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A Hybrid Variable Neighborhood Search Approach for the Multi-Depot
Green Vehicle Routing Problem
Mir Ehsan Hesam Sadatia,b, and Bülent Çataya,b
a Sabanci University, Faculty of Engineering and Natural Sciences, Istanbul, Turkey
b Smart Mobility and Logistics Lab, Sabanci University, Istanbul, Turkey
Abstract: The Multi-Depot Green Vehicle Routing Problem (MDGVRP) is an extension of the
well-known Green Vehicle Routing Problem (GVRP) where a fleet of alternative fuel-powered
vehicles (AFVs) are used to serve the customers. GVRP consists of determining AFV tours such
that the total distance travelled is minimum. The AFVs depart from the depot, serve a set of
customers, and complete their tours at the depot without exceeding their driving range and the
maximum tour duration. AFVs may refuel en-route at public refueling stations. In MDGVRP, the
AFVs are dispatched from different depot locations and may refuel during the day at any depot or
refueling station. We formulate MDGVRP as a mixed integer linear programming model and
develop a hybrid General Variable Neighborhood Search and Tabu Search approach by proposing
new problem-specific neighborhood structures to solve the problem effectively. We assess the
performance of our method using the GVRP dataset from the literature. Our results show that the
proposed method can provide high quality solutions in short computation times. Then, we extend
these instances to the multi-depot case and compare our solutions for small-size instances with the
optimal solutions. We also report our results for large-size problems and investigate the trade-offs
associated with operating multiple depots and adopting different refueling policies to provide
further insights for both academicians and practitioners.
Keywords: Green vehicle routing problem, multi-depot, variable neighborhood search, tabu
search, refueling.
Corresponding author. msadati@sabanciuniv.edu.
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1. Introduction
Transportation activities are the largest source of greenhouse gas (GHG) emissions in the US and
fourth largest globally (www.epa.gov). They were responsible for 27% of total (GHG) in the EU
in 2017. GHG emissions had increased by 2.2 % in 2017 compared to 2016, mainly because of
higher emissions from road transport. Almost 72% of the total transport-related GHG emissions
were caused by road transport, of which 9% were from light commercial vehicles and 19% from
heavy-duty vehicles (www.eea.europa.eu). In comparison to its 1990 levels, GHG emissions
should fall by around two thirds by 2050 in order to meet 60% emission reduction target of
European Commission (EC Transport White Paper, 2011). So, the shift to cleaner vehicles in
public and freight transport is a must in order to meet the targets and achieve the desired reductions
in GHG emissions to avoid dangerous levels of global warming.
Nowadays, many logistic companies add their fleets with green vehicles that run with alternative
fuel (such as ethanol, natural gas, electricity) instead of fossil fuel in order to lessen the adverse
effects of their operations on the environment. Consequently, route planning of these alternative
fuel vehicles (AFVs) has received increasing attention in the Vehicle Routing Problem (VRP)
literature over the last decade. In the classical VRP, the fuel tank capacity of the vehicles is
assumed unlimited since the vehicles can refuel easily and fast at any public gas station. However,
green vehicles need alternative fuel stations (AFSs) for refueling, which are scarce (Koç and
Karaoglan, 2016). The range anxiety of the AFVs and the scarcity of the AFSs make the resulting
VRP more complex to model and solve.
The Green Vehicle Routing Problem (GVRP) was proposed by Erdoğan and Miller-Hooks (2012)
as an extension of the Capacitated VRP (CVRP) where a fleet of AFVs was used to serve the
customers instead of internal combustion engine vehicles (ICEVs). In this study, we address the
Multi-Depot GVRP (MDGVRP), which is a relevant problem for many logistics companies that
operate multiple regional depots for their last-mile delivery operations and employ AFVs to cut
down their emissions and reduce fuel costs. In MDGVRP, the AFVs depart from the depot, serve
a set of customers, and complete their tours at the depot without exceeding their driving range and
the maximum tour duration. All customers must be served by exactly one AFV and AFVs may
refuel at depots and public refueling stations when they run out of fuel. The objective is to
minimize the total distance travelled. Note that the existence of multiple depots increases the
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difficulty of the problem significantly as each subset of customers assigned to a depot requires
solving a separate GVRP. Since the utilization AFVs with limited autonomy makes the GVRP a
challenging problem, its extension to MDGVRP brings even more complexity to the problem.
To solve this problem, we develop a hybrid algorithm that combines Variable Neighborhood
Search (VNS) with Tabu Search (TS). The proposed algorithm applies TS with multiple, problem
specific neighborhood structures as the local search mechanism within the general VNS (GVNS)
scheme and is referred to as GVNS/TS. We perform an extensive experimental study to investigate
the performance of the proposed method and provide insights for both researchers and
practitioners. Various metaheuristic algorithms have been successfully employed for solving
different VRP variants; however, VNS stands out with its simple algorithmic structure, ease of
implementation, and good performance despite the use of very few parameters. Furthermore,
allowing non-improving moves in a multiple neighborhood framework offers more potential for
achieving high quality solutions compared to other simple local search methods such as Variable
Neighborhood Descent (VND). These factors constitute the main motivation behind our
methodology selection.
Our contributions may be summarized as follows: (i) we introduce MDGVRP and formulate its
MILP model; (ii) we propose an efficient hybrid GVNS/TS method to solve it by implementing
new problem specific neighborhood structures; (iii) we present an extensive computational study
to evaluate the performance of the proposed method and provide managerial insights, (iv) we
present new benchmark results to the literature. The remainder of this paper is organized as
follows: Section 2 provides a review of related literature. Section 3 presents the mathematical
formulation of the problem. Section 4 details the proposed GVNS/TS solution methodology.
Section 5 presents the computational study and discusses the numerical results. Finally, conclusion
remarks are provided in the last section.
2. Literature review
In the GVRP introduced by Erdoğan and Miller-Hooks (2012) the AFVs were allowed to refuel
en-route in public refueling stations and the tank was assumed to be fully filled in constant time.
The authors modelled the mixed integer linear programming (MILP) formulation of the problem
where the objective is to minimize the total distance travelled and proposed two construction
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heuristics to solve it. Schneider et al. (2014) investigated the utilization of an electric vehicle (EV)
fleet within the context of green vehicles and introducing the Electric VRP with Time Windows
(EVRPTW). EVRPTW is similar to GVRP; however, the recharge duration was not constant and
was a linear function of the total energy transferred. The battery was assumed to be fully recharged
at the station. The authors proposed a Variable Neighborhood Search and Tabu Search approach,
and solved the problems they generated based on Solomon (1987) data as well as the GVRP data
of Erdoğan and Miller-Hooks (2012). Schneider et al. (2015) generalized GVRP by introducing
the Vehicle Routing Problem with Intermediate Stops and proposed an Adaptive Variable
Neighborhood Search (AVNS) to solve it. Bruglieri et al. (2016) proposed an efficient formulation
of GVRP by implicitly considering the AFSs and reducing the graph by eliminating dominated
AFSs that can be visited between each pair of customers.
Montoya et al. (2016) presented an efficient two-phase heuristic to tackle GVRP. In the first phase,
they construct a pool of routes by employing a set of “route-first cluster-second” heuristics that
optimally inserts the refueling stations in the routes. Then, in the second phase, they solve the set
partitioning problem using the routes stored in the pool. Koç and Karaoglan (2016) developed a
metaheuristic-based exact algorithm for solving GVRP. Their approach couples Simulated
Annealing with a branch-and-bound method.
