5323
Journal of Intelligent & Fuzzy Systems 40 (2021) 5323–5336
DOI:10.3233/JIFS-202129
IOS Press
Delivering and picking goods under time
window restrictions: an effective
evolutionary algorithm
Grigorios D. Konstantakopoulos∗ , Sotiris P. Gayialis, Evripidis P. Kechagias,
Georgios A. Papadopoulos and Ilias P. Tatsiopoulos
Sector of Industrial Management and Operational Research, School of Mechanical Engineering, National
Technical University of Athens, Athens, Greece
Abstract. Routing of vehicles and scheduling of deliveries play a crucial role in logistics operations as they affect both the
distribution cost and customer satisfaction. That is why researchers have intensively studied this problem in conjunction
with the multiple variables and constraints involved in the logistics operations. In this paper, the cases of time windows
and simultaneous pickups and deliveries, where goods are simultaneously delivered and collected from customers within a
predetermined time slot, are studied. The objective of our research is to create efficient routes that minimize both the number
of vehicles and the total distance travelled, as both of them affect the total distribution cost. Considering various plans of routes
that are differentiated by the number of routes and the sequence of visitations, can be beneficial for decision-makers, since
they have the opportunity to select the plan that better fits their needs. Therefore, in this paper we develop a multiobjective
evolutionary algorithm (MOEA) that integrates an improved construction algorithm and a new crossover operator for efficient
distribution services. Through the proposed MOEA a set of solutions (route plans), known as Pareto-optimal, is obtained,
while single biased solutions are avoided. The proposed algorithm is tested in two well-known datasets in order to evaluate
the algorithm’s efficiency. The results indicate that the algorithm’s solutions have small deviation from the best-published
and some non-dominated solutions are also obtained.
Keywords: Vehicle routing problem, time windows, simultaneous pickups and deliveries, multiobjective optimization, evolutionary algorithm, logistics
1. Introduction
Supply chain networks include multiple operations
such as distributing goods to retailers and consumers,
warehousing of products, and supplying raw materials to manufacturers. Additionally, they also include
the collection of products from end customers, in
order to be recycled, remanufactured, or repaired
[23]. Given the range of supply chain operations,
∗ Corresponding
author. Grigorios D. Konstantakopoulos, Sector of Industrial Management and Operational Research, School of
Mechanical Engineering, National Technical University of Athens,
Athens, Greece. E-mail: gkonpoulos@mail.ntua.gr.
the routing of vehicles and the scheduling of deliveries, constitute a small but essential part of them.
Therefore, executing efficient routing of vehicles and
scheduling of deliveries ensures that the transportation costs will be reduced and the provided services
will be improved. In this effort, researchers have
converted the multiple variables and constraints that
logistics companies face in real-life cases into variants of the Vehicle Routing Problem (VRP). This
interconnection between the logistics industry and
the research community has made the VRP one of
the most studied topics in operational research and
industrial engineering [17].
ISSN 1064-1246/$35.00 © 2021 – IOS Press. All rights reserved.
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G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
The simplest version of the VRP is known as the
Capacitated VRP and was first proposed by [11],
Dantzig and Ramser (1959). In this classical problem, a set of geographically scattered customers with
various demands must be served by a set of identical
vehicles and each customer is served by a single vehicle and only once. All vehicles start their trip from
the central depot and must return at the same location [2] after the deliveries are concluded. In each
stage of the distribution, the capacity of the vehicles
cannot be violated. Through time, more and more
VRP variants have been created following the needs
and requirements of logistics and distribution companies. According to [25] Konstantakopoulos et al.
(2020), among the most studied variants which reflect
most in real-world problems is the VRP with Time
Windows (VRPTW) and the VRP with Simultaneous Pickups and Deliveries (VRPSPD). In the case of
the VRPTW, the constraint that each customer must
be visited within a predetermined time interval (time
window) indicated by each customer, is added. Each
vehicle has the ability to arrive at a customer’s location before the opening of the time window and can
wait until service becomes possible. However, it is
not permitted to deliver goods outside the time window [5, 22]. In VRPSPD, the main variation from the
initial problem is that each customer can both receive
and return goods [28]. This variation creates an extra
constraint as at any stage of the route, where goods
are both delivered and collected, the capacity of the
vehicle cannot be violated. By combining these two
variants, this complex problem comes closer to reallife distribution cases that most logistics companies
face daily.
However, the conversion of real distribution cases
to variants of the VRP would have no value, if not
making the most of advances in technology and computer science. Both sectors have given researchers the
opportunity to develop more efficient algorithms to
address the variants of the VRP. Initially, exact algorithms were developed due to their ability to offer
high quality solutions. Nevertheless, the time complexity along with the limited number of customers
that exact algorithms can handle makes the specific
category less popular among others [19]. On the other
hand, heuristic algorithms offer good quality solutions within a short time-space, while having also the
ability to solve problems of any size. Some of the most
known and studied heuristics for the VRP are (i) the
Time-Oriented Nearest Neighbor (TONN) [31], (ii)
the Sweep [20] and (iii) the Savings [9] algorithm.
These algorithms, despite being more than 30 years
old, remain the basis of other more sophisticated and
improved algorithms and are still used in commercial
routing software [16].
Over the years, metaheuristics have gained the
interest of researchers and practitioners due to their
ability to produce high quality solutions in a minimum computational time. The center of attention
has shifted to these algorithms as they combine
advantages from both exact and heuristic algorithms.
Metaheuristics are separated into two main categories, population search and local search methods
[27]. Population-based approaches maintain reasonable solutions in memory, which are combined and
cooperate through a learning procedure for generating new solutions. On the other hand, local search
approaches aim at exploring the solution space by
moving the current solution to another promising
one in the neighborhood. Multiple algorithms of
both categories have been developed for addressing efficiently both the VRPTW and the VRPSPD.
According to [25] Konstantakopoulos et al. (2020),
among the most frequently developed algorithm for
these variants of the VRP are evolutionary and genetic
algorithms [4], particle swarm optimization [1, 22],
ant colony optimization [14], simulated annealing
[10, 33], large neighborhood search [26] and tabu
search [7, 34].
In most cases, the algorithms that address the
VRPTW and the VRPSPD produce biased solutions
either to the number of vehicles or the total distance
traveled, that are the two objectives. However, in
recent years, some researchers consider the VRP as
multiobjective which means that more than one solution can be considered optimal according to the values
of the two objective functions. More specifically,
Ombuki et al. (2006) [29] and Tan et al. (2006) [32]
were the first to propose a multiobjective genetic and
evolutionary algorithm, respectively, for the VRPTW,
offering new non-dominated solutions. Ghoseiri and
Ghannadpour (2010) [18] used goal programming for
the formulation and a genetic algorithm for the solving the multiobjective formulation of the VRPTW,
while Baños et al. (2013) [3] proposed a hybrid
algorithm that combined evolutionary strategies with
simulated annealing. Both [15], Garcia-Najera and
Bullinaria (2011) and [6], Chiang and Hsu (2014)
proposed MOEA’s which updated multiple nondominated solutions in Solomon’s instances. Instead,
researchers have not studied the multiobjective nature
of the VRPSPD, at the same extent. Gong et al. (2018)
[21] proposed an efficient bee evolutionary algorithm
for this problem, and was verified by simulating the
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G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
reverse logistics of an engineering machinery remanufacturing company. A characteristic feature of all
these algorithms is that they maintain the Paretooptimality concept, which means that in each case
a set of optimal solutions may result. The solutions
belonging in this set are called non-dominated solutions, as no solution can be considered better than the
others without further information, while solutions
that do not belong in that set are known as dominated
solutions.
In the present paper, we propose a MOEA that
uses the non-dominated sorting procedure proposed
by [12], Deb et al. (2002) for solving the vehicle routing problem with simultaneous pickups and deliveries
and with time windows (VRPSPDTW), while both
the number of vehicles and the total distance traveled must be minimized. The initial population of
solutions utilized by the MOEA are produced by an
improved variation of the TONN heuristic algorithm.
Additionally, the proposed MOEA incorporates a
new crossover operator and the swap node operator
(mutation). The MOEA is tested both in Solomon’s
benchmark instances for the VRPTW, as well as in
Salhi and Nagy’s for the VRPSPD, since there is no
dataset in the literature to combine both problems.
