pointed out before, since the rst publication on multi-depot

vehicle routing problem appeared in 1984. Since there, more than
2014
Year
145 papers have been published up to date in the scientic

literature about this problem and its variants. Between the middle
of the 1980s (when the rst works on the MDVRP were published)
Meta-heuristic and the end of the twentieth century, very few papers were pro-
posed in the literature (see Fig. 3): only 18 publications (excluding
those published in 2000), giving an average of 1.12 papers per year.
82%
Between 2000 and 2005, there was an increase in the number of

X
X
X
X
X
X
X
X
X
X
X
X
X
X
publication on MDVRP with an average of 4.33 publications per
year. This gives a total of 44 publications until 2005 inclusive.
Heuristic
The most impressive growing on the number of papers published

Solution method
35%
is observed between 2006 and 2014 with a total 103 publications,

X
X
X
X
X
giving an average of 10.4 papers per year in the period 20062010
and 12.75 paper per year in the period 20112014 inclusive.
Exact
12%
As can be seen, there is a clearly increasing trend showing the

X
X
growing interest in this eld. It is reasonable to expect that in
the coming years the MDVRP will receive an ever large amount
Other
41%
of attention. There are however some remarks to be made. As

X
X
X
X
shown in Fig. 4, most of the works have been focused on the min-
imization of cost, distance or time. The papers dealing with vehicle
Split delivery
load balancing are in fact papers that seek to optimize multiple

objectives simultaneously (very often cost and vehicle load). As
presented in previous sections of this review (see also Fig. 4), the
6%
majority of published works deals with the single objective

problem. While this problem is of theoretical interest, very often
Pick up & Delievery
decision-makers are faced to optimize multiple (contradictory)

objectives. Very few published works deals with multi-objective
problem. Hence, this gap in current research could be closed by
proposing efcient and effective solution approaches for multi-
objective environments.
12%
It is also interesting to study the different methodologies and

techniques that the authors apply in the reviewed literature.
Periodic
Fig. 5 shows two pie charts with this distribution: it rst classies
12%
the approaches as exact, heuristics and meta-heuristics algorithms,

X
while the second pie presents a distribution of the different

approximate algorithms employed in the reviewed literature. We
Capacitated
rst observe that exact algorithms (branch and bound, mathemat-

ical programming) are employed in 25% of reviewed papers. We
41%
have to consider that these techniques have proven to be useful

X
X
X
X
X
for simplied combinatorial optimization problems, specic

settings and/or small instances. Hence, a larger focus is needed
Heterogeneous eet
on approaches able to solve larger instances. Because of the NP-

hardness of the MDVRP, approximate heuristics have also been
proposed. Under the category of heuristic algorithms, we have
classied many different algorithms and ad-hoc methods that are
12%
specic and do not contain a well-known meta-heuristic template.

X
This represents 33% of the reviewed papers. For the other 42% a
meta-heuristic algorithm is proposed. Among the available proce-
Time windows
dures, Tabu Search (TS), genetic algorithms (GA) and simulated

Constraints
annealing (SA) have been the most employed in the reviewed

literature. The other meta-heuristics have been less employed:
29%
ant colony optimization and variable neighborhood search. Clearly,

X
X
X
there is a large opportunity for research here. Meta-heuristics have

Dharmapriya and Siyambalapitiya (2010)
long ago established themselves as state-of-the-art methodologies

Reviewed papers about the multi-objective MDVRP.
Rodrigues Pereira Ramos et al. (2014)
for the vast majority of vehicle routing problems.

