International Journal of Industrial Engineering, 22(6), 330-341, 2015
AN EFFECTIVE RANK BASED ANT SYSTEM ALGORITHM FOR SOLVING
THE BALANCED VEHICLE ROUTING PROBLEM
MajidYousefikhoshbakht, FarzadDidehvar*, FarhadRahmati
Department of Mathematics and Computer Science
Amirkabir University of Technology,
Tehran, Iran
*
Corresponding author’s e-mail: [email protected]
The vehicle routing problem (VRP) is the problem of designing optimal delivery from a given depot in order to satisfy the
customer population demand by a similar fleet of vehicles. It is noted that a considerable part of the drivers’ benefits is related
to their traveled distance; therefore, the balance of the route based on 'vehicles travelled distance' is important to obtain
drivers’ satisfaction. This paper presents a balance, based on the vehicles traveled route called balanced vehicle routing
problem (BVRP) and then, a model integer linear programming is proposed for solving the BVRP. Because this problem
belongs to NP-hard problems, an effective rank based ant system (ERAS) algorithm is proposed in this paper. In addition, a
number of test problems involving 10 to 199 customers have been considered and solved to show the efficiency of the
proposed ERAS. The computational results show that the proposed algorithm results are better than the results of classical
rank based ant system (RAS) and exact algorithm for solving the BVRP within a comparatively shorter time period.
Keywords:balanced vehicle routing problem; meta-heuristic; rank based ant system; NP-hard
(Received on September2, 2013; Accepted on May 14, 2015)
1. INTRODUCTION
The Capacitated Vehicle Routing Problem (CVRP) is one of the most important problems arising in designing the efficient
logistics networks. These networks are designed to facilitate the transfer of goods from distribution centers such as
warehouses or factories to a set of geographically dispersed customers. In addition, the VRP is an important problem of
Operational Research both from practical and theoretical points of view in which a customer’s network needs to be serviced
by one or more depots, to supply a set of customers (Yousefikhoshbakht, 2014). In this problem, all the customers correspond
to deliveries, the demands are deterministic, the vehicles are identical and are based at a single central depot, the travel cost
between each pair of customer locations is the same in both directions, i.e., the resulting cost matrix is symmetric, whereas in
some applications, as the distribution in urban areas with one-way directions imposed on the roads, the cost matrix is
asymmetric. Only the capacity restrictions for the vehicles are imposed, and the objective is to minimize the total cost (i.e., the
number of routes, their length or travel time) needed to serve all customers. The CVRP problem involves routing a fleet of
vehicles, each visiting a set of customers such that every customer is visited exactly once and exactly by one vehicle, with the
objective of minimizing the total distance traveled by all vehicles. Furthermore, if the total demand of all customers assigned
to the same vehicle does not exceed the capacity limit in the CVRP, then the feasibility of the route would always be hold, no
matter what the visiting sequence is.
The CVRP has been extensively studied since the early sixties and many new exact, heuristic and metaheuristic
approaches were presented in the past years. The VRP was first defined by Dantzig and Ramsermore than 50 years ago
(Dantzig et al., and Ramser, 1959) and it is considered as one of the most well-known combinatorial optimization tasks.
Different approaches for solving the CVRP have been explored during the past decades (Golden, et al., 2008). These
approaches range from the use of exact optimization methods for solving small-size problems with relatively simple
constraints to the use of approximation algorithms that provides near-optimal solutions for medium and large-size problems
with more complex constraints. Initially, in the late 1960s, several exact algorithms such as dynamic programming relaxation
approaches (Hadjiconstantinou et al., 1995) as well as branch and cut methods (Naddef et al., 2002) were developed for very
small numbers of variables and constraints. The largest problems which can be consistently solved by the most effective exact
algorithms proposed so far contain about up to 50 customers and one method that solved a 100-customer problem (Golden et
al., 1998), whereas larger instances may be solved only in particular cases. So instances with hundreds of customers, as those
arising in practical applications,mayonly be tackled by heuristic methods. Recently, several authors have been able to extend
ISSN 1943-670X
INTERNATIONAL JOURNAL OF INDUSTRIAL ENGINEERING
Yousefikhoshbakhtet al.
Balanced Vehicle Routing Problem
some formulations and lower bounds to a set of different variants of the VRP (Baldacci et al., 2009). As far as we know, this
is a first attempt at a more general exact framework for routing problems. For example, Baldacci et al. (2008) proposed an
exact algorithm based on branch and cut in which the authors are able to scope decent levels of performance for exact
methods, where exact solutions to the problems of up to 100 customers are found. In addition, the computational times are
great in some cases and adding new real constraints is a challenge in such specialized exact methodologies.
Since the exact algorithms could not perform well for large instances of the problem, mainly due to the computational
complexity, many researchers focused on approximation classes of algorithms. These algorithms searched for near optimal