The computational results indicate that the algorithm
is very efficient, offering competitive results compared with all the best-published of the literature.
However, since most studies do not focus on the multiobjective nature of the VRPSPDTW, neither do to
the multiple optimal solutions (non-dominated solutions) that have been proposed in each benchmark
instance over the years. This results in not evaluating
the efficiency of algorithms correctly since, in many
cases, the obtained solutions are not compared with
all best-published solutions in each instance. Therefore, the contribution of this paper is not limited to
addressing the VRPSPDTW as multiobjective in the
variation of the TONN algorithm and in the crossover
operator that is integrated into the algorithm. Our
research extends to the complete presentation of all
non-dominated solutions in Solomon’s and in Salhi
and Nagy’s dataset.
The next sections of the paper include the formulation of the problem, the presentation of the
proposed algorithm, its validation results and, finally,
the conclusions of the research. More specifically, the
mathematical formulation of the VRPSPDTW is presented in Section 2. Section 3 analyzes the structure
of the proposed algorithm, the improved variation of
the TONN algorithm, the new proposed crossover
operator, as well as the mutation operator. The com-
putational results are presented and compared with
the best-published solutions in benchmark instances
in Section 4. Finally, the conclusions are discussed in
Section 5.
2. Problem definition and mathematical
formulation
The VRPSPDTW is a variant of VRP which combines the parameter of time windows, as well as
the one of simultaneous pickups and deliveries set
by customers. More specifically, VRPSPDTW is
defined by a set of identical in terms of capacity
(Q) vehicles denoted by K and by a direct network G (N, A), where N is the set of nodes, and
A = (i, j) : i =
/ j, i, j ∈ N the set of arcs. Node 0
represents the central depot, while N * = N/ {0} represents the customers. Every arc (i, j), which is a path
from node i to node j, is characterized by a distance
which is indicated as dij and by a travel time tij that
has the same value as dij as the assumption that one
distance unit is covered in one unit of time is made.
Each client i is characterized by a delivery quantity
qi , a pickup quantity ri , a service time si for unloading and a time window (ei , li ), where ei is the opening
and li the closing time. The start of service of vehicle
k at customer j, is denoted by ajk and depends on the
time of arrival as vehicles arriving before the beginning of the time window must wait until service is
possible ei . If the vehicle arrives between ei and li ,
the service starts immediately, while no vehicle may
arrive after li . The decision variable xijk is equal to 1
if vehicle k executes the route from node i to node
j. Otherwise xijk is equal to 0. The second decision
variable, yik indicates the quantity of items on board
of vehicle k when departing from node i. The objective of the VRPTW is to service all customers in N * ,
while minimizing both the number of vehicles and
the total distance traveled. The mathematical model
of the VRPSPDTW is presented below:
k
Objective1 :
x0j
(1)
k∈K j∈N ∗
Objective2 :
dij xijk
(2)
xijk ≤ Q, ∀k ∈ K
(3)
k∈K (i,j)∈A
i∈N
qi
j∈N
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G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
ri
i∈N
(i,j)∈A
(5)
the time window, lj , while constraint (11) that the
earliest time for starting serving customer j is related
to the time of arrival, depending on whether the time
of arrival is before or after the start of the time window
ei .
(6)
3. Solution approach
(4)
xijk ≤ Q, ∀k ∈ K
j∈N
xijk yik + rj − qj ≤ Q, ∀k ∈ K
xijk = 1, ∀i ∈ N
k∈K j∈N
j∈N ∗
xilk −
i∈N
(7)
xljk = 0, ∀l ∈ N ∗ , ∀k ∈ K
(8)
k
xj0
= 1, ∀k ∈ K
(9)
j∈N
j∈N ∗
aik + si +
k
x0j
= 1, ∀k ∈ K
(10)
xijk tij ≤ lj , ∀i ∈ N, ∀k ∈ K
In order to address the VRPSPDTW that is presented above, an efficient multiobjective evolutionary
algorithm was developed. More specifically, Section
3.1 discusses the importance of multiobjective optimization and the Pareto-optimality concept. Section
3.2 presents the improved version of the TONN algorithm, that it is used for creating the initial population
of solutions. Section 3.3 describes the structure of the
chromosome applied in the MOEA. In Section 3.4 the
methodological procedure of the MOEA, along with
the pseudocode are thoroughly described. Finally, the
new proposed crossover operator, as well as the mutation operator, are described in Section 3.5.
j∈N
ajk = max
⎧
⎨
⎩
xijk aik + si + tij , ej
j∈N
∀i ∈ N, ∀k ∈ K
3.1. Multiobjective optimization
⎫
⎬
⎭
,
(11)
xijk ∈ {0, 1} , ∀ (i, j) ∈ A, ∀k ∈ K
(12)
yik ≥ 0, ∀i ∈ N, ∀k ∈ K
(13)
In the present paper, the VRPTW is encountered
as a multiobjective problem and the two objectives,
needed to be minimized, are the number of vehicles
and the total distance traveled and are described by
function (1) and (2) respectively. Constraints (3) and
(4) state that the total quantity of products delivered
and collected in total cannot exceed the capacity of the
vehicle, while constraint (5) states that, at any stage of
the distribution process, the total quantity of products
in the vehicles cannot exceed its capacity. Equation
(6) states that each customer must be served exactly
once and by one vehicle since xijk can obtain only
1, or 0 as values. The set of constraints (7), (8) and
(9) make sure that each vehicle starts from the depot
and visits and serves the customers that belong in the
route before it finally returns to the depot. Constraint
(10) indicates that for a trip from node i to node j,
no vehicle may arrive at customer j after the end of
In multiobjective optimization problems, a set of
optimal solutions can be obtained, instead of a single solution. The solutions belonging to this set are
non-dominated with respect to each other. The main
characteristic of the non-dominated solutions is that
while moving from one solution to another, there is
always a certain amount of sacrifice in one objective,
to achieve a certain amount of gain in another objective, as shown in Fig. 1. More specifically, no solution
between the non-dominated solutions X1 and X2 can
be considered better, since the mathematical relation
F1 (X1 ) > F1 (X2 ) and F2 (X1 ) < F2 (X2 ) is valid.
This means that moving from solution X1 to X2 , we
can observe a decrease in F1 objective with a simultaneous increase in F2 objective. Therefore, without
further information, no solution can be considered
better than the other and dominate. The set of all
the non-dominated solutions is called Pareto-optimal
front as shown in Fig. 1.
In the case of the VRP, the different obtained solutions are translated into different plans of routes and
deliveries. These different plans serve the interests of
multiple stakeholders that choose the most profitable
one. The choice is based on further information and
on the nature of each company. For example, different plans may be selected when the fleet of vehicles is
company-owned and when the distribution of goods
G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
Fig. 1. Dominated, non-dominated, and Pareto-optimal front.
is assigned to third party companies. Therefore, it
is crucial to consider the VRPSPDTW as multiobjective, where the two objectives are the number of
vehicles and the total distance traveled, in order to
cover the needs of multiple stakeholders. Through
this approach and in search of a cost-effective plan,
more than one solution may result. Therefore, it is
at the discretion of the decision-maker to select the
most profitable plan.
3.2. Construction method
The initial population of solutions contributes significantly to the efficiency of the proposed MOEA,
where it is used as an input. Therefore, an efficient
algorithm, in terms of time and solution’s quality, is
needed. The fact that the TONN construction algorithm meets these requirements is the main reason it
is selected for the construction procedure. The TONN
algorithm was first proposed by Solomon (1987) [9],
and belongs in the sequential methods that construct
one route at a time. More specifically, every route
starts from the depot (node 0) and is built by adding
the “closest” customer that has not been included in
any other route to the last visited node. If no other
customer can be added to the route, due to capacity
or time window constraints, a new route starts. The
routes construction procedure ends when no more
customers are left to add.
The fact that the TONN algorithm was first proposed for the VRPTW, which is only one of the
variants considered in our paper, leads to adaptations in the closeness function, in order to fit in the
VRPSPDTW. The closeness is now given by the metric cij = w1 dij + w2 vij + w3 uij , which involves the
distance dij between customers i and j, the capacity left to the vehicle k after serving customer j,
vij = Q − yik − qj and the urgency of delivery to
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customer j, uij = lj − (ai + si + tij ). In this case,
customer i is the previously served customer, while j
the one for whom the metric cij is minimized.