Leggieri and Haouari (2017) formulated a new mathematical model of GVRP and introduced a
reduction mechanism. They also presented a practical solution approach that could solve medium-
size instances optimally. Bruglieri et al. (2019) proposed a two-phase path-based solution method
for solving GVRP. In the first phase, they generate all feasible paths, and then in the second phase,
they combine them to generate the final routes by solving a MILP model. The authors also
proposed dominance rules to eliminate the dominated paths and decrease the size of the problem.
Various variants of EVRPTW have been studied following the study of Schneider et al. (2014). In
Keskin and Çatay (2016) and Bruglieri et al. (2015) the authors relaxed the full-recharge
restriction, and proposed mathematical programming models and solution methods that allow
partial recharge of the battery with any quantity. Desaulniers et al. (2016) developed a branch-
price-and-cut algorithm to solve EVRPTW by adopting full and partial strategies allowing single
and multiple recharges en-route.
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In Goeke and Schneider (2015), the authors extended EVRPTW by considering a mixed fleet of
internal combustion engine vehicles (ICEVs) and EVs. Hiermann et al. (2016) also considered a
mixed fleet that consisted of different types of EVs. Montoya et al. (2017) proposed nonlinear
charging functions whereas Felipe et al. (2014) allowed fast charging stations within the context
of Electric VRP (EVRP) without customer time windows. Keskin and Çatay (2018) also
investigated the impact of fast charging stations in EVRPTW and observed that they can
significantly improve the route plans when customer time windows are narrow.
Other variants of EVRP addressed in the recent literature involve battery swapping (Yang and Sun,
2015; Hof et al., 2017), heterogeneous fleet (Macrina et al., 2018; Hiermann et al., 2019), location
routing (Hof et al., 2017; Schiffer and Walther, 2017), non-linear charging function (Froger et al.,
2019), two-echelons (Jie et al., 2019), the effect of ambient temperature (Rastani et al., 2019),
capacitated recharging stations (Froger et al., 2017; Keskin et al., 2019), stochastic waiting times
at recharging stations (Keskin et al., 2021), and recharging during service (Cortés-Murcia et al.,
2019).
In a recent study, Koyuncu and Yavuz (2019) presented two alternative formulations that involved
various practical settings including refueling the vehicle at customer locations and stations,
different refueling policies, mixed fleet, and investigated their performances. For a detailed review
of the GVRP and EVRP literature we refer the reader to Erdelić and Carić (2019) and Asghari and
Al-e-hashem (2020).
Although the research on GVRP has been rapidly growing in the transportation science and
operations research community during the last decade, the logistics operations using multiple
depots have attracted little attention. Multi-depot VRP (MDVRP) is an extensively studied
problem in the VRP literature due to its practicality in many industries (Escobar et al., 2014b; Tu
et al., 2014; Masmoudi et al., 2016; Zhen et al., 2020; Sadati et al., 2020a; Sadati et al., 2020b). A
comprehensive review of MDVRP may be found in Montoya-Torres et al. (2014) and Ramos et
al. (2020).
3. Problem formulation
Our model extends the GVRP formulation of Erdoğan and Miller-Hooks (2012) to the multiple-
depot setting. For ease of understanding, we follow the same mathematical formulation and
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notation convention. Let denote a complete directed graph where and represent
the set of vertices and set of arcs, respectively. consists of three subsets: set of depots
, set of customers , and set of refueling stations
. is the service time (refuelling time) at customer (station) i.
Similar to Erdoğan and Miller-Hooks (2012), we assume that the fuel tank of the vehicle is filled
to full capacity and refueling time is constant. A nonnegative cost and travel time are
associated with each arc The fleet consists of identical vehicles with fuel consumption rate
and fuel tank capacity . The vehicles depart from and return to the same depot. The objective
is to design a set of routes that visit each customer once such that the total travel distance is
minimized. A maximum route duration is imposed on the journey time of the vehicles. A
refueling station may be visited multiple times by the same or different vehicles. So, for each
potential visit we add a copy of the station and create ′ ′′, where ′ ′ and ′
′.
Fig. 1 depicts an illustrative example which involves 12 customers (C1-C12), two refuelling
stations (S1 and S2), and two depots (D1 and D2). The depots can also be used for refuelling. We
set and . The values next to the cylindrical shapes indicate the fuel level of the tank
upon the arrival of the AFV at the corresponding node. AFV#1 departs from depot D1, serves
customers C1 thru C3, and then returns to D1 for refueling. After refueling, it continues its route
by visiting C4-C6 and returns to D1 at the end of its tour. AFV#2 departs from D2, serves C7 thru
C9, refuels at S2 before visiting C10, and then returns to its depot D2. Finally, AFV#3 departs
from D2, visits C11 and C12, and returns to D2 without needing any refuelling.
Fig. 1. An illustrative example of MDGVRP
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In what follows, we provide the MILP formulation of the problem. Table 1 summarizes the
mathematical notation.
Table 1. Mathematical notation
Sets:
Set of customers
Set of depots
Set of refueling stations
Set of depots, customers and refueling stations
′
Set of all refueling stations including dummy stations
Set of customers and all refueling stations ( ′
′
Set of all vertices (′
Parameters:
Distance from node to node
Travel time from node to node
Vehicle fuel tank capacity
Vehicle fuel consumption rate
Service time at node (service time if , refueling time if ′)
Maximum tour duration
A sufficiently large constant (
)
Decision variables:
1 if node is visited following node ; 0 otherwise
1 if node is visited by a vehicle assigned to depot ; 0 otherwise
Vehicle fuel level upon its arrival at node ( if ′)
Arrival time at node ( for due to initial refueling at depot)
min
′
′
(1)
subject to:
′
(2)
′
(3)
(4)
′
′
(5)
′
′
′
(6)
8
,
(7)
,
(8)
, , ,
(9)
, , ,
(10)
′, ,
(11)
′, ,
(12)
,
(13)
′, ,
(14)
′
(15)
′
(16)
′
(17)
,
(18)
′
(19)
′
(20)
The objective function (1) minimizes the total distance. Constraints (2) and (3) make sure that each
customer is visited exactly once. Constraints (4) assigns each customer to exactly one depot.
Constraints (5) restrict the visit to a station by at most once. Constraints (6) are the flow balance
constraints. Constraints (7) and (8) guarantee that a vehicle travels from a node to a depot or from
a depot to a node only if that node is assigned to that depot. Constraints (9) and (10) state that two
nodes (customer or refueling station node) must be assigned to the same depot if they are visited
consecutively by the same vehicle. Constraints (11)(13) keep track of time and ensure that the
maximum tour duration is not exceeded. They also prevent the formation of subtours. Constraints
(14) keep track of tank fuel level at each customer while constraints (15) guarantee that the vehicle
never runs out of fuel. Constraints (16) fill up the tank when the vehicle visits a refueling station.
Finally, constraints (17)(20) define the decision variables.
The Capacitated VRP (CVRP) is a special case of MDGVRP where a single depot is used instead
of multiple depots and the fuel tank capacity (driving range) of the vehicles is unlimited .
Since CVRP is NP-hard (Toth and Vigo, 2002), MDGVRP is NP-hard as well.