solutions at a reasonable computational cost, however, without any guarantees of optimality. These techniques are broadly
classified into Classical heuristics and metaheuristics. These groups of algorithms for the VRP are naturally divided into
constructive heuristics, improvement heuristics and two-phase heuristics. The constructive heuristics start from a null
solution and the arcs are selected serially until a feasible solution is achieved. The most known of these algorithms is the
saving algorithm presented by Clarke and Wright (1964) and sweep algorithm by Gillet–Miller (1974). Improvement
algorithms try to find an improved solution from a poorer solution which is usually generated by a constructive heuristic.
Nearly all improvements for algorithms replace a set of arcs to obtain a better solution. The 2-opt, 3-opt and Lin–Kernigham
heuristics are the most important heuristics that belong to the heuristics category of route improvement (Christofides et al.,
1979). In the two phase heuristics, known as cluster first-route second heuristics, the customers are first allocated to vehicles
and then a route is created from every cluster. In this category they, also, belong to the methods called route first-cluster
second in which first a giant traveling salesman problem (TSP) tour is constructed and then this route is decomposed into
feasible vehicle routes.
The VRP problem is a NP-hard problem and difficult to solve the large size of problems by exact methods within
acceptable computing times. Furthermore, solutions found by heuristic algorithms have a large disparity compared to the best
known or near optimal solutions. However, many recent studies have been focused on using advanced meta-heuristic
techniques to produce high quality solutions for larger problems, which unfortunately are very hard computationally for exact
algorithms (guaranteeing optimality). Recently many metaheuristic algorithms have been considered by researchers and
scientists for solving VRPs such as Tabu Search (TS) by Toth and Vigo (2003) , genetic algorithms by Baker and Ayechew
(2003), record to record travel by Li et al. (2005)(Li et al., 2005), honey bees mating optimization by Marinakis et al. (2008),
threshold accepting algorithms by Tarantilis et al. (2003) and Adaptive Memory based algorithms by Tarantilis et al. (2002).
Most of these methods focus on minimizing an aprioristic cost function subject to a set of well-defined constraints.
However, real-life problems tend to be complex enough so that not all possible costs, e.g., environmental costs, work risks,
etc., constraints and desirable solution properties, e.g., balanced traveled routes by drivers, time or geographical restrictions,
balanced work load across routes, solution attractiveness, etc., can be considered a priori during the mathematical modeling
phase. For that reason, The Balanced vehicle routing problem (BVRP) is considered in this paper as one of the most
significant versions of the CVRP. This problem has numerous applications in industrial and service problems such as delivery
management system (Kim et al., 2014), cutting stock problem (Eshghi et al., 2008), parallel reliability systems with
redundancy allocation (Ghafarian et al., 2014) and so on. The BVRP considers a balance of traveled routes for all the vehicles.
A considerable part of the drivers’ profits is related to their traveled distance; therefore, the balance of routes is important to
attract drivers' satisfaction. Besides, because the BVRP based on VRP belong to the class of NP-hard (Non-deterministic
Polynomial-time Hard) problems, this means that a polynomial time algorithm does not exist for it and the computational
attempt required to solve this problem increases exponentially with the size of the problem. So, these kinds of problems are
often solved with the help of metaheuristc techniques. As a result, the aim of this paper is to present an efficient ranked based
ant system, which is equipped with diversification and intensification mechanisms. Our meta-heuristic algorithm is combined
with several local searched algorithm including insert, swap and 2-opt algorithms. In this work we present several
modifications such as a new transition function and a modified updating pheromone method.
The rest of the paper is organized as follows: In Section 2 we first propose the mathematical model for BVRP and then
in section 3 the RAS are presented as an important part of our meta-heuristic then we explain the proposed algorithm which is
a modified version of RAS. In Section 4, the proposed algorithm is compared with other algorithms including RAS and the
exact algorithm in standard problems. Finally, the conclusions are summarized in Section 5.
2.MATHEMATICAL PROGRAMMING MODEL
The BVRP is described as follows. Let G  V , A  be a completed graph, where V  0,1, , n is the vertex set and
A  (i, j ):0  i, j  n  is the arc set. If the graph is not completed, we can replace the lack of any edge with an edge that is
of an infinite size. Vertex 0 represents the depot and N  1, . . . , n corresponds to the set of customers. Each customer i is
associated with a deterministic demand q i to be delivered (the depot is assigned to a demand q0  0. A non-negative distance
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ci j is associated with each arc (i, j )  A and represents the travel cost from node i to node j . The cost matrix is symmetric,
i.e., for all i, j  V , ci j  c j i . The use of the loop arc  i , i  is not allowed and defining ci i   for all i  V .
In this problem, a sequence of deliveries is generated for a homogeneous fleet of vehicles, in order to fulfill the