The second variation in the TONN is based on the
fact that time, distance and capacity are not in the
same scale, and that may affect the constructed solutions. Therefore, a min-max normalization procedure
[30], is applied in the dataset tij , dij , ei , li , si , qi .
This normalization technique provides linear transformation and fits data in a pre-defined boundary,
that in our case is [0, 1]. The formula used for the
normalization procedure is:
X = (X0 − Xmin ) / (Xmax − Xmin ) (D − C) + C
(14)
with X being the new value for variable X, [C,
D] the pre-defined boundary, X0 the current value
for variable X, and Xmin minimum and Xmax the
maximum value in dataset X.
This pre-processing data procedure improves the
accuracy and achieves the best performance in the
tested dataset. That is why in the case of cij metric,
in which the weights w1 , w2 and w3 are assigned to
dij , vij and uij respectively and sum up to 1, the normalization of data is essential and can offer improved
results. Therefore, the dataset used in the construction
algorithm involves normalized data.
3.3. Chromosome structure
The structure of the chromosome significantly
affects the algorithm in terms of efficiency and time
complexity, since it is involved in the crossover and
mutation operators. Therefore, similar to most studies
that propose evolutionary algorithms for the VRP, we
decided to represent the chromosome as an array with
length equal to the total number of customers. Each
object of the chromosome is a gene and is represented
by an integer which refers to the customer’s node
(id). The sequence of the genes determines the order
of customers’ visitation. When a sequence cannot be
continued due to the violation of capacity or time window constraints, a new route begins. An example of a
chromosome with 9 customers is presented in Fig. 2,
as well as the order of visitations in each route.
Fig. 2. Chromosome Structure.
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3.4. Evolving solutions with the proposed
multiobjective evolutionary algorithm
After the TONN algorithm generates the initial
population (R0 ) of size N, the MOEA is applied
for evolving solutions and obtain the Pareto-optimal
front. More specifically, in each generation t, taking into account the parents’ population Rt set,
the algorithm implements the crossover operator in
all N parents and generates N offsprings saved in
Qt (offsprings’ population). The proposed crossover
operator is thoroughly described in Section 3.5. Consequently, the chromosomes belonging to the union
set Rt ∪ Qt (entire population) are sorted according
to the non-dominated sorting and create Pareto fronts
(sets) F1 , F2 , . . . FK . This procedure is used for sorting the solutions in sets (F1 , F2 , . . . FK ) according to
the pareto dominance principle. This means that each
set Fi contains solutions, with F1 being the best nondominated set that is represented as Pareto-optimal
front in Fig. 1. Accordingly, F2 set contains solutions which are dominated only by those in set F1 .
The set Rt+1 , which contains the parents of the next
generation starts being filled with solutions of sets
F1 , F2 , and so on, until N chromosomes fill the Rt+1
set. Through this procedure, the best solutions in each
iteration are maintained and used for evolving the
solutions. The described procedure is followed for
a specific number of iterations, which is the maximum number of generations. The pseudocode of the
MOEA is shown in Fig. 3.
3.5. Crossover and mutation operators
Multiple crossover operators, such as those presented in [31], have been developed over the years
for the VRP. The most efficient crossover operator
that is implemented in most genetic and evolutionary
algorithms is the Best Cost Route Crossover (BCRC)
that is thoroughly described in Ombuki et al. (2006)
[22]. More specifically, from the P set, two parents
(Pi , Pi+1 ) are selected, and from each one, a single
route is also selected. The customers that belong to
each route are then removed from the opposite parent.
More specifically, customers selected from parent Pi
are removed from parent Pi+1 , and reversely. Since
all customers must be served, they must also be
contained in the chromosome. The insertion of a customer is made in the position of the chromosome that
minimizes the cost. After the insertion procedure, two
new chromosomes, known as offsprings, are generated. Through this procedure, the set Qt is filled with
Fig. 3. Pseudocode of the MOEA.
N new solutions, before the non-dominated sorting
procedure is applied and consequently set Pt+1 arises.
The crossover operator proposed and applied in
the present paper works similarly to the BCRC and
is called Sequential Crossover (SC). The differences
between the two crossover operators are noted in two
points.
• In the selection of customers, as in SC, a random
number of consecutive genes are selected from
each chromosome, instead of a single route as
in BCRC. In this way, the diversity and consequently the search space increases.
• In the reinsertion procedure the customers can
only be inserted in specific positions. More
specifically, they can be inserted in the position
of the chromosome for which the total distance
traveled is minimized. However, they can also be
inserted in the position of the chromosome for
which the total distance traveled is minimized
while simultaneously the number of vehicles is
not increased (compared to the parent). This procedure also increases the diversity and the search
capabilities of the algorithm.
The mutation operator followed in our paper is
a swap node operator. In this case, two customers
G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
(genes) of the chromosome are randomly selected
and swapped. The main contribution of this procedure is the differentiation of chromosomes, which
may help in escaping from local optimum. Finally,
both the mutation and the crossover operator are not
applied in each generation and in each set of parents
(Pi , Pi+1 ). Instead, each operator has a specific probability to be applied, that is defined at the beginning
as shown in Fig. 3.
4. Computational tests and results
The proposed algorithm cannot be simultaneously
tested in time windows and pickups and deliveries due to the lack of a dataset containing both
attributes. Therefore, the algorithm is tested in two
different datasets, the one of Solomon which is for
the VRPTW and the one of Salhi and Nagy which is
for the VRPSPD. In Solomon’s dataset, the instances
are separated in those with geographically clustered
customers’ data (C1 and C2), with randomly generated geographical customers data (R1 and R2) and
with mixed geographical customers data (RC1 and
RC2). Also, problem sets C1, R1 and RC1 have short
scheduling horizon and narrow time windows, while
set C2, R2, and RC2 have long scheduling horizon
and wide time windows. The proposed algorithm was
coded in Python and ran in a computer with the following specifications: Intel Core i7 – 8550 U 1.8 GHz
with 16GB memory. Additionally, the dataset of Salhi
and Nagy (1999) [32], is based on the dataset of
Christofides et al. (1979) [8]. More specifically, all
data remain the same except the demand that is separated in a delivery and in a pickup quantity. Finally,
the evolutionary algorithm parameters used in the
proposed algorithm are the same with those presented
in [22] so that our results be representative and comparable.
•
•
•
•
Population size = 300
Number of generations = 350
Crossover probability = 0.80
Mutation probability = 0.10
Another parameter that is defined is the maximum
computational time. In our research, the time limit
is set at 10 minutes which is considered reasonable
computational time for cases with 100 customers.
4.1. Results in the VRPTW
The VRPTW has been widely studied and, therefore, for most of Solomon’s instances a set of
5329
non-dominated solutions (Pareto-optimal front) has
arisen over the years from multiple researchers, leading to the best published results in these instances.
The computational results obtained from the proposed MOEA are compared, in Tables 1 and 2,
with the best-published ones in the literature to
evaluate their efficiency. More specifically, Table 1
contains instances with short scheduling horizon, narrow time windows, and small vehicles’ capacities,
while Table 2 contains instances with long scheduling horizon, wide time windows and large vehicles’
capacities. In each table and for each instance of
the dataset, the set of non-dominated solutions that
are characterized by the total distance traveled (TD)
and the number of vehicles (NV) are presented.
Column “MOEA” presents the results obtained by
the proposed algorithms, column “Best-Published”
showcases the best-published results. Finally, the
column “Deviation (%)” presents the percentage difference between best-published and best obtained
results.
We consider that our results are competitive compared to best published in the literature. The deviation
presented in both tables refers to the total traveled
distance when the number of vehicles, both in the
best-published results and in those of the MOEA
are equal; otherwise it is not calculated. The mean
deviation is 2.64%, a criterion that proves the effectiveness of the algorithm. However, the MOEA seems
to have better efficiency with geographically clustered customers’ data (C datasets), irrespective of the
scheduling horizon and the range of the time windows, as the deviation of the obtained results with
the best-published range from 0.00% to 0.73%.