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4. Solution methodology
We present a hybrid GVNS/TS algorithm for solving MDGVRP. The basic component of the
algorithm is General Variable Neighborhood Search (GVNS) developed by Mladenović and
Hansen (1997). The neighborhood structures in GVNS/TS are changed systematically through the
VNS approach and the local search is applied by implementing TS, which was originally
developed by Glover (1986). VNS has a simple algorithmic structure and involves very few
parameters; however, it is one of the most successful metaheuristics for solving hard combinatorial
optimization problems. It has been applied to various routing problems such as the Electric VRP
with Time Windws (Schneider et al., 2014), VRP with Simultaneous Pickup and Delivery with
Time Limit (Polat et al., 2015), VRP with Two-Dimensional Loading Constraints (Wei et al.,
2015)VRP with Divisible Deliveries and Pickups (Polat, 2017), Clustered VRP (Hintsch and
Irnich, 2018), and the Dial-a-Ride Problem with Electric Vehicles (Masmoudi et al., 2018). The
combination of VNS with TS was also used in the VRP context for solving the Capacitated
Location Routing Problem (Escobar et al., 2014a), VRP with clustered backhauls and 3D loading
constraints (Bortfeldt et al., 2015), VRP with Drones and En-route Operations (Schermer et al.,
2019), and Long-Term Car Pooling Problem (Mlayah et al., 2020). A recent survey shows that TS
and VNS are the two most frequently employed metaheuristic approaches for solving the VRP and
its variants. We refer the reader to Elshaer and Awad (2020) for details.
4.1. General Variable Neighborhood Search
In our proposed method, we first generate an initial solution by using Clarke and Wright (CW)
savings algorithm (Clarke and Wright, 1964). An initial feasible solution for MDGVRP is created
by first assigning customers to their nearest depot and then applying CW to each depot and the
assigned customers. We implement the parallel version of CW where multiple vehicle routes are
constructed simultaneously while respecting the driving range and tour duration limitations.
Next, we implement GVNS/TS by means of a set of neighborhood structures
which comprises the shaking phase, and another set of neighborhood structures
which comprises the local search phase. In shaking, solution ′ is generated randomly in the first
neighborhood of , and then TS is performed by applying the first neighborhood to obtain a
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new solution
. If
is feasible and its objective value is better than the incumbent solution ,
is replaced by
and TS is performed by continuing with neighborhood . Otherwise, is
incremented by 1 and TS is performed for
by next neighborhood structure ( ). This
procedure is continued until all TS neighborhood operators are explored ( ) by restarting
from neighborhood whenever an improved feasible solution is obtained. During the TS
procedure, if a new incumbent solution is obtained, the index is reset to 1 to start from the first
shaking neighborhood operator. Otherwise, is incremented by 1 ( ) and GVNS/TS
restarts from the incumbent solution . This procedure is repeated and terminates if the solution
does not improve for a pre-specified number of consecutive iterations () or a pre-
specified limit on the number of iterations () has been reached. The pseudocode of GVNS/TS
is presented in Algorithm 1.
Algorithm 1. The pseudocode of GVNS/TS
// Set of the neighborhood structures for shaking
// Set of the neighborhood structures for tabu search
1: Set // Generate initial solution using Clarke and Wright savings algorithm
2: Set
3: repeat
4: for to do
5: select a random solution ′from the ℎ neighborhood of // Shaking
6: for to do
7: Find the best neighbor
of ′in ′ // Tabu Search
8: if
is feasible and
9: Update
10: Set
11: Set
12: else
13: Set
14: end if
15: Set ′
16: end for
17: if
18: Set
19: else
20: Set
21: end if
22: Set
23: end for
24: until Stopping condition
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4.2. Shaking
At each iteration of the shaking phase of GVNS/TS a random solution is generated using one of
the neighborhood operators . We use the following five neighborhoods: 1-0 Move,
1-1 Exchange, 2-2 Exchange, 1-2 Exchange and 1-1-1 Exchange. 1-0 Move and 1-1 Exchange are
well-known neighborhood structures in the literature. The former removes a customer and inserts
between two consecutive customer and the latter swaps two customer nodes. 2-2 Exchange swaps
two pairs of consecutive customers whereas 1-2 Exchange swaps one customer with two
consecutive customers. All four are applied as both intra-route and inter-route moves. 1-1-1
Exchange is a three-way inter-route operator that exchanges the positions of three customers on
three different routes simultaneously. All of these neighborhood operators are explored in a cyclic
sequential order starting from 1-0 Move and following the order above.
4.3. Strategic oscillation for handling infeasible solutions
GVNS/TS is carried out using strategic oscillation of Glover (2000) which allows infeasible
solutions that violate tour duration and vehicle driving range constraints. Strategic oscillation is
based on the idea of accepting infeasible solution spaces in the hope of finding a better feasible
solution in the following iterations and has been successfully employed in many studies (Cordeau
et al., 1997; Toth and Vigo, 2003). It accepts such an infeasible solution by penalizing its objective
function value as follows:
′ (21)
refers to the value of objective function (1) associated with solution S and
′ (22)
. (23)
where and indicate the violation in driving range and maximum tour duration,
respectively. and are positive penalty coefficients corresponding to infeasibilities, and
. If the current solution is feasible, then we set ′ . After each iteration, the
penalty coefficients are updated to find feasible and infeasible solutions with approximately the
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same frequency. We set if is infeasible with respect to driving range constraint;
and otherwise
. Likewise, if is infeasible or feasible with respect to tour duration
constraint, we set or
, respectively, where is the parameter to update
and.
4.4. Tabu search
TS with multiple neighborhood structures has been effectively employed in the literature (see e.g.
Schneider et al., 2014; Soto et al., 2017; Qiu et al., 2018). So, at each iteration of GVNS/TS, we
perform a local search by applying TS using four neighborhood operators: 2-opt, 1-AddStation, 1-
DropStation and 1-SwapStation. 2-opt is a well-known operator whereas the others are problem
specific. Considering each unused station, 1-AddStation examines all the arcs where it can be
inserted and performs the best insertion. “Best insertion” here refers to the insertion of a station
into an arc which decreases the cost of the current solution most or increases it least, if no cost
improving insertion exists. In 1-DropStation, a station is removed from the route by connecting its
predecessor node to its successor node. “Best removal” is performed such that the removal of the
station from the solution decreases the cost of the current solution most or increases it least, if no
cost improving removal exists. Finally, 1-SwapStation applies 1-AddStation and 1-DropStation
simultaneously. All neighborhood operators are illustrated in Fig. 2-Fig. 4.
TS is applied by exploring the first move and a new solution
is obtained from the current
solution . If
is a feasible solution with a better objective function value than the incumbent
solution ,
will replaced by and TS is performed by reinitializing the neighborhood index to
one ( ). Otherwise, is increased by 1 and TS starts from
with next neighborhood structure
( ). This procedure is repeated until all neighborhood operators are explored ( ) in
cycling order. The TS terminates after iterations. These four neighborhood operators are
explored in a cyclic sequential order starting from 2-opt and end with 1-SwapStation.
In order to prevent cycling in exploring the solution space, tabu conditions are used associated
with each neighborhood. The tabu condition of each move is defined as follows: A (the) node(s)
whose position is (are) changed (swapped) by a move cannot be displace (re-swapped) by the same
move.
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(a)
(b)
Fig. 2. 1-AddStation operator: (a) current solution, (b) solution after 1-AddStation move
(a)
(b)
Fig. 3. 1-DropStation operator: (a) current solution, (b) solution after 1-DropStation move
(a)
(b)
Fig. 4. 1-SwapStation operator: (a) current solution, (b) solution after 1-SwapStation move
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The tabu condition is repealed at the end of tabu duartion or if the aspiration criterion is satisfied,
i.e. if the incumbent solution is improved. Furthermore, we implement the diversification strategy
of Gendreau et al. (2008) that aims at searching the unexplored regions of the solution space. This
procedure is as follows: let
be the current and
be the new solution obtained by the
neighborhood.