requirement of a known client. The objective of the problem is to determine the optimal combination of the vehicles and the
corresponding routes, so as to minimize the operating cost of all vehicles such that some constraints must be respected as
follows.
(1) Each route covered by the vehicles starts and ends at the depot.
(2) Each customer is visited exactly once by exactly one vehicle.
(3) Each route is exactly assigned to one vehicle.
(4) Each route covered by each vehicle divides another route obtained by another vehicle is between 1   and 1   in
which 0    1 is parameter set by user.
(5) The total demand of all customers served in a route cannot exceed the capacity of the vehicle assigned to that route.
(6) The number of the private vehicles used in the solution cannot exceed from defined number m.
, Table 1 lists a quick reference for the notation to show a clear definition of the notations shown in the paper. Some
symbols in this table are used in the mathematical formulation. The exact MIP formulation for the BVRP is given as follows.
Table 1.Glossary of mathematical symbols
Symbol
Explanation
Basic completed graph
The arc set.
Customers set
G
A
N
V  {0, N }
Vertex set, 0 represents the depot
qi
Demand of customer i
ci j
Travel distance, and travel cost between two nodes i and j
Q
Capacity of the vehicle
k
xi j take value 1 if a vehicle of type k travels directly from
k
xi j
K
n
customer i to customer j , and
0 otherwise
yi j
the quantity of goods that a vehicle k is carrying when leaves
customer i to service customer j.
Nu
Maximal number of the private vehicles
n
Min  cikj xikj
)1(
k 1 i  0 j  0
Subject to
K
n
 x
k
ij
 1,
j  1, 2,..., n
) 1(
j  1, 2,..., n , k  1, 2,..., K
) 2(
k 1 i  0
n
n
x x
i 0
k
ij
i 0
k
ji
 0,
) 3(
n
n
n
n
1    ( cikj xikj ) / ( cilj xilj )  1  
i 0 j 0
K
n
K
n
 y   y
k
ij
k 1 i  0
k  1, 2,..., K , l  k  1, 2,..., n
i 0 j 0
k
ji
 qj ,
j  1, 2,..., n
) 4(
i, j  0,1,..., n , i  j , k  1, 2,..., K
) 5(
k 1 i  0
q j xikj  yikj  (Qk  qi ) xikj
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yikj  0, xikj 0,1
i, j  0,1,..., n , k  1, 2,..., K
) 6(
Constraints (1) makes certain that each customer is visited exactly once, constraints (2) states that a vehicle of the same
type of one visiting a customer will also be departing from it. The balance restriction for traveled routes ofvehicles available
is guaranteed by constraints (3).Equality equations (4) ensure that the demand of all customers is fully satisfied. Constraint (5)
states that the vehicle capacity is never exceeded. Restrictions (6) force that the flow is non-negative and each arc in the
network has the value 1 if it is used and 0 otherwise.
3. SOLUTION METHOD
In this section at first the classic rank based ant system (RAS) is explained and then the modification of RAS is explained in
detail as the proposed algorithm.
3.1 Ant Colony Algorithm
The ant colony optimization (ACO) was firstly proposed by Dorigo et al. and used for the combinational optimization
problems in 1991 (YousefiKhoshbakht et al., 2008) especially the traveling salesman problem (TSP). This algorithm was
inspired by the behavior of real ant colonies in nature and is one of the oldest meta-heuristic algorithms that have recently
attracted a lot of attention by researchers. The ACO was developed well lately; modifying the method of updating the local
and global pheromones, and the distribution of ants on the nodes, are some examples of it. These developments lead to more
efficient algorithms such as EAS, ant colony system (ACS), rank based ant system (RAS), and max-min ant system (MMAS).
RAS algorithm which was proposed by Bullenheimer in 1997 is another version of the ACO (Bullnheimer et al., 1997).In
RAS, the artificial ants simulate the transitions from one point to another point. The ant maintains a tabu list in memory that
defines the set of points still to be visited when it is at point. The ant chooses to go from point to point during a tour with a
k
probability given in the equation (1). In this equation, Node j that is next to node i , among the unvisited node J i , is selected
by ant k in the route. According to the following transition rule, the probability of each node being visited is:
 ij (t )ij (t )
(7)
Pi jk (t ) 
 rJ  ir (t )ir (t )
k
i
Where
j
*
: The unvisited node j is maximized in J i for which [ i r (t )][i r (t )] .
k
 i j (t ) : The amount of pheromone that is on the edge joining nodes i and j.
i j (t ) : The heuristic information for the ant visibility measure defined here as the reciprocal of the distance between
node i and node j.
 ,  : Two control parameters that represent the relative importance of the amount of pheromone on the edge between
node i and node j. compared to the ant visibility value respectively.
In contrast to ACO, the pheromone of all edges belonging to the route obtained by ants will not be updated in RAS. The
pheromone updating of RAS includes only global updating rules (equation (8)). The pheromone updating equation was meant
to simulate the change in pheromone amount due to the addition of a new pheromone deposited by ants on the visited edges,
and the pheromone evaporation. It results in the new pheromone trail being a weighted average between the old pheromone
value and the amount of pheromone deposited. In other words, this algorithm improved the AS algorithm through ranking the
solutions constructed by ants. What distinguishes this algorithm from the other algorithms is the fact that in RAS the amount
of pheromone release is based on the rank of the ants in finding solutions.
 1
 i j (t  1)  (1   ) i j (t ) 
 

