A Abstract – In recent years, VRP

A Review on
Metaheuristic Algorithm in Multi Depot Vehicle Routing Problem


Mohd Shahizan Othman *,a, sherylaidah binti samsuddinb, Lizawati Mi Yusuf c

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of Computing, Universiti Teknologi Malaysia, 81310, UTM Johor Bahru, Johor,



a,*[email protected], b [email protected], c [email protected],

Abstract – In recent years, VRP has attracted high
attention in real life problem. VRP has many variants and MDVRP is a part of
it. MDVRP is arises with rapid development in logistic and transportation.
Hence, to solve the problem, there is a need to apply metaheuristic in order to
get a better result. This reviewed paper has discussed about Metaheuristic and
hybrid algorithm in MDVRP. Based on the previous work, it showed that mostly
hybrid algorithm in metaheuristic algorithm did help in give good solution for


Keywords: Metaheuristic, Multi depot, VRP, MDVRP




In recent years, Logistics and transportation cost became
rising and critical issue in all the fields. The most challenging strategy is
to optimize product distribution from suppliers to users following satisfying
constraints. Goods transportation processed has undergone drastic increased
because of increasing in community of users and growing number of on-line
shopping stores. VRP is one of the most
studied problem in field of logistic and transportation (Allahyari, Salari, & Vigo, 2015)


VRP is evolution from the classic
Traveling Salesman Problem (TSP). In TSP, The number of vehicle used to handle
a group of customer with the optimal design of routes can be one or more than
that. The difference between TSP and VRP is the TSP aim to find a shortest path
where the vehicle must visit and return to the city where it started whereas
VRP aim to minimize the total cost for serving customers. Nonetheless, Laporte
(2007) highlighted that the VRP is practically more challenging to solve
compare to a TSP (Annouch, Bouyahyaoui, Bellabdaoui, & Mohammed,


The VRP was first presented by Dantzig and Ramser in 1959 (Calvete et al, 2004). The VRP plays an important
role in reducing the costs of transportation in logistic distribution and
considered as one of the most important combinatorial optimization problems.
Based on definition about VRP, we can conclude that there are four elements or
components that involve in VRP which are vehicle, customer, depot and goods (Dondo & Cerdá, 2007; Goksal, Karaoglan, &
Altiparmak, 2013; Student & Sharma, 2015). Generally, in distribution
and pickup of goods, VRP will determined a minimum-cost set of routes.


Many variants of VRP are studied to address the
variety of conditions in real world applications such as the capacitated VRP
(CVRP), the VRP with time windows (VRPTW), the heterogeneous fleet VRP (HVRP),
the VRP with pickup and delivery (VRPPD), and multiple depots VRP (MDVRP) (Goksal et al., 2013). The MDVRP is said to be more
demanding and sophisticated compared to VRP. MDVRP is the extended version of
VRP where more than one depot are used. The variants of multi-depots known to
be NP-hard problems because it have many constraints which consume a high
computational time to find optimal solutions for large problems.

There are three solution methods to solved problems
in MDVRP which are exact, heuristics and metaheuristic method. This paper
provided a review about the metaheuristic algorithms for solving multi-depot
vehicle routing problem (MDVRP) and hybrid algorithm in MDVRP.



The VRP and its variants is said to be NP-hard problem. MDVRP is one of
the variant where it is one of the most widely used variations in VRP problem (Karakati? & Podgorelec, 2015) and main optimization
problems in the logistics field. Compared with VRP that has single-depot, MDVRP
is said to be more practical and challenging in real life problem because it
involved more than one depot. It is a challenging task for decision makers to
find out which user are served by which depots without exceeding the capacity
constraints since there are a more depots involved (Calvet, Ferrer, Gomes, Juan, & Masip, 2016). Same like VRP, MDVRP also
have many variants. Among others are time windows, heteregenous fleet,
capacitated, periodic, pick-up and delivery, split delivery and etc.

Time Windows – 
Related to every customer where it define a time interval in which the
customer should be provided (Calvete et al., 2004).

Heteregenous Fleet –  To minimize the transportation cost, the
number and the type of vehicle used must be determined and which customer must
be serviced by which vehicle also what sequence to follow should be specified (Dondo & Cerdá, 2007).

Capacitated – The capacity of each vehicle is

Periodic – each depot has different period (Crainic et al, 2012).

Pick-up and Delivery –  Every location have goods for pickup and
delivery along with multiple delivery locations that have possibility not the
depots (Kachitvichyanukul et al, 2015).

Split Delivery – Within the set of customer
nodes, the optimal location of the depot specified  (Ray. et al.,















Figure 1 : Different variants of VRP
and MDVRP (Montoya et al, 2015)

Basically, MDVRP aims to minimize the
total delivery distance or time spent in serving all customers. To get the
customer satisfaction, the delivery time should be not take time too long.
Besides that, MDVRP also aims to minimize the number of vehicles needed. If the
number of vehicle reduced, then it also reduced the total operation cost. No
matter which type of objectives is defined, the ultimate goal of the MDVRP is
to increase the efficiency of the delivery.