As for geographically random customer data (R
and RC), multiple non-dominated solutions at each
instance exist as the search space is more extensive, especially in cases with long scheduling horizon
and wide time windows (C2 and RC2). The MOEA,
which integrates the proposed crossover operator, has
the ability to search a vast solution space to find
non-dominated solutions. As a result, two new nondominated solutions are proposed in Solomon’s R202
and R210 instances. In R202 instance, a decrease
in the total distance traveled is accomplished, while
in R210 instance, two non-dominated solutions are
updated, as shown in Fig. 4. More specifically, in
R210 instance, the proposed solution with objective values (NV = 4, TD = 920.13) improves both the
solution with objective values (NV = 4, TD = 924.79),
since the distance traveled is decreased, as well as the
solution with objective values (NV = 5, TD = 922.30),
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G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
Table 1
Solomon’s benchmark instances with short scheduling horizon and narrow time windows
Problem
Instance
C101
C102
C103
C104
C105
C106
C107
C108
C109
R101
R102
R103
R104
R105
R106
R107
R108
R109
R110
NV
10
10
10
10
10
10
10
10
10
20
18
14
–
10
–
15
–
13
–
11
–
10
–
12
13
–
MOEA
TD
828.94
828.94
828.94
828.94
828.94
828.94
828.94
828.94
828.94
1667.33
1499.67
1237.50
–
1006.48
–
1391.86
–
1287.07
–
1084.15
–
994.89
–
1177.76
1167.19
–
NV
Best-Published
TD
10
10
10
10
10
10
10
10
10
18
17
13
9
10
14
15
12
13
10
11
9
10
11
12
–
10
827.3
827.3
826.3
822.9
827.3
827.3
827.3
827.3
827.3
1607.7
1434
1175.67
1007.24
974.2
1377.11
1346.12
1251.98
1234.6
1104.66
1051.84
958.66
942.9
1194.73
1101.99
–
1118.59
Deviation
(%)
Problem
Instance
0.20
0.20
0.32
0.73
0.20
0.20
0.20
0.20
0.20
–
–
–
–
3.31
–
3.40
–
4.25
–
3.07
–
5.51
–
6.88
–
–
R110
Fig. 4. Updated Pareto-optimal front on R210 problem instance.
since both the number of vehicles and the total distance traveled are decreased.
The solutions in Solomon’s instances which
improve the results and offer new non-dominated
solutions are presented in the Appendix.
R111
R112
RC101
RC102
RC103
RC104
RC105
RC106
RC107
RC108
NV
–
12
–
–
12
–
10
–
15
–
–
14
–
12
10
–
14
15
–
–
–
13
–
12
–
11
MOEA
TD
–
1122.48
–
–
1099.70
–
965.12
–
1649.50
–
–
1504.56
–
1292.51
1165.00
–
1575.46
1558.06
–
–
–
1417.86
–
1248.62
–
1209.42
NV
Best-Published
TD
11
12
10
11
12
9
10
14
15
12
13
14
11
12
10
13
14
15
16
11
12
13
11
12
10
11
1054.23
1068
1096.72
1084.76
1048.7
982.14
953.63
1650.14
1619.8
1554.75
1470.26
1461.33
1261.67
1196.12
1135.48
1629.44
1542.29
1519.29
1518.6
1424.73
1394.43
1371.69
1222.16
1212.83
1139.82
1117.53
Deviation
(%)
–
5.10
–
–
4.86
–
1.20
–
1.83
–
–
2.96
–
8.06
2.60
–
2.15
2.55
–
–
–
3.37
–
2.95
–
8.22
address the VRPSPDTW, is tested in the dataset
of Salhi and Nagy, and the results obtained from
the proposed MOEA algorithm are compared to the
best-published. As in the case of time windows, the
algorithm manages to produce new non-dominated
solutions in instances CMT2X, CMT6X, CMT7X,
CMT8X, CMT9X, CMT10X. In each of these 6
cases, the number of vehicles is reduced at the
expense of increased total distance traveled, as shown
in Table 3. In addition, the mean TD deviation
between the best-published results and the best
obtained ones from the MOEA was found to be
8.50%. As shown in Table 3, the TD deviation for
each one of the problem instances is not calculated
for all solutions. Instead, the TD deviation is calculated only when the MOEA and the best-published
results have equal NV values.
The solutions in Salhi and Nagy’s instances which
improve the results and offer new non-dominated
solutions are given in the Appendix.
4.2. Results in the VRPSPD
The VRPSPD is studied less frequently than the
VRPTW, despite being the most common problem
distribution and logistics companies face in their daily
operations. The proposed MOEA, which is able to
5. Conclusions
To sum up, in this paper, we examined the vehicle routing problem with simultaneous pickups and
G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
5331
Table 2
Solomon’s Benchmark Instances with Long Scheduling Horizon and Wide Time Windows
Problem
Instance
C201
C202
C203
C204
C205
C206
C207
C208
R201
R202
R203
R204
R205
R206
R207
R208
NV
3
3
3
3
3
3
3
3
4
5
–
–
–
–
–
4
5
3
4
5
–
–
3
4
–
3
4
5
3
4
5
–
3
–
–
3
MOEA
TD
591.56
591.56
591.56
590.6
588.88
588.5
588.5
588.32
1290.75
1202.7
–
–
–
–
–
1085.64
1014.96
1026.22
902.25
891.03
–
–
751.79
746.01
–
1048.74
1021.15
1004.27
970.22
897.25
892.18
–
854.95
–
–
731.84
NV
3
3
3
3
3
3
3
3
4
5
6
7
8
9
3
4
5
3
4
5
6
2
3
4
5
3
4
5
3
4
5
2
3
4
2
3
Best-Published
TD
589.1
589.1
591.17
590.6
586.4
586
585.8
585.8
1252.37
1193.29
1171.2
1173.75
1150.92
1148.48
1191.7
1079.39
1041.1
939.5
901.2
890.5
874.87
825.52
749.42
743.23
735.86
994.43
959.74
954.16
906.14
889.39
879.89
890.61
812.76
797.99
725.75
706.86
Deviation
(%)
Problem
Instance
0.42
0.42
0.07
0.00
0.42
0.43
0.46
0.43
3.06
0.79
–
–
–
–
–
0.58
–2.51
9.23
0.12
0.06
–
–
0.32
0.37
–
5.46
6.40
5.25
7.07
0.88
1.40
–
5.19
–
–
3.53
R209
deliveries and time windows, aiming to find an
effective solution to the problem by proposing an
evolutionary algorithm that solves the multiobjective
VRPSPDTW. The first step to solving this problem
was the application of an improved TONN construction algorithm for generating the initial population of
solutions. Consequently, these solutions were integrated into the proposed MOEA in order to find the
best routes plan, that minimizes the two objectives:
the number of vehicles and the total distance traveled.
These two objectives may be contradictory and this
is the reason that some instances of the problem may
have multiple possible solutions. Pareto-optimality
concept was used in order to balance the results for the
different objectives and find the best overall solution
to the problem. Besides addressing the VRPSPDTW
R210
R211
RC201
RC202
RC203
RC204
RC205
RC206
RC207
RC208
NV
3
4
5
–
4
–
–
–
3
4
4
5
7
8
–
4
5
6
–
4
5
3
–
4
5
6
–
3
4
5
6
–
4
–
–
3
4
MOEA
TD
956.00
881.64
875.63
–
920.13
–
–
–
838.85
770.54
1473.09
1329.59
1284.48
1282.64
–
1200.07
1129.46
1125.22
–
958.55
938.04
801.18
–
1340.53
1304.35
1210.00
–
1301.87
1121.53
1094.2
1087.27
–
1023.02
–
–
859.52
805.66
NV
3
4
5
3
4
5
6
2
3
4
4
5
7
8
3
4
5
8
3
4
5
3
4
4
5
6
7
3
4
5
7
3
4
5
6
3
4
Best-Published
TD
909.16
864.15
859.39
938.58
924.79
922.30
912.53
885.71
767.82
755.82
1406.91
1279.65
1273.51
1272.28
1365.65
1162.54
1118.66
1099.54
1049.62
945.08
926.82
798.41
788.66
1297.19
1236.78
1187.98
1161.81
1146.32
1081.83
1068.77
1054.61
1061.14
1001.85
982.58
966.37
828.14
783.035
Deviation
(%)
5.15
2.02
1.89
–
–0.50
–
–
–
9.25
1.95
4.70
3.90
0.86
0.81
–
3.23
0.97
–
–
1.43
1.21
0.35
–
3.34
5.46
1.85
–
13.57
3.67
2.38
–
–
2.11
–
–
3.79
2.89
as a multiobjective problem, another contribution of
the paper was proven to be the newly developed
crossover operator (Sequential Crossover) which significantly affects the efficiency of the results obtained
from the MOEA.