i j
(t )  
gb
i j
(t )
(8)
 1
Where:
 : A parameter called evaporation rate in the range  0,1 regulating the reduction of pheromone on the edges.
 i j

Q

(   ). L (t )
(t )  
0

(i , j )  S 
(i , j )  S
(9)
k
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Q : A constant coefficient determined by the user.
 : The number of ants which have been ranked and on their edges the pheromone has been deposited.
 : The variable indicating ranking index from 1 to   1 .
S : The edges traversed by an ant which has gained the  th rank in finding the best solution.

L (t ) : The length of the path traversed by the  th ant.
The amount of the global pheromone release for the best ant is calculated by the equation (12):
Q

(i, j )  best route
 . g b
gb
(10)
 i j (t )   L (t )
0
(i, j )  best route


3.2 Algorithm based on Modified Rank
A hybrid efficient RAS (ERAS) is proposed to solve BVRP in this paper which is based on the rank based ant system (RAS).
This proposed algorithm achieves performance improvements through the introduction of new mechanisms based on ideas
not included in the original AS. The ERAS algorithm improved the RAS algorithm through several modifications as a method
of pheromone update and a new transition function and several local searches like insert, swap and 2-opt heuristics are used to
more improve the RAS. These lead to avoid premature convergence and then escape from local optimum points. The
proposed algorithm has now a considerable amount of applications obtaining world class performance on problems like the
traveling salesman problem, quadratic assignment, vehicle routing, sequential ordering, scheduling, routing in Internet-like
networks, and so on. The proposed steps are described in more details as follows:
3.2.1 Construct solutions
In the proposed algorithm, for n groups, m ants are initially positioned on depot and each ant of the colony efforts to build a
solution represented as a single route. In other words, each ant initially selects a node from a feasible set of nodes by random
at the beginning of the trip according to the proposed ERAS. After that, some nodes may no longer be feasible for this ant
including the selected node. Therefore, after selecting each node, the feasible set of nodes that can be visited by the ant should
be updated and this set progressively becomes smaller. After the selection of initial node, the next node j is selected from
k
node i in the route by ant k among the unvisited nodes J i , according to the following transition rule which shows the
probability of each city being visited by the equation (11). It is noted that this solution consists of a number of vehicle routes
in which all of them are considered parallel. In other words, all m ants in a group build m routes simultaneously.
 ij (t )ij (t )
k
(11)
Pi j (t ) 
j  jik