Sequencing each
route in every depots

Appoint customer
in each depot to routes

Appoint customer
to depot





1. The hierarchy of decision in the MDVRP


To solved MDVRP, three types of possible solution applied
which are exact method, heuristic and metaheuristic method. Exact method
introduced based on branch and cut and only can solve a small instances problem
(Mingozzi, Giorgi, & Baldacci, 1999). It only can solve instances
with 25 up to 50 customers. Example of exact method are Branch-and-cut
algorithms, branch and price algorithm, column generation and integer
programming (Azi, Gendreau, & Potvin, 2010; Ma et al., 2017). Reeves stated that heuristic
is a technique that seek near-optimal solutions at a reasonable computational
cost but it does not guarantee optimality 15. The heuristic method is more
flexible and can solved more instances than the exact method (Pop et al, 2011). Example of heuristic method
are Nearest Neighbour and a Clarke-Wright based heuristic (El-Sherbeny, 2010). For metaheuristic, it is
effective techniques applicable to a problems involving large instances.
Metaheuristic and heuristics can solve larger problem but metaheuristic give
more deep search of the objective space compare to traditional heuristics.
During search process, it allow fewer and even unreasonable intermediate
solutions. That is why it rarely stranded in local optimum 18. Example of
metaheuristic are Ant Colony Optimization (ACO), Genetic Algorithm (GA), Simulated
Annealing (SA) and Tabu search (TS) etc. (Montoya-Torres et al., 2015).




Nowadays, most of the MDVRP problem is solved by using three approach
which are exact methods, heuristic and metaheuristic methods. What
distinguishes between these three methods is the former guarantee the
optimality of the solution found, while the closing usually provide a
high-quality solution faster. For exact method, usually it applied in solving
small instances problem such as work on (Contardo & Martinelli, 2014b) where it involved only 23
instances (Contardo & Martinelli, 2014a) and work on Baldacci et al,
(2009) solving for 20 up to 50 instances (Baldacci & Mingozzi, 2009). Moreover, since MDVRPs are
NP hard, exact methods are not suitable to get optimal solutions. That is why
heuristic algorithms have been selected to solve the MDVRPs at a faster rate
thus giving computationally effective solutions. However, metaheuristic method
have received high attention among researchers (Montoya-Torres et al., 2015) and it can solve more larger
instances than heuristic and exact method. That is why many researchers applied
metaheuristic algorithm for solving MDVRP.

3.1 Metaheuristics algorithm


A repetitive generation process that leads a subordinate
heuristic by bringing together various concept for exploiting and exploring the
search space is defined as metaheuristic (Osman & Laporte, 1996). They are constructed
for optimization problem in which the optimization methods and classical
heuristic failed to be efficient and effective (Osman & Laporte, 1996). In higher-level frameworks,
fundamental heuristic will be integrate in metaheuristic. User have a broad and
continually growing number of metaheuristics at their disposal. There are four
componnets in metaheuristic which are initial space solution, search engines,
learning and guidance method and management of information structures (Osman, 2001). Almost all
metaheuristics share the following characteristics; they are nature inspired,
they make use of stochastic components, they do not use the gradient or Hessian
matrix of the objective function, they have several parameters that need to be
fitted to the problem at hand.


There are two basic components in any
metaheuristic algorithm which are diversification and intensification. This two
components are very important because the behaviour metaheuristic is determined
by them (Blum & Roli, 2003). Diversification is to produce various solution to make it
able to explore the search space on the global scale, Whereas intensification
is to give focal point on the search in a local area by exploiting the
information that are current better solution establish in this area. This is an
integration with the collection of the better solution. The collection of the
best guarantee that the solution will converge to the optimality. Furthermore,
the diversification through randomisation prevent the solution being trapped at
local optima, at the same time the diversity of the solution will increases.
Usually, the global optimality is feasible through good integration of these
two elements (Yang, 2011).

advantages of metaheuristis are metaheuristics give decision
makers with powerful tools that acquire great features solutions, in a
reasonable computational effort (Osman, 2001).


Table 1 shows variants
in MDVRP and its solution method using metaheuristic algorithm which have been
used by other researchers in solving NP-hard combinatorial problems. From Table 1 below, it shows that most
widely studied variant is time windows. It can be concluded that time window is
one of the challenging and practical problem in logistic management.


Furthermore, from table 1, it also shows that
mostly researchers choose to combined metaheuristics algorithm with other
metaheuristic algorithm or another word known as hybrid metaheuristic method in
order to solved multi-depot vehicle routing problem. It shows that hybrid
metaheuristic algorithms strategy in solving multi depot vehicle routing
problem has been used many times as it can figure out high dimensional problems
quickly in order to get a better solution.