The algorithm’s efficiency was validated using the
dataset of Solomon for the VRPTW and the dataset of
Salhi and Nagy (1999) for the VRPSPD, and the outcomes obtained were compared to the best-published
results, indicating the high efficiency of the algorithm
in terms of solution’s quality.
The next step of our research is to apply the algorithm in real-life cases, in which even 1000 customers
must be served. In order to be able to solve large scale
instances with up to 1000 customers, parameters of
the algorithm such as the population size and the num-
5332
G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
Table 3
Comparison of results for the Salhi and Nagy dataset
Problem CMT
1X
2X
3X
4X
5X
6X
7X
8X
9X
10X
11X
12X
13X
14X
NV
3
6
7
5
7
10
5
9
8
9
12
13
14
15
17
5
6
11
10
MOEA
NV
481.13
771.01
730.21
753.83
897.65
1130.28
606.33
1040.67
1036.52
955.21
1744.19
1741.36
1698.21
1354.51
1937.02
904.52
712.21
1757.99
918.25
3
–
7
5
7
10
6
11
467
–
676
703
847
1025
557
919
9
–
–
–
15
19
4
5
11
10
896
–
–
–
1215
1571
868
667
1560
826
ber of generations must also be studied so that the
algorithm becomes functional and efficient in a reasonable computational time. Finally, for evaluating
the efficiency of the algorithm in real-life cases, travel
times and distances between nodes must be collected
from online maps and consequently be exploited as
inputs in the algorithm.
Funding
The present work was co-funded by the European Union and Greek national funds through
the Operational Program “Competitiveness,
Entrepreneurship and Innovation” (EPAnEK),
under the call “RESEARCH-CREATEINNOVATE”
(Project Code: T1EDK-00527 and Acronym:
SMARTRANS).
References
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[1]
[2]
[3]
Best-published
TD
TD
T.J. Ai and V. Kachitvichyanukul, A particle swarm
optimization for the vehicle routing problem with simultaneous pickup and delivery, Comput Oper Res 36 (2009),
1693–1702. doi:https://doi.org/10.1016/j.cor.2008.04.003
R. Baldacci, A. Mingozzi and R. Roberti, Recent exact
algorithms for solving the vehicle routing problem under
capacity and time window constraints, Eur J Oper Res 218
(2012), 1–6. doi:10.1016/j.ejor.2011.07.037
R. Baños, J. Ortega, C. Gil, A.L. Márquez and F. De Toro,
A hybrid meta-heuristic for multi-objective Vehicle Routing
Problems with Time Windows, Comput Ind Eng 65 (2013),
286–296. doi:10.1016/j.cie.2013.01.007
[13]
[14]
[15]
Deviation
(%)
3.03
–
8.02
7.23
5.98
10.27
–
–
–
6.61
–
–
–
11.48
–
–
–
12.69
11.17
J. Berger, M. Barkaoui and O. Bräysy, A Route-Directed
Hybrid Genetic Approach For The Vehicle Routing Problem
With Time Windows, INFOR Inf Syst Oper Res 41 (2003),
179–194. doi:10.1080/03155986.2003.11732675
D. Chen, E. Cao and W. Lai, A differential evolution
algorithm for pickups and deliveries problem with fuzzy
time windows, J Intell Fuzzy Syst 30 (2016), 267–277.
doi:10.3233/IFS-151752
T.C. Chiang and W.H. Hsu, A knowledge-based evolutionary algorithm for the multiobjective vehicle routing problem
with time windows, Comput Oper Res 45 (2014), 25–37.
doi:10.1016/j.cor.2013.11.014
W.C. Chiang and R.A. Russell, A Reactive Tabu Search
Metaheuristic for the Vehicle Routing Problem with
Time Windows, INFORMS J Comput 9 (1997), 417–430.
doi:doi:10.1287/ijoc.9.4.417
N. Christofides, A. Mingozzi and P. Toth, The vehicle routing problem, Wiley, Chichester, (1979).
G. Clarke and J.W. Wright, Scheduling of Vehicles from a
Central Depot to a Number of Delivery Points, Oper Res 12
(1964), 568–581. doi:10.1287/opre.12.4.568
Z.J. Czech and P. Czarnas, Parallel simulated annealing for
the vehicle routing problem with time windows, in: 10th
Euromicro Work. Parallel, Distrib. Network-Based Process., 2002 376–383. doi:10.1109/EMPDP.2002.994313
G.B. Dantzig and J.H. Ramser, The Truck Dispatching Problem, Manage Sci 6 (1959), 80–91. doi:10.1287/mnsc.6.1.80
K. Deb, A. Pratap, S. Agarwal and T. Meyarivan,
A fast and elitist multiobjective genetic algorithm:
NSGA-II, IEEE Trans Evol Comput 6 (2002), 182–197.
doi:10.1109/4235.996017
J. Dethloff, Vehicle routing and reverse logistics: The vehicle routing problem with simultaneous delivery and pick-up,
OR-Spektrum 23 (2001), 79–96. doi:10.1007/PL00013346
L.M. Gambardella, É. Taillard and G. Agazzi, Macsvrptw: A multiple colony system for vehicle routing
problems with time windows, in: New Ideas Optim 1999
63–76.
A. Garcia-Najera and J.A. Bullinaria, An improved multiobjective evolutionary algorithm for the vehicle routing
G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
problem with time windows, Comput Oper Res 38 (2011),
287–300. doi:10.1016/j.cor.2010.05.004
S.P. Gayialis and I.P. Tatsiopoulos, Design of an IT-driven
decision support system for vehicle routing and scheduling,
Eur J Oper Res 152 (2004), 382–398. doi:10.1016/S03772217(03)00031-6
S.P. Gayialis, G.D. Konstantakopoulos and I.P. Tatsiopoulos, Vehicle Routing Problem for Urban Freight
Transportation: A Review of the Recent Literature, in: Oper.
Res. Digit. Era–ICT Challenges, Springer P, Springer, 2019
89–104.
K. Ghoseiri and S.F. Ghannadpour, Multi-objective vehicle
routing problem with time windows using goal programming and genetic algorithm, Appl Soft Comput 10 (2010),
1096–1107. doi:10.1016/j.asoc.2010.04.001
M.G. Giaglis, I. Minis, A. Tatarakis and V. Zeimpekis, Minimizing logistics risk through real-time vehicle routing and
mobile technologies, Int J Phys Distrib Logist Manag 34
(2004), 749–764. doi:10.1108/09600030410567504
B.E. Gillett and L.R. Miller, A Heuristic Algorithm for the
Vehicle-Dispatch Problem, Oper Res 22 (1974), 340–349.
doi:10.1287/opre.22.2.340
G. Gong, Q. Deng, X. Gong, L. Zhang, H. Wang and
H. Xie, A Bee Evolutionary Algorithm for Multiobjective Vehicle Routing Problem with Simultaneous Pickup
and Delivery, Math Probl Eng 2018 (2018), 2571380.
doi:10.1155/2018/2571380
Y.J. Gong, J. Zhang, O. Liu and C.H.S. Hung, Optimizing
the Vehicle Routing Problem With Time Windows: A Discrete Particle Swarm Optimization Approach, IEEE Trans
Syst Man Cybern Part C Appl Rev 42 (2012), 224–267.
doi:10.1109/TSMCC.2011.2148712
K. Govindan, H. Soleimani and D. Kannan, Reverse logistics and closed-loop supply chain: A comprehensive review
to explore the future, Eur J Oper Res 240 (2015), 603–626.
doi:10.1016/j.ejor.2014.07.012
R.P. Hornstra, A. Silva, K.J. Roodbergen and L.C. Coelho,
The vehicle routing problem with simultaneous pickup and
delivery and handling costs, Comput Oper Res 115 (2020),
104858. doi:https://doi.org/10.1016/j.cor.2019.104858
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
5333
G.D. Konstantakopoulos, S.P. Gayialis and E.P. Kechagias,
Vehicle routing problem and related algorithms for logistics
distribution: a literature review and classification, Oper Res
Int J (2020). doi:10.1007/s12351-020-00600-7
G.D. Konstantakopoulos, S.P. Gayialis, E.P. Kechagias,
G.A. Papadopoulos and I.P. Tatsiopoulos, A Multiobjective
Large Neighborhood Search Metaheuristic for the Vehicle Routing Problem with Time Windows, Algorithms 13
(2020). doi:10.3390/a13100243
N. Labadie, C. Prins and C. Prodhon, Metaheuristics for
Vehicle Routing Problems, John Wiley & Sons (2016).