(
t
)

(
t
)
 rJ i r i r
k
i
Where
 i j (t ) : The amount of pheromone on the edge joining nodes i and j .
 i j (t ) : The heuristic information for the ant visibility measure is defined as the modified savings of combining the two
nodes in one tour as opposed to serve them during two different tours. The modified savings of combining any two customers
i and j are computed as i j (t )  aci 0  bc0 j  cci j  d ci 0  c0 j where node 0 is the depot, a, b, c and d are parameters which
are set by user and ci j denotes the distance between nodes i and j .
 ,  : The controlparameter.
3.2.2 Local Search
A successful application of ACO needs a powerful technique for search intensification and diversification. The intensification
is a comprehensive exploration of some area of solution space, typically in the neighborhood of a good solution. The
diversification is leading the search to the promising regions of the solution space that is still unexplored. In the proposed
algorithm, local updating pheromone is used for diversification and several effective local searches are employed to achieve
intensification of the search. A local search starts with an initial solution and searches within neighborhoods for finding better
solutions. The proposed ERAS uses three types of neighborhood moves including the insert move, swap move and the 2-Opt
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move on both single route and multiple routes processes. In the insert move, a customer is removed from its original route and
inserted in that route or in a randomly selected route. In the swap move, two customers are randomly selected and exchanged
with each other, from the same route or the two different routes. Finally, in the 2-Opt move, two non-adjacent edges are
replaced by two other edges. It should be noted that there are several routes for connecting nodes and producing the tour
again, but a state is acceptable that satisfies the constraints of problem.
In the ERAS, after the all ants have constructed their solutions, the best solutions found unit now and the best solution in
the current iteration are improved by applying these several local searches. It is noted that because a local search is a
time-consuming procedure, we will only apply these local searches to the both best solutions that have not been improved yet.
The idea here is that a better solution may have a better chance to find a global optimum. For achieving this goal, insert
exchange, swap exchange and 2-opt move are used with the probability  ,  , and  respectively so that      1 . Note
that the 2-opt local search is a powerful global search algorithm and the insert and swap exchanges are more applied for local
search in the feasible space. Thus n times local searches are performed in each iteration in which the probability of using the
2-opt, insert and swap exchanges are considered  ,   0.40 and   0.20.
3.2.3 Updating pheromone
In the proposed algorithm not only the global trial updating is used as same as RAS but also local trial updating are applied for
achieving the diversification as mentioned before. The equation (12) is used for local trail updating. Ants update the amount
of pheromone on the visited edge while traversing between nodes i and j based on this equation, whenever an ant traverses an
edge  i , j  , its pheromone trail  i j is decreased. The purpose of local trail updating is to decrease the amount of pheromone
after an ant traverses an  i, j  edge so that the visited edges are less attractive to the following ants in future iterations.
Furthermore, local updating not only encourages an increase in the exploration of edges that have not been visited yet, but
also helps avoid poor stagnation situations.
 i j (t  1)  (1   ). i j (t )   0
(12)
Where
 : A parameter called evaporation rate in the range  0,1 regulating the reduction of pheromone on the edges.
 i j (t ) : The amount of pheromone that is on the edge joining nodes i and j .
 0 : Initial amount of pheromone valued by users.
It should be noted that the global pheromone updating method is performed by using  different ants that produced the
best solutions at the end of each iteration. Each time an ant travels an edge, the local pheromone update mechanism is applied
and the pheromone of that edge decreases so that it forces other ants to travel other edges. When ants complete their travels,
the global pheromone updating method is applied based on the best solutions found by the ants so that in the next iteration ants
pay more attention to the best solution found so far.
In order to prevent from local optimization and increase the probability of obtaining a higher-quality solution, upper and
lower limits from min-max AS (MMAS)  a, b  are fixed to the updating equation. If the amount of pheromone is not within
the minimum a and the maximum b while pheromone is being updated, then a or b amount is allocated to it by the equation
(13).
ti j  a
a  b / 4