If the objective function value of the new solutions is greater than equal to the objective function
value of current solution (
), then
′
′
is added to ′
as a
penalty, where and represent the number of routes and the number of customers visited in
,
respectively. indicates the number of times that customer/station is moved by operator until
then and is the total number of iterations carried out so far.
Since an exhaustive search of these neighborhoods require significant computational effort we
implement granular neighborhood search of Toth and Vigo (2003). In granular search, we accept
the arcs with shorter distances (cost) than the threshold value defined as
, where denotes
the objective function value of the initial solution. The acceptable arcs are rebuilt after iterations
by updating the granularity threshold to ′′
′′, where , ′ and ′ are the objective function
value of the best solution obtained by GVNS/TS, the number of routes, and number of customers,
respectively.
5. Experimental study
In this section, we first tune the parameters of GVNS/TS by performing initial experiments using
a subset of the GVRP instances generated by Erdoğan and Miller-Hooks (2012). Next, we evaluate
the performance of GVNS/TS using the whole set of GVRP instances by comparing its results
with those from the literature. Then, we modify the GVRP instances to create MDGVRP instances
and solve them with GVNS/TS. For small-size data set, we use CPLEX to evaluate its
performance. For large-size data set, we report our best solutions as benchmarks for future
research. All experiments were carried out on a computer equipped with Intel Core i7-8700 3.2
GHz CPU and 32 GB RAM. The algorithm was coded in C# of Microsoft® Visual Studio 2019.
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5.1. Parameter tuning
We selected 10 instances randomly from the GVRP data set and performed 10 runs for each by
considering different values for each parameter. The selected instances are 20c3sU1-U2, 20c3sU9-
U10, 20c3sC1-C2, 20c3sC9-C10, S1_10i6s, and S2_10i6s. For each parameter, we calculated the
average percentage deviation (over 10 runs) from the best solution (Δ%) and set the value of the
corresponding parameter to the value that produced the smallest Δ%. We repeated this procedure
until parameters had been tuned. The parameters, their considered values, and the average percent
deviation for each value are reported in Table 2. The values selected are indicated in bold. If two
parameter values yielded the same deviation, we favored the smaller value.
Table 2. Parameter tuning
Parameter
Definition
Values
(Δ%)
Tabu duration
1
(0.12)
2
(0.05)
5
(0.02)
10
(0.00)
20
(0.00)
40
(0.00)
Penalty for fuel capacity
violation
0.2
(0.04)
0.3
(0.04)
0.5
(0.00)
1
(0.00)
2
(0.00)
-
Penalty for maximum tour
duration violation
0.2
(0.03)
0.3
(0.02)
0.5
(0.00)
1
(0.00)
2
(0.00)
-
Parameter to update
-
-
0.1
(0.09)
0.25
(0.01)
0.5
(0.00)
0.75
(0.00)
Total number of iterations
3000
(0.90)
5000
(0.42)
10000
(0.00)
12000
(0.00)
15000
(0.00)
-
Number of tabu search
iterations
10
(0.14)
20
(0.08)
25
(0.08)
50
(0.00)
100
(0.00)
200
(0.00)
Maximum number of non-
improving iterations
100
(0.26)
200
(0.10)
300
(0.05)
500
(0.00)
1000
(0.00)
2000
(0.00)
5.2. Computational results for GVRP instances
We first tested the performance of GVNS/TS using the GVRP instances of Erdoğan and Miller-
Hooks (2012) as well as Andelmin and Bartolini (2017) instances. In standard GVRP instances
some customers may be infeasible and cannot be visited because of the given driving range (fuel
tank capacity) of the vehicle and maximum tour duration constraints. Therefore, these infeasible
customers are identified and discarded from the problem. So, we adopted the same approach.
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5.2.1. Numerical results on small-size GVRP instances
The small-size data set of Erdoğan and Miller-Hooks (2012) includes 40 instances involving 20
customers and the number of refueling stations vary between 2 and 10. Table 3 provides our results
and compare them with those obtained by the state-of-the-art methods appeared in the literature.
In this table, the first three columns “Instance”, “n”, and “v” provide the instance name, the
number of customers, and fleet size, respectively. The column “Optimal” shows the optimal
solutions reported by Bruglieri et al. (2019) whereas the three columns under “GVNS/TS” report
our solutions where “Best”, “Avg”, and “t(s)” denote the best solution, average solution, and CPU
time of the best solution in seconds, respectively. In the following columns, “SSG”, “SSH”, and
“MGMV” refer to Hybrid Variable Neighborhood Search/Tabu Search of Schneider et al. (2014),
Adaptive Variable Neighborhood Search of Schneider et al. (2015), and Multi-Space Sampling
Heuristic of Montoya et al. (2016), respectively. The two columns “t(s)” and “Δ%” under each
algorithm show the CPU time of the best solutions and percentage deviations of our solutions from
those of the best solutions of the corresponding algorithms, respectively. All results are best of 10
runs and all CPU times are in seconds.
The results in Table 3 show that our proposed GVNS/TS was able to find the optimal solution in
all of 40 instances. Moreover, the average and best solutions are same, i.e., GVNS/TS converged
to the optimal solution in all 10 runs, which is a clear indication of the robustness of the algorithm.
The comparison with SSG, SSH, MGMV reveals that our method outperforms SSG and performs
as good as SSH and MGMV. Although the average results are not reported in the paper, we note
that our overall average for 40 instances slightly outperforms SSH and MGMV by 0.15% and
0.01%, respectively (SSG did not provide their average results). On the other hand, our CPU times
are significantly lower. While average run time is 1.26 seconds, the run times of SSG, SSH, and
MGMV are 39.00, 10.20, and 4.20 seconds, respectively.
For a fair comparison, we specify the CPU (Single Thread Performance) used by each method and
report its CPU Mark provided by Passmark below the table (www.cpubenchmark.net). The speed
of the CPU is proportional to the reported value. According to these values, we see that our CPU
is 2.52 times (15134/6007) faster than that of MGMV whereas our average run time is 3.34 times
faster (4.20/1.26). Similarly, while our CPU is 4.07 times faster than that of SSG and SSH, our
average run time is 30.95 and 8.10 times faster than that of SSG and SSH, respectively. So, we
17
conclude that our proposed method outperforms all benchmark methods in terms of computational
effort.