G. Nagy and S. Salhi, Heuristic algorithms for single
and multiple depot vehicle routing problems with pickups and deliveries, Eur J Oper Res 162 (2005), 126–141.
doi:https://doi.org/10.1016/j.ejor.2002.11.003
B. Ombuki, B.J. Ross and F. Hanshar, Multi-objective
genetic algorithms for vehicle routing problem with time
windows, Appl Intell 24 (2006) 17–30. doi:10.1007/s10489006-6926-z
S. Salhi and G. Nagy, A cluster insertion heuristic for
single and multiple depot vehicle routing problems with
backhauling, J Oper Res Soc 50 (1999), 1034–1042.
doi:10.1057/palgrave.jors.2600808
M.M. Solomon, Algorithms for the vehicle routing and
scheduling problems with time window constraints, Oper
Res 35 (1987), 254–265.
K.C. Tan, Y.H. Chew and L.H. Lee, A hybrid multiobjective
evolutionary algorithm for solving vehicle routing problem
with time windows, Comput Optim Appl 34 (2006), 115.
M. Woch and P. Łebkowski, Sequential Simulated Annealing for the Vehicle Routing Problem with Time Windows,
Decis Mak Manuf Serv 3 (2009), 87–100.
M. Yousefikhoshbakht, F. Didehvar and F. Rahmati, A
combination of modified tabu search and elite ant system
to solve the vehicle routing problem with simultaneous
pickup and delivery, J Ind Prod Eng 31 (2014), 65–75.
doi:10.1080/21681015.2014.893928
5334
G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
Appendix
The new solutions proposed in Solomon’s benchmark instances are:
R202: Number of vehicles = 5, Total distance traveled = 1014.96
Vehicle 1: 0 −→ 69 −→ 1 −→ 30 −→ 90 −→ 10 −→ 32 −→ 70 −→ 31 −→ 52 −→ 0
Vehicle 2: 0 −→ 45 −→ 47 −→ 36 −→ 63 −→ 64 −→ 11 −→ 19 −→ 62 −→ 88 −→ 7 −→ 18 −→
6 −→ 99 −→ 96 −→ 95 −→ 97 −→ 43 −→ 2 −→ 13 −→ 58 −→ 0
Vehicle 3: 0 −→ 92 −→ 37 −→ 42 −→ 15 −→ 14 −→ 38 −→ 44 −→ 16 −→ 61 −→ 86 −→ 84 −→ 83
−→ 8 −→ 82 −→ 48 −→ 49 −→ 46 −→ 17 −→ 60 −→ 5 −→ 93 −→ 85 −→ 91 −→ 100 −→ 98 −→
59 −→ 94 −→ 89 −→ 0
Vehicle 4: 0 −→ 27 −→ 50 −→ 33 −→ 65 −→ 71 −→ 29 −→ 76 −→ 3 −→ 79 −→ 78 −→ 81 −→ 9
−→ 51 −→ 20 −→ 66 −→ 35 −→ 34 −→ 68 −→ 24 −→ 80 −→ 77 −→ 12 −→ 0
Vehicle 5: 0 −→ 28 −→ 26 −→ 39 −→ 67 −→ 23 −→ 72 −→ 73 −→ 21 −→ 40 −→ 53 −→ 87 −→
57 −→ 41 −→ 22 −→ 75 −→ 56 −→ 4 −→ 0
R210: Number of vehicles = 4, Total distance traveled = 920.13
Vehicle 1: 0 −→ 2 −→ 57 −→ 15 −→ 42 −→ 14 −→ 44 −→ 38 −→ 86 −→ 16 −→ 61 −→ 99 −→ 59
−→ 87 −→ 43 −→ 41 −→ 22 −→ 74 −→ 56 −→ 4 −→ 55 −→ 25 −→ 54 −→ 26 −→ 58 −→ 0
Vehicle 2: 0 −→ 95 −→ 92 −→ 98 −→ 85 −→ 5 −→ 45 −→ 36 −→ 47 −→ 48 −→ 82 −→ 7 −→ 88
−→ 62 −→ 19 −→ 11 −→ 64 −→ 49 −→ 46 −→ 8 −→ 84 −→ 17 −→ 83 −→ 60 −→ 94 −→ 13
−→ 97 −→ 37 −→ 100 −→ 91 −→ 93 −→ 96 −→ 0
Vehicle 3: 0 −→ 27 −→ 69 −→ 1 −→ 30 −→ 65 −→ 71 −→ 33 −→ 50 −→ 76 −→ 3 −→ 79 −→ 81
−→ 9 −→ 34 −→ 78 −→ 29 −→ 24 −→ 80 −→ 68 −→ 77 −→ 0
Vehicle 4: 0 −→ 28 −→ 12 −→ 39 −→ 67 −→ 23 −→ 75 −→ 72 −→ 73 −→ 21 −→ 40 −→ 53 −→ 6
−→ 89 −→ 18 −→ 52 −→ 31 −→ 10 −→ 63 −→ 90 −→ 20 −→ 51 −→ 35 −→ 0
The new solutions proposed in Salhi and Nagy’s benchmark instances are:
CMT2X: Number of vehicles = 6, Total distance traveled = 771.01
Vehicle 1: 0 −→ 8 −→ 35 −→ 53 −→ 14 −→ 19 −→ 54 −→ 13 −→ 57 −→ 15 −→ 5 −→ 37 −→ 20
−→ 70 −→ 60 −→ 71 −→ 36 −→ 69 −→ 61 −→ 22 −→ 0
Vehicle 2: 0 −→ 58 −→ 10 −→ 38 −→ 65 −→ 11 −→ 66 −→ 59 −→
0
Vehicle 3: 0 −→ 26 −→ 67 −→ 34 −→ 68 −→ 73 −→ 62 −→ 28 −→ 74 −→ 21 −→ 47 −→ 48 −→
30 −→ 0
Vehicle 4: 0 −→ 4 −→ 29 −→ 45 −→ 27 −→ 52 −→ 46 −→ 7 −→ 72 −→ 39 −→ 9 −→ 12 −→ 0
Vehicle 5: 0 −→ 2 −→ 33 −→ 63 −→ 1 −→ 43 −→ 42 −→ 64 −→ 41 −→ 56 −→ 23 −→ 24 −→ 49
−→ 16 −→ 0
Vehicle 6: 0 −→ 75 −→ 6 −→ 51 −→ 17 −→ 40 −→ 3 −→ 44 −→ 32 −→ 50 −→ 18 −→ 25 −→ 55
−→ 31 −→ 0
CMT5X: Number of vehicles = 5, Total distance traveled = 606.33
Vehicle 1: 0 −→ 46 −→ 11 −→ 2 −→ 20 −→ 35 −→ 36 −→ 3 −→ 28 −→ 22 −→ 0
Vehicle 2: 0 −→ 45 −→ 33 −→ 39 −→ 10 −→ 30 −→ 34 −→ 50 −→ 9 −→ 49 −→ 0