1
(13)

t i j  b 
ti j  b
 100(1   )
t i j
otherwise

3.2.4 Termination Condition
In the proposed algorithm, when the best solution is iterated t times, convergence condition is satisfied and the algorithm will
be stopped so the best solution is selected and shown by it. A pseudo-code of the ERAS is presented in Figure 1.
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Procedure ERAS for solving BVRP
S*:=∅;
// S * is the best solution found yet //
*
f :=∞
// f* is the value of the S*//
n:= the number of nodes;
// n is the number of ants //
Initialize pheromone trails as a;
While do // main cycle //
Begin
S**:=∅;
// S ** is the best solution found in current iteration //
**
f :=∞;
// f** Is the value of the S** //
S:= none;
// S is a matrix and population of solutions //
For i := 1to n do
Begin
Construct a feasible solution Si and gain f(Si) as value of the Si;
Local updating pheromone trails; S=S∪Si;
If f(Si)< f** f**= f(Si); S**=Si; end;
Apply insert, swap and 2-opt algorithms for the best know solution and the best solution in the current iteration.
If these two solutions will be improved, the new solutions are replaced instead of the old solutions.
Global update pheromone trails;
If the amount of pheromone on an arc is below than a or upper than b, a and b are replaced respectively.
Until the best solution is iterated t times.
Show S* and f*
End // procedure //
Figure 1.The proposedalgorithmfor BVRP
4. COMPUTATIONAL RESULTS
The ERAS was programmed in Matlab 7 and executed on a Pentium IV 3.6 GHz machine with 8 GB of RAM running
Windows XP Home Basic Operating system. The performance of the proposed algorithm was tested on two sets of
benchmark problems for the BVRP. In this section, we first discuss the parameter setting of the ERAS algorithm and then the
detailed computational results are discussed.
4.1 Parameter settings
Parameter setting is a science itself and should generally be accomplished on instances different from those actually used for
computational comparisons in metaheuristic algorithms. Because the proposed algorithm is metaheuristic, the solutions
produced by the proposed algorithm were dependent on the seed used to generate the sequence of pseudo-random numbers
and on the different values of the parameters of the algorithm like every meta-heuristic algorithm. Many parameters exist in
our proposed algorithm, and the values of them directly or indirectly affect the quality of the final solution. Apart from that
the goal should be to find some robust parameters that allow an algorithm to find high quality solutions for a wide range of
problem instances with different features. In other words, a parameter setting procedure is necessary to reach the best balance
between the quality of the solutions obtained and the required computational attempt. We should mention that there is no way
of defining the most effective values of the parameters but in this paper, we try to test some parameters and select the best of
them for solving the BVRP. Our ERAS feature the following parameters that need to be adjusted:
Q : A constant coefficient in the equation (9).
 : A number of ants which have been ranked and global updating pheromone have been deposited on their edges in the
equation (9)
 ,  : Control parameters in the equation (11) that represent the relative importance of the amount of pheromone on the
edge  i, j  compared to the ant visibility value respectively.
 : The evaporation rate parameter of pheromone in equation (12) regulates the reduction of pheromone on all edges
 0 : Initial pheromone of all edges in equation (12)
a, b, c and d : parameters of ant visibility value in the equation (13).
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t : The number of iterations of the best solution in the algorithm as the termination condition.
When tuning the parameters, the instance C1 was determined as the test problem. Then, the algorithm with each
parameter combination for this instance was tested for 5 times. The ranges of eleven parameters were set in the Table 2. In this
table, all of the parameter values have been determined on the first instance, i.e., C1 by the numerical experiments. Then to
determine the parameters value several alternative values for each parameter were tested while all the others were held
constant and the ones that were selected gave the best computational results concerning both the quality of the solution and
the computational time needed to achieve this solution. The results confirm that our parameter setting worked well. It is also
possible that better solutions could exist.
Table 2.Parameter setting for the ERAS
Parameters
Candidate Valeu
The best value
Q
50, 100, 200, 300
50

3, 6, 9, 12, 15
9

1, 2, 3, 4, 5
1
a
1, 2, 3, 4, 5
0.1, 0.2, 0.3, 0.4. 0.5
1, 2, 3, 4
5
0.1
1
b
1, 2, 3, 4
1
c
1, 2, 3, 4
2
d
1, 2, 3, 4
2
0
1 / 1000(1   ),1 / 900(1   ),1 / 800(1   ),1 / 700(1   )
1 / 800(1   )
t
3, 5, 10, 15, 20
15