Table 3. Results for small-size Erdoğan and Miller-Hooks data
Instance
Optimal
GVNS/TS
SSG
SSH
MGMV
Best/Avg
t(s)a
t(s)b
Δ%
t(s)b
Δ%
t(s)c
Δ%
20c3sU1
20
6
1797.49
1797.49
1.26
41.40
0.00
9.60
0.00
4.80
0.00
20c3sU2
20
6
1574.77
1574.78
0.63
38.40
0.00
9.00
0.00
4.20
0.00
20c3sU3
20
6
1704.48
1704.48
0.63
38.40
0.00
7.80
0.00
4.20
0.00
20c3sU4
20
5
1482.00
1482.00
1.26
39.00
0.00
10.20
0.00
4.20
0.00
20c3sU5
20
6
1689.37
1689.37
1.26
40.20
0.00
10.80
0.00
4.20
0.00
20c3sU6
20
6
1618.65
1618.65
0.63
40.20
0.00
9.00
0.00
4.20
0.00
20c3sU7
20
6
1713.66
1713.66
1.26
38.40
0.00
11.40
0.00
4.20
0.00
20c3sU8
20
6
1706.50
1706.50
1.26
40.20
0.00
9.60
0.00
4.20
0.00
20c3sU9
20
6
1708.81
1708.82
0.63
39.60
0.00
11.40
0.00
4.20
0.00
20c3sU10
20
4
1181.31
1181.31
0.63
38.40
0.00
13.80
0.00
4.20
0.00
20c3sC1
20
4
1173.57
1173.57
0.63
37.20
0.00
22.80
0.00
4.20
0.00
20c3sC2
19
5
1539.97
1539.97
0.63
34.80
0.00
12.60
0.00
4.80
0.00
20c3sC3
12
3
880.20
880.20
0.63
15.00
0.00
9.00
0.00
2.40
0.00
20c3sC4
18
4
1059.35
1059.35
0.63
31.80
0.00
13.80
0.00
3.60
0.00
20c3sC5
19
7
2156.01
2156.01
0.63
36.00
0.00
8.40
0.00
6.00
0.00
20c3sC6
17
8
2758.17
2758.17
0.63
42.60
0.00
8.40
0.00
4.80
0.00
20c3sC7
6
4
1393.99
1393.99
0.02
10.80
0.00
2.40
0.00
3.60
0.00
20c3sC8
18
9
3139.72
3139.72
1.26
37.20
0.00
4.80
0.00
7.20
0.00
20c3sC9
19
6
1799.94
1799.94
0.63
36.00
0.00
9.60
0.00
6.00
0.00
20c3sC10
15
8
2583.42
2583.42
0.63
27.00
0.00
5.40
0.00
4.20
0.00
S1_2i6s
20
6
1578.12
1578.12
1.26
42.60
0.00
9.60
0.00
4.20
0.00
S1_4i6s
20
5
1397.27
1397.27
1.26
45.00
0.00
9.60
0.00
4.20
0.00
S1_6i6s
20
5
1560.49
1560.49
1.26
43.80
0.00
12.00
0.00
4.20
0.00
S1_8i6s
20
6
1692.32
1692.32
1.26
44.40
0.00
10.20
0.00
4.20
0.00
S1_10i6s
20
4
1173.48
1173.48
1.26
42.60
0.00
14.40
0.00
4.20
0.00
S2_2i6s
20
6
1633.10
1633.10
0.63
45.00
0.00
11.40
0.00
5.40
0.00
S2_4i6s
19
6
1505.07
1505.07
0.63
52.80
-1.82
8.40
0.00
5.40
0.00
S2_6i6s
20
7
2431.33
2431.33
1.89
46.80
0.00
7.80
0.00
4.20
0.00
S2_8i6s
16
7
2158.35
2158.35
1.26
34.20
0.00
5.40
0.00
3.60
0.00
S2_10i6s
16
5
1585.46
1585.46
1.26
36.60
-19.05
9.00
0.00
3.60
0.00
S1_4i2s
20
6
1582.20
1582.21
1.26
37.80
0.00
7.80
0.00
4.20
0.00
S1_4i4s
20
5
1460.09
1460.09
1.26
40.80
0.00
9.60
0.00
4.20
0.00
S1_4i6s5
20
5
1397.27
1397.27
1.26
45.00
0.00
9.60
0.00
4.20
0.00
S1_4i8s
20
5
1397.27
1397.27
1.26
49.20
0.00
10.20
0.00
4.20
0.00
S1_4i10s
20
5
1396.02
1396.02
1.89
51.00
0.00
13.80
0.00
4.20
0.00
S2_4i2s
18
4
1059.35
1059.35
0.63
30.60
0.00
13.80
0.00
3.60
0.00
S2_4i4s
19
5
1446.08
1446.08
0.63
36.00
0.00
12.60
0.00
5.40
0.00
S2_4i6s5
20
5
1434.14
1434.14
1.26
41.40
0.00
12.00
0.00
4.80
0.00
S2_4i8s
20
5
1434.14
1434.14
1.26
45.00
0.00
12.00
0.00
4.80
0.00
S2_4i10s
20
5
1434.13
1434.13
1.26
46.80
0.00
14.40
0.00
5.40
0.00
Average
1635.43
1635.43
1.26
39.00
-0.52
10.20
0.00
4.20
0.00
a: Intel Core i7-8700 with 3.2 GHz (CPU Mark: 2696)
b: Intel Core i5-750 with 2.67GHz (CPU Mark: 1146)
c: Intel Xeon E5410 with 2.33 GHz (CPU Mark: 997)
18
5.2.2. Numerical results on large-size GVRP instances
The large-size instances proposed by Erdoğan and Miller-Hooks (2012) involve 111 to 500
customers and 21 to 28 refueling stations. The results are given in Table 4. Similar to
Table 3, the first three columns provide the instance name, the number of customers, and fleet size,
respectively. The columns “BKS” and “Ref.” indicate the best-known solution reported in the
literature and the corresponding reference, respectively. The columns under “GVNS/TS” report
our best solutions (“Best”), average solutions (“Avg”), the CPU times of the best solutions in
seconds (“t(s)”), and percentage deviation of best (“Best Δ%”), and average (“Avg Δ%”) solutions
from BKSs, respectively.
Table 4. Results for large-size Erdoğan and Miller-Hooks data
GVNS/TS
Instance
BKS
Ref.
Best
Avg
t(s)
Best Δ%
Avg Δ%
111c_21s
109
17
4770.47
SSH
4772.43
4801.42
224
0.04
0.65
111c_22s
109
17
4774.65
MGMV
4778.29
4802.69
193
0.08
0.59
111c_24s
109
17
4767.14
SSH
4768.20
4785.26
168
0.02
0.38
111c_26s
109
17
4767.14
SSH
4768.20
4781.11
145
0.02
0.29
111c_28s
109
17
4765.52
SSH
4767.03
4779.81
194
0.03
0.30
200c_21s
192
31
8839.62
MGMV
8848.71
8902.73
395
0.10
0.71
250c_21s
237
37
10482.52
MGMV
10496.11
10548.74
476
0.13
0.63
300c_21s
283
44
12367.60
MGMV
12386.39
12452.28
977
0.15
0.68
350c_21s
329
50
14073.34
MGMV
14100.29
14180.61
1303
0.19
0.76
400c_21s
378
59
16660.20
MGMV
16695.89
16751.32
1543
0.21
0.55
450c_21s
424
65
18241.48
MGMV
18289.31
18332.08
1751
0.26
0.50
500c_21s
471
73
20496.50
MGMV
20562.18
20665.92
2153
0.32
0.83
Average
10417.18
10436.09
10482.00
794
0.13
0.57
We observe that the average deviation of the solutions obtained by GVNS/TS from BKS is only
0.13% in the best of 10 runs and 0.57% on the average of 10 runs. Our average CPU time is 794
seconds. So, we conclude that the proposed algorithm converges to very good solutions in
reasonable computation times.
Next, we test the performance of our GVNS/TS using the dataset created by Andelmin and
Bartolini (2017) by removing some customers from the large-size instances of Erdoğan and Miller-
Hooks (2012). It consists of 40 instances involving 50 to 100 customers and categorized as AB1
19
and AB2 subsets. The customer and refueling stations are same in both subsets; however, the speed
and fuel consumption rate of the vehicles in AB2 instances are 60 miles per hour (mph) and 0.2137
gallons per mile, respectively whereas they are set to 40 mph and 0.2 gallons per mile in AB1 and
original data. Similar to original instances, some customers may be infeasible in the AB1 data set
and cannot be visited because of the given driving range (fuel tank capacity) of the vehicle and
maximum tour duration constraints. On the other hand, all customers are feasible in the AB2 data
set.