Vehicle 3: 0 −→ 5 −→ 38 −→ 16 −→ 29 −→ 21 −→ 31 −→ 26 −→ 7 −→ 23 −→ 6 −→ 0
Vehicle 4: 0 −→ 32 −→ 1 −→ 27 −→ 48 −→ 8 −→ 43 −→ 24 −→ 14 −→ 25 −→ 13 −→ 18 −→ 0
Vehicle 5: 0 −→ 12 −→ 47 −→ 17 −→ 37 −→ 15 −→ 44 −→ 42 −→ 19 −→ 40 −→ 41 −→ 4 −→ 0
CMT7X: Number of vehicles = 9, Total distance traveled = 1040.67
Vehicle 1: 0 −→ 75 −→ 4 −→ 67 −→ 34 −→ 46 −→ 52 −→ 27 −→ 13 −→ 0
Vehicle 2: 0 −→ 26 −→ 12 −→ 40 −→ 17 −→ 51 −→ 6 −→ 68 −→ 2 −→ 0
Vehicle 3: 0 −→ 7 −→ 35 −→ 8 −→ 54 −→ 19 −→ 14 −→ 53 −→ 11 −→ 0
Vehicle 4: 0 −→ 45 −→ 29 −→ 48 −→ 70 −→ 47 −→ 21 −→ 74 −→ 28 −→ 30 −→ 0
Vehicle 5: 0 −→ 33 −→ 73 −→ 1 −→ 43 −→ 41 −→ 42 −→ 22 −→ 62 −→ 0
Vehicle 6: 0 −→ 16 −→ 63 −→ 23 −→ 56 −→ 49 −→ 24 −→ 31 −→ 10 −→ 0
Vehicle 7: 0 −→ 38 −→ 65 −→ 66 −→ 59 −→ 58 −→ 25 −→ 55 −→ 18 −→ 0
Vehicle 8: 0 −→ 3 −→ 44 −→ 32 −→ 50 −→ 9 −→ 39 −→ 72 −→ 71 −→ 60 −→ 20 −→ 0
Vehicle 9: 0 −→ 57 −→ 15 −→ 5 −→ 37 −→ 36 −→ 69 −→ 61 −→ 64 −→ 0
G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
CMT8X: Number of vehicles = 8, Total distance traveled = 1036.52
Vehicle 1: 0 −→ 94 −→ 59 −→ 85 −→ 93 −→ 95 −→ 13 −→ 87 −→ 97 −→ 100 −→ 98 −→ 37
−→ 92 −→ 91 −→ 0
Vehicle 2: 0 −→ 27 −→ 28 −→ 68 −→ 79 −→ 81 −→ 9 −→ 71 −→ 66 −→ 65 −→ 51 −→ 53
−→ 58 −→ 0
Vehicle 3: 0 −→ 40 −→ 26 −→ 12 −→ 76 −→ 50 −→ 77 −→ 3 −→ 33 −→ 34 −→ 78 −→ 89 −→
5 −→ 0
Vehicle 4: 0 −→ 96 −→ 99 −→ 61 −→ 16 −→ 44 −→ 86 −→ 17 −→ 84 −→ 83 −→ 18 −→ 30 −→
32 −→ 90 −→ 10 −→ 62 −→ 52 −→ 0
Vehicle 5: 0 −→ 28 −→ 26 −→ 39 −→ 67 −→ 23 −→ 72 −→ 73 −→ 21 −→ 40 −→ 53 −→ 87 −→
57 −→ 41 −→ 22 −→ 75 −→ 56 −→ 4 −→ 0
Vehicle 6: 0 −→ 2 −→ 57 −→ 42 −→ 15 −→ 14 −→ 38 −→ 43 −→ 41 −→ 22 −→ 23 −→ 75 −→
72 −→ 0
Vehicle 7: 0 −→ 73 −→ 74 −→ 21 −→ 56 −→ 39 −→ 67 −→ 25 −→ 55 −→ 80 −→ 88 −→ 63 −→
64 −→ 11 −→ 0
Vehicle 8: 0 −→ 82 −→ 48 −→ 47 −→ 49 −→ 36 −→ 46 −→ 19 −→ 7 −→ 60 −→ 8 −→ 45 −→ 0
CMT9X: Number of vehicles = 12, Total distance traveled = 1744.19
Vehicle 1: 0 −→ 112 −→ 147 −→ 6 −→ 94 −→ 95 −→ 59 −→ 93 −→ 85 −→ 98 −→ 37 −→ 92 −→
00 −→ 75 −→ 4 −→ 67 −→ 34 −→ 46 −→ 52 −→ 27 −→ 13 −→ 0
Vehicle 2: 0 −→ 97 −→ 117 −→ 53 −→ 105 −→ 26 −→ 149 −→ 109 −→ 12 −→ 132 −→ 69 −→ 1
−→ 30 −→ 62 −→ 0
Vehicle 3: 0 −→ 27 −→ 146 −→ 52 −→ 127 −→ 31 −→ 88 −→ 148 −→ 10 −→ 108 −→ 90 −→ 32
−→ 111 −→ 58 −→ 40 −→ 0
Vehicle 4: 0 −→ 21 −→ 73 −→ 72 −→ 74 −→ 133 −→ 22 −→ 75 −→ 56 −→ 139 −→ 110 −→ 28
−→ 89 −→ 18 −→ 0
Vehicle 5: 0 −→ 118 −→ 60 −→ 5 −→ 84 −→ 8 −→ 17 −→ 113 −→ 86 −→ 141 −→ 13 −→ 137
−→ 87 −→ 144 −→ 57 −→ 115 −→ 0
Vehicle 6: 0 −→ 2 −→ 145 −→ 41 −→ 15 −→ 96 −→ 99 −→ 104 −→ 100 −→ 138 −→ 0
Vehicle 7: 0 −→ 119 −→ 44 −→ 16 −→ 14 −→ 142 −→ 42 −→ 43 −→ 116 −→ 77 −→ 79 −→ 129
−→ 76 −→ 0
Vehicle 8: 0 −→ 33 −→ 81 −→ 102 −→ 50 −→ 68 −→ 101 −→ 70 −→ 122 −→ 20 −→ 51 −→ 9
−→ 120 −→ 0
Vehicle 9: 0 −→ 135 −→ 35 −→ 103 −→ 106 −→ 7 −→ 82 −→ 48 −→ 124 −→ 47 −→ 36 −→ 143
−→ 49 −→ 0
Vehicle 10: 0 −→ 150 −→ 80 −→ 134 −→ 24 −→ 29 −→ 121 −→ 34 −→ 78 −→ 3 −→ 54 −→ 83
−→ 114 −→ 46 −→ 123 −→ 19 −→ 107 −→ 0
Vehicle 11: 0 −→ 130 −→ 55 −→ 25 −→ 39 −→ 23 −→ 67 −→ 4 −→ 91 −→ 61 −→ 45 −→ 0
Vehicle 12: 0 −→ 38 −→ 140 −→ 125 −→ 11 −→ 63 −→ 126 −→ 64 −→ 131 −→ 128 −→ 71 −→
136 −→ 65 −→ 66 −→ 0
CMT9X: Number of vehicles = 13, Total distance traveled = 1741.36
Vehicle 1: 0 −→ 112 −→ 147 −→ 94 −→ 95 −→ 59 −→ 93 −→ 85 −→ 98 −→ 37 −→ 92 −→ 97 −→ 0
Vehicle 2: 0 −→ 89 −→ 117 −→ 53 −→ 105 −→ 26 −→ 149 −→ 109 −→ 12 −→ 0
Vehicle 3: 0 −→ 132 −→ 69 −→ 1 −→ 30 −→ 62 −→ 52 −→ 127 −→ 31 −→ 88 −→ 148 −→ 10 −→ 108
−→ 90 −→ 32 −→ 111 −→ 0
Vehicle 4: 0 −→ 58 −→ 40 −→ 21 −→ 73 −→ 72 −→ 74 −→ 133 −→ 22 −→ 75 −→ 56 −→ 139 −→ 110
−→ 0
Vehicle 5: 0 −→ 28 −→ 27 −→ 146 −→ 18 −→ 118 −→ 60 −→ 5 −→ 84 −→ 8 −→ 6 −→ 0
Vehicle 6: 0 −→ 17 −→ 113 −→ 86 −→ 141 −→ 13 −→ 137 −→ 87 −→ 144 −→ 57 −→ 115 −→ 0