We understand that the most influential parameters in the proposed algorithm that directly affect the quality of the final
solution, are the power parameters of pheromone amount on the edge  i , j  , power parameter for the amount of ant visibility
value, the evaporation rate of pheromones the termination condition of finishing the algorithm. Based on the gained results,
the algorithm with the smaller weight parameter (  ) of pheromone trails possesses higher performance. This may be
attributed to that the initial pheromone trails are large values. If using the large control factor of pheromone trail, the effect of
visibility value is weakened and it results in a premature convergence. In addition, the solutions qualities of the algorithms
with   5 are better than 1,2, 3, 4 and5. From the test results, it can be found that by setting the evaporation factor to 0.1, the
proposed algorithm can yield better solutions. This can be attributed to that if pheromone evaporation is too rapid; it is easier
to result in the search to be trapped in local minimum. In other words, the smaller evaporation factor can ensure the sufficient
diversity of search space and guide following ants to explore better solutions. Thus, the combinations of optimal parameters
are determined: { = 1,  =5,  = 0.1}.
Clearly, increasing the number of iterations of the best solution for termination function ( t ) devoted to solving the
problem should ultimately (after ‘enough’ iterations) improve solution quality. It is noted that the number of iterations
assigned to solving the problem has shown to be important for the solution quality that can be obtained by the algorithm. If the
number of problematic iterations is too small, the problems will be solved insufficiently well and the algorithm may converge
to a poor solution. If the number of problem iterations is too large, the algorithm will waste resources and converge too
slowly. The settings chosen provide a good compromise with respect to this tradeoff and have shown to work well
independent of the problem instance solved. However, as discussed in this section, the number of iterations devoted to solving
the problem should be kept small to find solutions within reasonable time, in particular for problems of real world size. So
there is a tradeoff between solution quality found at the end of the runtime and solution quality found after short time.
Therefore, the best value for t is 15.
4.2 Results on benchmark instances
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Our proposed meta-heuristic algorithm was tested on two small and large size sets of BVRP benchmark problems. In the first
set, a new set consisting of eight tests numbered from D1 to D8 with sizes from 10 to 45 nodes without the depot were derived
from the well-known Christofides’s benchmark for Vehicle Routing Problem (VRP) (Christofides et al., 1979). Some of the
examples randomly located over a square with service time. Besides, they have a fixed fleet with capacity restrictions and
with route length restrictions. Euclidean distances are used in the all problems. In this set, the proposed ERAS was compared
to the solver Cplex 12.4 in AIMMS as an exact algorithm and classic RAS. AIMMS is an advanced development environment
for building advanced planning systems and optimizing the problems in applied research studies.
As seen in table 3, the proposed ERAS provides solutions, which are better than the solutions provided by RAS and
Cplex 12.4, that uses branch and bound algorithm for solving MILP models. With more details, the ERAS are better than the
RAS in D4, D5, D7 and D8 and are better than the Cplex 12.4 in D6, D7 and D8. In addition, the proposed approach gives
better solutions within significantly shorter time frame when compared with CPLEX 10.0. Due to the NP-hard nature of
BVRP, MILP approach cannot solve the problem within an acceptable time frame. On the contrary, ERAS perform well on
NP-hard problems. Therefore, the proposed approach provided better solutions than MILP. Considering these results and
CPU times, it can be stated that, ERAS based proposed approach is efficient and performs well.
Table 3.Computational results for Exact, RAS and ERAS
Instance n m
Q
D
RAS
Time (Sec) ERAS Time (Sec)
Exact Algorithm
Time (Sec)
BKS
D1
10 3
120
∞
89.53
2.61
89.53
3.41
89.53
2
545.33
D2
15 5
100
∞
99.93
3.92
99.93
4.82
99.93
1452
884.43
D3
20 3
130
∞
121.62
5.01
121.62
6.29
121.62
6432
862.87
D4
25 4
120
∞
169.31
22.61
168.78
21.62
168.78
6582
1134.54
D5
30 4
100
∞
256.6
41.90
249.66
32.81
249.66
3198
1402.21
D6
35 4
120
200 295.53
91.71
295.53
85.88
298.51
20435
598.03
D7
40 5
120
160 349.21
95.45
335.63
94.82
341.80
98401
965.31
D8
45 5
120
200 423.18
99.61
396.41
99.71
499.51
243691
934.81
It is noted that the commercial linear programming software including ILOG and Cplex could not find optimal solutions
for the large-scale of the problems, like BVRP and hence can be used to evaluate the accuracy of the proposed model.