In Table 5, we provide our results for all AB instances and compare them with the results reported
by Andelmin and Bartolini (2017) and Bruglieri et al. (2019). In this table, the first three columns
provide the instance name, the number of customers and fleet size, respectively. The following
columns “AB” and “BMPP” refer to Andelmin and Bartolini (2017) and Bruglieri et al. (2019),
respectively. Similar to the previous tables, the two columns under each algorithm show the best
solution and CPU times of the corresponding algorithms in seconds; and the following five
columns under “GVNS/TS” report our best solutions, average solutions, the CPU times of the best
solutions in seconds, and percentage deviation of our best and average results from BKSs,
respectively.
All results are best of 10 runs and all CPU times are in seconds. Note that Andelmin and Bartolini
(2017) proposed an exact approach which provided the optimal solution in most of the instances.
Bold values in AB and GVNS/TS “Best” columns indicate that the given result is optimal. The
results show that GVNS/TS found the optimal solution in 16 instances. Moreover, the average
deviation of the solutions obtained by GVNS/TS from those of AB is only 0.06% and 0.18% for
the best and average of 10 runs, respectively. When we compare our results with those of BMPP
we observe that GVNS/TS outperforms BMPP in all of instances and the average improvement is
2.22%. Furthermore, the average CPU time of GVNS/TS is 82 seconds compared to 8594 seconds
of AB and 1225 seconds of BMPP. Although our CPU is faster these differences in run times are
remarkable. Overall, these results show the superior performance of GVNS/TS in solving GVRP
instances with regard to both computational time and solution quality. In what follows, we test its
performance on small-size MDGVRP instances.
20
Table 5. Results for Andelmin and Bartolini data
Instance
AB
BMPP
GVNS/TS
Best
t(s)a
Best
t(s)b
Best
Avg
t(s)
Best Δ%
Avg Δ%
AB101
50
9
2566.62
1391
2584.88
211
2566.62
2571.05
13
0.00
0.17
AB102
50
10
2876.26
2266
2908.77
381
2876.26
2880.18
14
0.00
0.14
AB103
50
10
2804.07
1621
2869.37
379
2804.07
2807.35
14
0.00
0.12
AB104
47
9
2634.17
488
2660.33
185
2635.14
2637.70
16
0.04
0.13
AB105
73
14
3939.96
11082
3961.24
637
3939.96
3941.99
37
0.00
0.05
AB106
74
13
3915.15
3781
3953.20
610
3917.59
3920.40
27
0.06
0.13
AB107
75
13
3732.97
10905
3884.79
899
3737.18
3740.70
26
0.11
0.21
AB108
75
13
3672.40
4443
3760.92
609
3675.79
3678.30
20
0.09
0.16
AB109
75
13
3722.17
5679
3851.97
893
3723.29
3728.65
32
0.03
0.17
AB110
75
13
3612.95
9806
3753.00
796
3635.42
3640.81
38
0.62
0.77
AB111
71
14
3996.96
8484
4007.44
851
4003.28
4007.17
40
0.16
0.26
AB112
100
18
5487.87
11030
5844.05
2192
5495.82
5497.99
126
0.14
0.18
AB113
100
17
4804.62
12276
4980.98
1829
4810.16
4815.15
139
0.12
0.22
AB114
100
18
5324.17
11198
5505.54
2656
5330.29
5335.95
131
0.11
0.22
AB115
100
17
5035.35
13236
5338.85
2773
5040.68
5044.48
127
0.11
0.18
AB116
100
16
4511.64
11371
4600.60
2028
4520.19
4522.85
146
0.19
0.25
AB117
99
18
5370.28
12006
5439.55
2030
5380.49
5384.69
142
0.19
0.27
AB118
100
19
5756.88
13120
5857.64
2037
5760.19
5763.25
148
0.06
0.11
AB119
98
19
5599.96
11226
5682.33
2512
5601.29
5607.19
124
0.02
0.13
AB120
96
19
5679.81
11343
5846.51
1806
5679.81
5684.59
122
0.00
0.08
AB201
50
6
1836.25
1263
1882.75
423
1836.25
1839.52
21
0.00
0.18
AB202
50
6
1966.82
500
2008.90
443
1966.82
1969.66
16
0.00
0.14
AB203
50
6
1921.59
6149
1983.83
482
1921.59
1925.92
17
0.00
0.23
AB204
50
6
2001.70
1773
2048.12
367
2001.70
2006.03
26
0.00
0.22
AB205
75
9
2793.01
11056
2855.81
615
2793.01
2796.22
45
0.00
0.11
AB206
75
9
2891.48
12255
2976.43
594
2891.48
2894.74
35
0.00
0.11
AB207
75
8
2717.34
3145
2791.66
581
2717.34
2720.32
34
0.00
0.11
AB208
75
8
2552.18
3926
2619.92
411
2555.31
2559.09
20
0.12
0.27
AB209
75
8
2517.69
6814
2550.89
535
2517.68
2522.55
44
0.00
0.19
AB210
75
8
2479.97
9802
2538.98
573
2479.89
2482.05
33
0.00
0.08
AB211
75
9
2977.63
8358
3008.80
613
2979.19
2981.24
53
0.05
0.12
AB212
100
11
3341.43
11568
3409.74
1933
3341.43
3346.54
164
0.00
0.15
AB213
100
10
3133.24
11149
3207.99
1541
3133.24
3137.54
180
0.00
0.14
AB214
100
11
3384.28
12910
3457.66
1883
3384.28
3386.61
146
0.00
0.07
AB215
100
11
3480.52
12029
3537.11
2404
3481.98
3484.72
178
0.04
0.12
AB216
100
10
3221.78
13026
3301.22
1865
3222.16
3227.49
142
0.01
0.18
AB217
100
11
3714.94
12759
3797.22
1827
3715.19
3719.99
168
0.01
0.14
AB218
100
11
3658.17
14707
3727.27
1834
3661.29
3666.69
135
0.09
0.23
AB219
100
11
3790.71
12482
3862.63
1929
3795.09
3800.12
192
0.12
0.25
AB220
100
11
3737.88
11318
3807.88
1840
3741.26
3744.01
164
0.09
0.16
Average
3579.07
8594
3666.67
1225
3581.74
3585.54
82
0.06
0.18
a: Intel Core i7-5600U with 2.66 GHz (CPU Mark: 1859)
b: Intel Core i7-5500U with 2.4GHz (CPU Mark: 1632)
21
5.3. Computational results for MDGVRP instances
To generate MDGVRP instances we modified the GVRP instances proposed by Erdoğan and
Miller-Hooks by converting some of refueling stations to depots. For small-size MDGVRP, we
selected two depots, one is the original depot and the other is the first refueling station in the data.
In large-size MDGVRP, we selected the first two refueling stations as depots in addition to the
original depot.
5.3.1. Results for small-size data
To test the performance of GVNS/TS, we solved the small-size MDGVRP instances on IBM ILOG
CPLEX v.12.9.0 with a two-hour time limit. We used C# to call the CPLEX library using the same
computer. The results are reported in Table 6. In this table, “” represents the number of depots.