Table 1 : Variants in MDVRP and
its solution method using metaheuristic algorithm




(G. Wang & Lin, 2017)

Hybrid mosquito-host seeking

(Shimizu, Sakaguchi, & Yoo, 2016)

MD VRP  + simultaneous pickup and delivery

Weber basis saving method +
modified TB

(Shimizu et al., 2016)


Hybrid method

(Li, Li, & Pardalos, 2016)


Hybrid GA

(Yalian, 2016)

Improved ACO

(Kachitvichyanukul et al., 2015)


PSO + multiple social learning

(Xu, Jiang, & Branch, 2014)


Improved variable neighbourhood
search algorithm

(Contardo & Martinelli, 2014b)

Exact algorithm

(Liu & Yu, 2013)


(T. J. Wang & Wu, 2013)

Cloud adaptive PSO

(Ala?a & Dridi, 2013)



(Dharmapriya, Siyambalapitiya, & Kulatunga,



(Vidal, Crainic, Gendreau, Lahrichi, & Rei,


Hybrid GA

(Zhen & Zhang, 2009)


Hybrid algorithm

(Wen & Meng, 2008)


Improved PSO


classification of metaheuristics


Metaheuristic approaches can be classified into several
condition regarding how memory is exploited and the search path followed by


a)      Trajectory
methods vs discontinuous method

In different metaheuristics, one of
important thing is whether they pursue one single search trajectory corresponding to a closed walk on the neighborhood graph or
whether larger jumps in the neighborhood graph are allowed. Local search algorithms that
carry out more complex transition which are consist of simpler moves also known
as trajectory method. Usually, these method allow not efficient solutions to be
able to escape from local minima


b)      Population
based vs single point search

The main point that differentiate
trajectory methods and discontinuous walk method is whether population of
search points or one single search point is used. In second case, at each
iteration of algorithm, only one single solution is employed. to
develop new solution that are predicted to give good fitness, Population-based
meta-heuristic methods will associate a number of solutions in order to achieve
it but at the same time, the good benefits from the old one is still shared. It
is a repetitive procedures that continuously take over solutions with better
ones. (Roeva et al.
2014). The
advantage of population-based algorithm, it give an efficient way for the
exploration of the search space. Yet, the manipulation of population will give
a best final performance.


c)      Memory
usage vs. memoryless method

Future search direction very influenced
by the use of search experience (memory, in the widest sense). Usually, short
term memory is used to block revisiting newly found solutions and to prevent
cycling, whereas long term memory is used for intensification and
diversification features
A markov process is performed by memory less algorithm in the process to
complete next process where it is the present state of the search process.


One vs. various neighbourhood

Local search algorithm mostly is a one
single neighborhood structures that describes the type of allowed moves. Certain
initial solution is started by local search algorithm and the process happens
repeatedly to change the current solution to be a good solution. It is known as
neighborhood of the current solution.


e)      Dynamic
vs. static objective function

algorithms customized the evaluation of the single search states when the
algorithm is running. so far, a static objective function is used in introduced


f)       Nature-inspired
vs. non-nature inspiration

of metaheuristics algorithm is inspired by naturally from phenomena  and these algorithm have been created by
mimicking the most fortunate processes in nature, involving chemical and
physical process, and biological systems (Yang, 2011). Several example are particle swarm
optimization, ant colony optimization, bee colony optimization and firefly
algorithm. From these phenomena, advantages can be taken for algorithmic
approaches to be used as efficient solution of combinatorial optimization


3.3 hybridization method in


Metaheuristics is known as a powerful tool to solve any
hard optimisation problems In order to get a better and successful solutions,
various method are hybridized (Piotrowski
& Napiorkowski, 2018). However, to improved
metaheuristic algorithm for a given problem and to make it more powerful, metaheuristic
hybrid optimisation techniques should be applied. With this combination, there
is advantages that can be taken form both algorithms (Beheshti,
Hejazi, & Mirmohammadi, 2014). 


Table 2 shows several hybrid algorithm that has been used by researchers
in solving problem in multi depot. It can been seen that genetic algorithm and
local search method mostly used for hybrid algorithm in metaheuristics. It
shows that genetic algorithm and local search is much better for hybridization
process with any other metaheuristic algorithms.


Table 2 : Hybrid algorithm  in MDVRP




(G. Wang & Lin, 2017)


+ 3-opt local optimization

(Shimizu et al., 2016)

Simultaneous pick up and

modified TB + Weber basis
saving method

(Li et al., 2016)


GA + Adaptive local search

(Allahyari et al., 2015)

Covering tour

GRASP + iterated local search +

(Chaichiratikul, 2013)

Multi depot pick up and

Memetic algorithm

(Vidal et al., 2012)


GA  + Adaptive diversity control

(Dharmapriya et al., 2012)

TW and split delivery


(Zhen & Zhang, 2009)


ACO + local search


Time window: TW; Mosquito host seeking: MHS; tabu search:
TB; genetic algorithm: GA; simulated annealing: SA; ant colony algorithm: ACO;




This paper
is a review about metaheuristic algorithm in MDVRP. The paper has discussed
about MDVRP, solution method in MDVRP, metaheuristic algorithm, classification
of metaheuristic and hybridization method in metaheuristic. From this review
paper, it can be conclude that, metaheuristic algorithm is a better method to
be used in order for solving MDVRP, but, from this review paper also proved
that, hybrid metaheuristic algorithm is more better for solving MDVRP.









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