Therefore, in the second large set, the ERAS are only compared to EAS. This set contains between 50 to199 nodes as well as
the depot and can be downloaded from the library website's following address
URL: http://mscmga.ms.ic.ac.uk/jeb/orlib/vrpinfo.html
In the second set shown in table 4, the first 10 instances include the customers that are randomly distributed around the
depot. In the last four instances, the customers appear in clusters and the depot is not centered. Columns 2-5 show the problem
size n, the number of vehicles m, the capacity of vehicles Q and the maximum tour length D. It should be noted that in these
examples, the time limit has not been included due to the difference in Euclidean distance between customers and the time
required to provide service for each of them. Therefore, just the limits for the maximum capacity for vehicles and the
maximum travel for each vehicle have been included in these examples.
Since the RAS and ERAS are meta-heuristic algorithms, the results are reported for ten independent runs and the best
solutions are shown in the table 4. The information is consisting of the number of customers, the solution costs are obtained
from the best solution costs of RAS and the best solution costs of ERAS. Finally, the BKS was shown in the last column. As
can be seen in this table, the proposed algorithm finds the optimal solution for 14 out of 14 problems. The results indicate that
the proposed algorithm is a competitive approach compared to the RAS. For instances C6 and C8 both the ERAS and RAS are
same. However, in other instances the proposed algorithm finds the BKS and much better that RAS. The performance
comparison of results shows that the proposed algorithm clearly yields better solutions than that of RAS.
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Table 4.Computational results for the Christofides’s benchmark
Instance
n
m
Q
D
RAS
ERAS
BKS
C1
50
5
160
∞
549.21
545.33
545.33
C2
75
10
140
∞
892.41
884.43
884.43
C3
100
8
200
∞
899.21
862.87
862.87
C4
150
12
200
∞
1285.08
1134.54
1134.54
C5
199
17
200
∞
1592.33
1402.21
1402.21
C6
50
6
160
200
598.03
598.03
598.03
C7
75
11
140
160
989.65
965.31
965.31
C8
100
9
200
230
934.81
934.81
934.81
C9
150
14
200
200
1311.93
1264.61
1264.61
C10
199
18
200
200
1583.76
1461.72
1461.72
C11
120
7
200
∞
1274.69
1201.42
1201.42
C12
100
10
200
∞
921.49
904.37
904.37
C13
120
11
200
720
1784.86
1721.80
1721.80
C14
100
11
200
1040
939.41
931.81
931.81
A simple criterion to measure the efficiency and the quality of an algorithm is to compute the relation average of PD of
its solution from the another algorithm on specific benchmark instances. We compute the PD by using equation (14) where
**
*
c ( s ) and c ( s ) are the best solution found by RAS and ERAS respectively for the given instances. A zero gap indicates that
a same solution is found by both algorithms.
PD  100  [c( s )  c( s )] / c( s ) (14)
**
*
*
In figure 2 it is reported that the average of percentage deviation (PD) for RAS and ERAS is compared to the BKS. From
this table we conclude that the ERAS method has the best deviation from the BKS. As shown in this figure, the best algorithm
is the proposed algorithm which has found the best solutions for all 14 examples and has shown a better performance than
RAS.
15
10
5
D1
D2
D3
D4
D5
D6
D7
D8
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
0
Figure 2.Comparison of PD for 14 instances between RAS and ERAS
5. CONCLUSIONS
An important version of the VRP called BVRP is considered in this paper. This version of VRP has significant applications in
transportation system, especially when the balance of the route based on vehicles travelled distance is important for the
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Balanced Vehicle Routing Problem
company. The BVRP has not been studied in the existing literature. A mathematical formulation is presented for this problem
and an effective RAS was proposed due to its NP-hard nature. Since there were no existing benchmarks, this study generated
some test problems. The efficiency of the proposed algorithm is compared with classic RAS and the exact algorithm in a
range of test instances, including small-scale instances and large-scale ones. From the comparison with the results of
proposed algorithm with exact algorithm and RAS, the ERAS can provide better solutions within a comparatively shorter
time period and is more efficient than the RAS and the exact algorithm especially for large-size problems which the exact
algorithm may fail to get a feasible solution. For the future research, one important direction is to work on the similar but more
difficult variant, the BVRP incorporating with soft or hard time window, i.e., time windows must be obeyed, or that can be
violated at a BVRP with time window is very complex and more similar to the real-life scenarios. Efficient algorithm like
combination of the proposed algorithm and tabu search needs to be designed for this problem.
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