Vehicle 7: 0 −→ 2 −→ 145 −→ 41 −→ 15 −→ 96 −→ 99 −→ 104 −→ 100 −→ 138 −→ 0
Vehicle 8: 0 −→ 119 −→ 44 −→ 16 −→ 14 −→ 142 −→ 42 −→ 43 −→ 116 −→ 77 −→ 79 −→ 129 −→
76 −→ 0
Vehicle 9: 0 −→ 33 −→ 81 −→ 102 −→ 50 −→ 68 −→ 101 −→ 70 −→ 122 −→ 20 −→ 51 −→ 9 −→
120 −→ 0
Vehicle 10: 0 −→ 135 −→ 35 −→ 103 −→ 106 −→ 7 −→ 82 −→ 48 −→ 124 −→ 47 −→ 36 −→ 143 −→
49 −→ 0
Vehicle 11: 0 −→ 150 −→ 80 −→ 134 −→ 24 −→ 29 −→ 121 −→ 34 −→ 78 −→ 3 −→ 54 −→ 83 −→
114 −→ 46 −→ 123 −→ 19 −→ 107 −→ 0
Vehicle 12: 0 −→ 130 −→ 55 −→ 25 −→ 39 −→ 23 −→ 67 −→ 4 −→ 91 −→ 61 −→ 45 −→ 0
Vehicle 13: 0 −→ 38 −→ 140 −→ 125 −→ 11 −→ 63 −→ 126 −→ 64 −→ 131 −→ 128 −→ 71 −→ 136 −→
65 −→ 66 −→ 0
5335
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G.D. Konstantakopoulos et al. / Delivering and picking goods under time window restrictions
CMT9X: Number of vehicles = 14, Total distance traveled = 1698.21
Vehicle 1: 0 −→ 112 −→ 94 −→ 95 −→ 59 −→ 85 −→ 93 −→ 98 −→ 100 −→ 37 −→ 92 −→ 0
Vehicle 2: 0 −→ 97 −→ 117 −→ 137 −→ 53 −→ 105 −→ 26 −→ 149 −→ 12 −→ 109 −→ 68 −→
150 −→ 80 −→ 116 −→ 0
Vehicle 3: 0 −→ 40 −→ 76 −→ 77 −→ 102 −→ 27 −→ 146 −→ 0
Vehicle 4: 0 −→ 89 −→ 147 −→ 6 −→ 96 −→ 99 −→ 104 −→ 118 −→ 5 −→ 84 −→ 61 −→ 28
−→ 111 −→ 138 −→ 0
Vehicle 5: 0 −→ 50 −→ 1 −→ 132 −→ 69 −→ 101 −→ 32 −→ 90 −→ 31 −→ 127 −→ 52 −→ 0
Vehicle 6: 0 −→ 13 −→ 87 −→ 144 −→ 57 −→ 115 −→ 2 −→ 145 −→ 41 −→ 22 −→ 133 −→ 0
Vehicle 7: 0 −→ 21 −→ 73 −→ 72 −→ 74 −→ 75 −→ 56 −→ 39 −→ 139 −→ 4 −→ 110 −→ 58
−→ 18 −→ 106 −→ 0
Vehicle 8: 0 −→ 7 −→ 82 −→ 124 −→ 47 −→ 123 −→ 19 −→ 107 −→ 48 −→ 60 −→ 83 −→ 0
Vehicle 9: 0 −→ 114 −→ 8 −→ 125 −→ 45 −→ 17 −→ 113 −→ 16 −→ 141 −→ 88 −→ 148 −→
62 −→ 10 −→ 108 −→ 0
Vehicle 10: 0 −→ 131 −→ 128 −→ 20 −→ 30 −→ 79 −→ 129 −→ 120 −→ 81 −→ 3 −→ 0
Vehicle 11: 0 −→ 9 −→ 103 −→ 71 −→ 135 −→ 33 −→ 70 −→ 122 −→ 51 −→ 66 −→ 65 −→ 0
Vehicle 12: 0 −→ 136 −→ 35 −→ 78 −→ 54 −→ 130 −→ 134 −→ 24 −→ 29 −→ 121 −→ 34 −→ 0
Vehicle 13: 0 −→ 42 −→ 142 −→ 14 −→ 119 −→ 44 −→ 91 −→ 86 −→ 15 −→ 43 −→ 38 −→ 140
−→ 46 −→ 0
Vehicle 14: 0 −→ 55 −→ 25 −→ 67 −→ 23 −→ 126 −→ 63 −→ 11 −→ 64 −→ 49 −→ 143 −→ 36
−→ 0
CMT9X: Number of vehicles = 17, Total distance traveled = 1937.02
Vehicle 1: 0 −→ 112 −→ 156 −→ 183 −→ 94 −→ 95 −→ 59 −→ 85 −→ 93 −→ 98 −→ 92
−→ 91 −→ 0
Vehicle 2: 0 −→ 27 −→ 167 −→ 190 −→ 31 −→ 162 −→ 101 −→ 176 −→ 102 −→ 157 −→ 81 −→
120 −→ 0
Vehicle 3: 0 −→ 53 −→ 105 −→ 28 −→ 111 −→ 196 −→ 68 −→ 150 −→ 80 −→ 177 −→ 170 −→
12 −→ 184 −→ 154 −→ 138 −→ 0
Vehicle 4: 0 −→ 152 −→ 58 −→ 40 −→ 13 −→ 87 −→ 97 −→ 151 −→ 100 −→ 193 −→ 141 −→
44 −→ 119 −→ 142 −→ 15 −→ 147 −→ 0
Vehicle 5: 0 −→ 89 −→ 166 −→ 118 −→ 5 −→ 84 −→ 173 −→ 61 −→ 16 −→ 191 −→ 192 −→
37 −→ 0
Vehicle 6: 0 −→ 146 −→ 52 −→ 127 −→ 88 −→ 182 −→ 148 −→ 159 −→ 62 −→ 10 −→ 189 −→
108 −→ 0
Vehicle 7: 0 −→ 180 −→ 149 −→ 26 −→ 195 −→ 130 −→ 134 −→ 163 −→ 24 −→ 132 −→ 69 −→
1 −→ 0
Vehicle 8: 0 −→ 50 −→ 76 −→ 116 −→ 77 −→ 158 −→ 3 −→ 79 −→ 129 −→ 169 −→ 55 −→ 109
−→ 0
Vehicle 9: 0 −→ 6 −→ 96 −→ 99 −→ 104 −→ 117 −→ 137 −→ 178 −→ 115 −→ 2 −→ 144 −→ 172
−→ 42 −→ 0
Vehicle 10: 0 −→ 18 −→ 153 −→ 106 −→ 194 −→ 7 −→ 82 −→ 48 −→ 123 −→ 19 −→ 107 −→ 175
−→ 0
Vehicle 11: 0 −→ 198 −→ 21 −→ 72 −→ 197 −→ 75 −→ 133 −→ 22 −→ 171 −→ 74 −→ 73 −→ 110
−→ 179 −→ 0
Vehicle 12: 0 −→ 155 −→ 4 −→ 139 −→ 39 −→ 187 −→ 165 −→ 60 −→ 83 −→ 199 −→ 114 −→ 8
−→ 174 −→ 125 −→ 45 −→ 17 −→ 113 −→ 0
Vehicle 13: 0 −→ 70 −→ 30 −→ 160 −→ 128 −→ 188 −→ 161 −→ 103 −→ 9 −→ 164 −→ 34 −→ 33
−→ 0
Vehicle 14: 0 −→ 185 −→ 78 −→ 135 −→ 35 −→ 136 −→ 71 −→ 66 −→ 20 −→ 122 −→ 51 −→ 0
Vehicle 15: 0 −→ 32 −→ 131 −→ 181 −→ 63 −→ 126 −→ 90 −→ 57 −→ 145 −→ 41 −→ 186 −→ 56
−→ 23 −→ 67 −→ 0
Vehicle 16: 0 −→ 121 −→ 29 −→ 25 −→ 54 −→ 65 −→ 11 −→ 124 −→ 47 −→ 168 −→ 36 −→ 143
−→ 49 −→ 64 −→ 0
Vehicle 17: 0 −→ 43 −→ 14 −→ 38 −→ 140 −→ 86 −→ 46 −→ 0