Three columns under CPLEX indicate the upper bound (“UB”), percentage optimality gap
(“Gap%”), and run time in seconds (“t(s)”), respectively. The columns “Best”, “Best Δ%”, “t(s)”
under GVNS/TS indicate the best solution obtained after 10 runs of GVNS/TS, the percentage gap
between our best solution and the CPLEX upper bound, and CPU time of the best solution in
seconds, respectively. Similar to the GVRP instances, some customers in the MDGVRP problems
cannot be served because of the limited driving range and maximum tour duration restrictions. So,
these infeasible customers are discarded from the corresponding instance.
We observe that CPLEX could solve 14 instances out of 40 to optimality within the given time
limit of two hours. The average optimality gap is 15.3% and the average run time 5311 seconds.
GVNS/TS finds the optimal solution in 14 instances and match the CPLEX upper bound in 21
instances. In the reaming five instances GVNS/TS provided better solutions than the CPLEX upper
bound (indicated in bold). Furthermore, the average computation time of GVNS/TS is only 4.27
seconds. Note that the deviation of the average of the average solutions of GVNS/TS from the
CPLEX UB in 10 runs is only -0.04%. These results further validate the performance and
robustness of the proposed GVNS/TS algorithm.
22
Table 6. Results for small-size MDGVRP instances
CPLEX
GVNS/TS
Instance
UB
Gap%
t(s)
Best
Best Δ%
t(s)
20c3sU1
20
2
6
1646.65
18.01
7200
1646.65
0.00
3.00
20c3sU2
20
2
5
1399.65
0.00
345
1399.65
0.00
3.31
20c3sU3
20
2
6
1640.55
0.00
2953
1640.55
0.00
3.48
20c3sU4
20
2
5
1338.35
0.00
5340
1338.35
0.00
3.27
20c3sU5
20
2
5
1395.60
11.10
7200
1395.6
0.00
2.46
20c3sU6
20
2
6
1402.06
0.00
1981
1402.06
0.00
2.97
20c3sU7
20
2
5
1518.18
0.00
2516
1518.18
0.00
3.45
20c3sU8
20
2
6
1493.79
0.00
5935
1493.79
0.00
3.89
20c3sU9
20
2
6
1663.46
25.69
7200
1663.46
0.00
3.95
20c3sU10
20
2
4
1115.89
0.00
51
1115.89
0.00
3.15
20c3sC1
20
2
4
1110.55
26.83
7200
1110.55
0.00
2.86
20c3sC2
19
2
5
1178.42
0.00
1564
1178.42
0.00
2.76
20c3sC3
12
2
3
880.20
0.00
16
880.20
0.00
1.90
20c3sC4
20
2
7
1640.45
21.12
7200
1640.45
0.00
3.56
20c3sC5
18
2
6
1760.20
69.66
7200
1760.20
0.00
3.49
20c3sC6
17
2
7
2270.48
9.31
7200
2213.44
-2.51
2.89
20c3sC7
6
2
3
911.01
0.00
1
911.01
0.00
0.80
20c3sC8
18
2
9
2852.19
0.00
897
2852.19
0.00
7.33
20c3sC9
19
2
5
1473.56
11.57
7200
1473.56
0.00
2.63
20c3sC10
18
2
9
2703.73
51.97
7200
2600.02
-3.84
6.19
S1_2i6s
20
2
6
1391.42
19.83
7200
1391.42
0.00
3.52
S1_4i6s
20
2
5
1194.63
10.09
7200
1194.63
0.00
3.86
S1_6i6s
20
2
5
1471.78
14.94
7200
1471.78
0.00
4.75
S1_8i6s
20
2
5
1393.45
8.70
7200
1393.45
0.00
3.14
S1_10i6s
20
2
4
1165.36
7.75
7200
1165.36
0.00
2.65
S2_2i6s
20
2
5
1509.96
36.59
7200
1509.96
0.00
5.52
S2_4i6s
19
2
4
1404.42
19.80
7200
1404.42
0.00
3.25
S2_466s
20
2
7
2341.91
0.00
389
2341.91
0.00
9.85
S2_8i6s
17
2
6
1554.22
15.46
7200
1509.22
-2.90
4.16
S2_10i6s
16
2
4
1625.28
48.11
7200
1625.28
0.00
5.78
S1_4i2s
20
2
6
1379.57
0.00
2038
1379.57
0.00
4.79
S1_4i4s
20
2
5
1263.85
0.00
1196
1263.85
0.00
2.99
S1_4i6s5
20
2
5
1194.63
10.04
7200
1194.63
0.00
4.09
S1_4i8s
20
2
5
1194.63
17.48
7200
1194.63
0.00
4.63
S1_4i10s
20
2
5
1194.63
19.43
7200
1194.63
0.00
5.71
S2_4i2s
20
2
7
1640.45
19.55
7200
1640.45
0.00
4.01
S2_4i4s
20
2
7
1640.35
21.33
7200
1640.35
0.00
4.31
S2_4i6s5
20
2
5
1315.61
28.09
7200
1315.61
0.00
7.12
S2_4i8s
20
2
5
1345.27
30.55
7200
1315.61
-2.20
8.75
S2_4i10s
20
2
5
1321.57
38.99
7200
1315.61
-0.45
10.72
Average
1498.45
15.30
5311
1492.41
-0.30
4.27
23
5.3.2. Results for large-size data
The results for 12 large-size MDGVRP instances are presented in Table 7. We report the best
(“Best”) and average (“Avg”) solutions over 10 runs, CPU time of the best solution in seconds
(“t(s)”), and the percentage deviation of the average solution from the best solution (“Δ%”). The
CPU time on the average is 995 seconds and the deviation of the overall average of average
solutions from best solutions is 0.21%. This small deviation shows the robustness of the proposed
method in solving large-size problems.
Table 7. Results for large-size MDGVRP instances
Instance
Best
Avg
t(s)
Δ%
111c_21s
109
3
14
3084.35
3084.35
278
0.00
111c_22s
109
3
14
3067.00
3069.18
293
0.07
111c_24s
109
3
14
3067.00
3067.00
269
0.00
111c_26s
109
3
14
3064.57
3066.12
269
0.05
111c_28s
109
3
14
3067.00
3067.00
282
0.00
200c_21s
194
3
23
5255.96
5260.89
653
0.09
250c_21s
240
3
27
6344.42
6380.92
801
0.58
300c_21s
289
3
35
8438.27
8470.21
1130
0.38
350c_21s
337
3
41
10170.07
10200.32
1340
0.30
400c_21s
386
3
46
11285.80
11312.87
1419
0.24
450c_21s
434
3
52
12610.96
12680.59
2304
0.55
500c_21s
483
3
59
14614.24
14647.20
2896
0.23
Average
7005.80
7025.55
995
0.21
5.3.3. Effect of using tabu search in local search
Allowing non-improving moves in a multiple neighborhood framework offers more potential for
achieving high-quality solutions compared to those given by VND in the local search phase. To
investigate the effect of TS on the performance of the algorithm, we repeated our tests by replacing
TS with VND and referred to this algorithm as GVNS/VND. So, we re-solved all 12 large-size
MDGVRP instances using GVNS/VND by removing the basic TS feature that allows non-
improving moves and terminating the search at each neighborhood of 2-opt, 1-AddStation, 1-
DropStation, and 1-SwapStation when it has converged to the local optimum. These four
neighborhoods are explored in a cyclic sequential order starting from 2-opt and end with
1-SwapStation. While GVNS/VND stops exploring a neighborhood when the solution can
24
no longer improve, GVNS/TS continues exploring that neighborhood by accepting non-improving
moves if they are not tabu. The run time of GVNS/VND is expected to be shorther than that of
GVNS/TS. Note that a perfect comparison is not possible because the CPU times differ from one
run to another. To allow similar run times for a fair comparison, we increased the iteration number
limit of GVNS/VND and set a time limit equal to the run time of the best solution obtained with
GVNS/TS.