Ombuki-Berman and Hanshar (2009)
Tavakkoli-Moghaddam et al. (2010)
Ghoseiri and Ghannadpour (2010)

Venkatasubbaiah et al. (2011)
Martnez-Salazar et al. (2014)
5. Conclusions and lines for future research

Adelzadeh et al. (in press)
Hasanpour et al. (2009)
Jiang and Ding (2010)

Lin and Kwok (2005)
Finding an optimal (or at least very good) route for vehicles

Weise et al. (2010)
Li and Liu (2011)
delivering products to customers is one of the key functions in

Tan et al. (2006)
Lau et al. (2009)
Lau et al. (2010)
Mirabi (2014)
any logistics systems. Since the rst works proposed in literature

% of papers
in the late 1959, the literature on vehicle routing problems has

Reference
been highly increasing. A lot of works, as well as state-of-the-art

surveys, have been published. In this paper, we focused on one of
the less reviewed variants: the vehicle routing problem with multi-
ple depots (MDVRP). We presented a complete analysis of scientic
Table A2
2011
2014
2005
2006
2009
2010
Year
literature since the rst works about this problem were published
in the mid 1980s.
Early works on MDVRP mainly proposed exact algorithms References

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Computers & Industrial Engineering 79 (2015) 115129

Contents lists available at ScienceDirect

Computers & Industrial Engineering

A literature review on the vehicle routing problem with multiple depots

1. Introduction Gutirrez-Franco, & Halabi, 2009; Ozrat & Ozkarahan, 2010;

Khairuddin, & Mourad, 2008; Mafoli, 2002) or its different vari-

constraints time windows Qi xijk 6 Pk k 1; .. .; K 5

yi yj M Nxijk 6 N M 1 For 1 6 i j 6 N and 1 6 k 6 K 9

xijk 2 f0; 1g 8i; j;k 10

Fig. 3. Distribution of published papers per year for the MDVRP.

Fig. 4. Distribution of objective functions.

well-known algorithms non-dominated sorting genetic algorithms

Year Reference Constraints Solution method

J.R. Montoya-Torres et al. / Computers & Industrial Engineering 79 (2015) 115129

J.R. Montoya-Torres et al. / Computers & Industrial Engineering 79 (2015) 115129

employed by the enterprises associated with the research. The

multi-objective MDVRP with time windows and split delivery is

Makui, and Mazloomi (2010) studied the multi-objective problem

Multi-Depot Location-Routing Problem. Traditionally, this problem

X is solved sequentially: rst, the location of depots is addressed and

a scatter search algorithm that seeks to maximize total demand

showed that the proposed multi-objective scatter search (MOSS)

combining these solutions to create new ones. Then, a subset is

non-randomly selected from this reference set and selected

(2010) rst determines the minimum number of required facilities

and then evaluates all combinations of facilities by incorporating

Jiang and Ding (2010) minimized the distribution cost, the

measurement model and an immune algorithm. The procedure is

size and total distance deviation by using a genetic algorithm

coupled with goal programming. Finally, Venkatasubbaiah,

tation problems from literature, and compared with previous

as to minimize the number of vehicles and the total travel distance.

Adelzadeh et al. (in press) proposed a mathematical model and a

nez-Salazar et al. (2014) studied the location-routing problem

both a Scatter Tabu Search procedure and the well-known NSGA-

customer waiting time for service. Finally, Rodrigues Pereira

Adelzadeh et al. (in press)

Ramos, Gomes, and Barbosa-Pvoa (2014) simultaneously evalu-

Vahed et al. (in press)

ated travel distance and level of carbon emissions in a multi-depot

Luo and Chen (2014)

Xu and Jiang (2014)

vehicle routing problem found in waste collection supply chain.

Sitek et al. (2014)

As pointed out before, since the rst publication on multi-depot

145 papers have been published up to date in the scientic

Between 2000 and 2005, there was an increase in the number of

The most impressive growing on the number of papers published

is observed between 2006 and 2014 with a total 103 publications,

As can be seen, there is a clearly increasing trend showing the

of attention. There are however some remarks to be made. As

load balancing are in fact papers that seek to optimize multiple

majority of published works deals with the single objective

decision-makers are faced to optimize multiple (contradictory)

It is also interesting to study the different methodologies and

the approaches as exact, heuristics and meta-heuristics algorithms,

while the second pie presents a distribution of the different

rst observe that exact algorithms (branch and bound, mathemat-