Table 8. Comparison of TS and VND
Instance
Best Δ%
Avg Δ%
111c_21s
2.40
4.01
111c_22s
6.79
6.97
111c_24s
2.82
3.93
111c_26s
1.22
1.69
111c_28s
0.94
1.54
200c_21s
1.69
2.57
250c_21s
0.46
0.58
300c_21s
0.63
0.66
350c_21s
0.60
0.77
400c_21s
1.12
1.56
450c_21s
0.87
0.97
500c_21s
1.29
1.37
Average
1.74
2.22
The results are provided in Table 8 where the columns “Best Δ%” and “Avg Δ%” show the
percentage deviation of the GVNS/VND best and average objective function values over 10 runs
from those of GVNS/TS. A positive Δ% value indicates that TS performs better than VND. Note
that our setting slightly favored GVNS/VND with an overall average run time of 995 seconds
compared to 968 seconds for GVNS/TS. Despite this drawback, the results show that GVNS/TS
outperforms GVNS/VND in all of the instances and the contribution of TS to the solution quality
is on average 1.74% and 2.22% for the best and average objective function values, respectively,
as compared to the performance of VND. Furthermore, we observe that TS can enhance the
performance of the algorithm by as much as 6.79% and 6.97% (instance 111c_22s) in the best and
average solutions, respectively. From these results, we conclude that the algorithm greatly benefits
from TS in achieving high-quality solutions.
25
5.4. Trade-off analysis
In this section, we consider five large-size MDGVRP instances (111c_21s, 200c_21s, 300c_21s,
400c_21s and 500c_21s) which involve the same stations but different number of customers, and
investigate the trade-offs with respect to operating different numbers of depots and allowing
refueling only at depots.
5.4.1. Influence of the number of depots
Originally, the number of depots is . Now, we consider by keeping the
original depot and assuming that the first refueling stations in the GVRP data are depots.
Our aim is to investigate the trade-offs between operating more depots and several performance
factors such as the number of customers served, fleet size and total distance traveled. Note that
when , the problem reduces to standard GVRP.
The results are summarized in Fig. 5 . Fig. 5(a) shows that increasing the number of depots
improves the service level as more customers can be reached by the availability of additional
depots. In other words, the driving range of AFVs is less restrictive in reaching the customers.
Furthermore, in Fig. 5(b) we observe that a smaller fleet is needed when more depots are utilized.
For instance, in instance 500c_21s, increasing the number of depots from one to five decreases the
fleet size from 73 to 47 vehicles. So, the cost of operating more depots can be compensated by
using less AFVs. Finally, Fig. 5(c) shows how different number of depots affect the total distance
(fuel cost) of the solution. The more the depots are operated the less is the total distance that the
vehicles travel, despite the fact that more customers can be served with more depots. This is
expected because increasing the number of depots allow customers located in that neighborhood
to be served from their nearest depot. Overall, the results in Fig. 5 reveal the benefits of operating
multiple depots. However, a cost-benefit analysis is needed to assess the actual figures in a real
business environment.
26
(a) Influence on service level
(b) Influence on fleet size
(c) Influence on total distance
Fig. 5. Influence of different numbers of depots on performance factors
5.4.2. Influence of refueling only at depots
In this study, we allowed AFVs to refuel at depots and public stations. Even though numerous
public AFSs may exist in the region, not all of them are truck friendly (Trey et al., 2016). In
addition, many companies that employ AFVs prefer refueling them at their own facilities because
of inefficient utilization of drivers’ time and security concerns of their cargo (Morganti and
Browne, 2018). So, refueling at depots may be the common situation in real-world logistics
operations, considering the limited AFS infrastructures in most regions. In order to investigate the
0
100
200
300
400
500
600
1 2 3 4 5
Service Level
m
0
10
20
30
40
50
60
70
80
12345
Fleet Size
m
0
5000
10000
15000
20000
25000
12345
Total Distance
m
27
influence of refueling only at company-owned depots on the routing decisions we solved the same
five MDGVRP instances discussed in the previous section by setting and removing all
refueling stations.
Fig. 6 summarizes the results according to the same performance factors, i.e. number of customers
served, fleet size and total distance traveled. In this figure, the results are given in terms of the
ratio of the performance factor values observed by allowing refueling at public AFSs vs. refueling
at depots. For example, in the case of fleet size, the value provided represents
. In all
instances of different sizes, we observe that all the ratios are greater than one. In other words,
limiting the refueling at depots decreases the number of customers served, the fleet size and total
distance. Even though the last two indicate improvements, a decrease in the number of customers
suggests a lower service level. So, the reductions in fleet size and distance are achieved by visiting
fewer customers. This is due the fact that the driving range of the AFVs does not permit service to
some distant customers if the vehicles are not refueled at public stations available in their regions.
Furthermore, we notice an increasing trend in the ratio values as the problem size gets larger. Since
all five instances involve the same stations, we can conclude that the availability of public AFSs
is a key business factor for logistics operators that serve hundreds of customers daily.
Fig. 6. Refueling at depots only vs. refueling at public stations
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
111c_21s 200c_21s 300c_21s 400c_21s 500c_21s
28
6. Concluding remarks
In this paper, we introduced a variant of the well-known Green Vehicle Routing Problem, namely
Multi-Depot Green Vehicle Routing Problem where the customers are served from a set of depots
using a fleet of AFVs. To solve the problem, we developed a hybrid GVNS/TS approach that
combines the General Variable Neighborhood Search method (GVNS) with Tabu Search (TS). In
our GVNS/TS, the neighborhood structures are systematically changing through the VNS
approach and a TS-based local search procedure is implemented to exploit the search space. We
validated the performance of the proposed method by solving GVRP instances and benchmarking
our results to those of the state-of-the-art methods proposed in the literature. Then, we modified
the GVRP data to generate MDGVRP instances and solved them using GVNS/TS. We compared
the results of small-size problems against (near-)optimal solutions obtained with CPLEX. Our tests
on both GVRP and MDGVRP data showed that GVNS/TS is robust and able to find high-quality
solutions in reasonable computation times. We also provided managerial insights about the
influence of operating different number of depots and allowing refueling only at depots on the
routing decisions and customer service levels.
In this study, we assumed that the fuel tanks of the AFVs are filled to capacity in constant time.
Future research on this topic may address other realistic GVRP variants where different types of
AFVs may be used with additional restrictions and limitations. Among those, Electric Vehicle
Routing Problems (EVRPs) arise with challenging attributes such as long recharging durations at
stations. The authors are planning to extend their method to solve EVRP with Time Windows as
well as its multi-depot variants which would require new mechanisms to cope with service time
window restriction and variable recharging time due to partial charging. Since it is not practical to
convert the whole fleet to AFVs in short time, further research on this topic may focus on the
utilization of a mixed fleet that may consist of ICEVs and different types of AFVs. Since vehicle
types are associated with different total costs of ownership and fuel costs, the investigation of
mixed fleets within this context may provide valuable managerial insights for logistics operators.
29
Acknowledgments
The authors wish to thank the three anonymous reviewers for providing valuable comments and
suggestions throughout the review process of the paper.
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