An Adjustable Robust Formulation and a Decomposition Approach for
the Green Vehicle Routing Problem with Uncertain Waiting Time
at Recharge Stations
Luigi Di Puglia Pugliese
1 a
, Francesca Guerriero
2 b
and Giusy Macrina
2 c
1
Istituto di Calcolo e Reti ad Alte Prestazioni, Consiglio Nazionale delle Ricerche, 87036, Rende, Italy
2
Dipartimento di Ingegneria Meccanica, Energetica e Gestionale, Universit
´
a della Calabria, 87036, Rende, Italy
Keywords:
Vehicle Routing Problem, Electric Vehicle, Uncertain Parameters, Robust Optimization, Decomposition
Approach.
Abstract:
We investigate the problem of routing a fleet of electric vehicles in an urban area, which must serve a set
of customers within predefined time windows. We allow partial battery recharge to any available recharge
stations, located in the area. Since the recharge stations could be busy when a driver arrives for charging
operations, the time spent can be seen as the sum of recharge times and waiting times. We model the waiting
times as uncertain parameters and we assume to do not know their distribution. Hence, we address the problem
under the robust optimization framework by modelling the realization of the uncertain parameter with the
budget of uncertainty polytope. We propose an adjustable robust formulation, then we implement a row-
and-column generation solution approach based on a two-stage reformulation of the problem, to provide a
robust routing against infeasibility with respect to time collect requirements. We test the proposed approach
on benchmark instances and we analyze the behavior of the considered transportation system with respect to
the uncertain parameter. In addition, we investigate the price of robustness and the reliability of the robust
solutions obtained.
1 INTRODUCTION
The worrying effects of the road traffic emissions on
the air quality has become an issue of global impor-
tance. Hence, the need to provide sustainable trans-
portation plans is the main objective of many coun-
tries. In recent years, several political decisions and
regulations concerning the reduction of the green-
house and polluting emissions have been proposed
and have already become law (i.e., Kyoto Protocol
(1997) and the European plan on climate change
(2008)). Certainly, the greenhouse issue has become
a political topic with high priority and the definition
of sustainable logistics systems is an essential choice.
Hence, a “green” management of transportation sys-
tems plays a key role. Among the strategies that
can be adopted, the use of Battery Electric Vehicles
(BEVs) represents a promising alternative to the tradi-
tional Internal Combustion Engine Vehicles (ICEVs)
a
https://orcid.org/0000-0002-6895-1457
b
https://orcid.org/0000-0002-3887-1317
c
https://orcid.org/0000-0001-6762-3622
for reducing negative externalities, such as noise and
pollution. In fact, one of the benefits associated with
BEVs is the absence of the CO
2
emissions. However,
there are also several critical aspects related to some
technical issues. The BEVs have a poor battery life
compared to the ICEVs autonomy, the vehicles need
to be recharged during the routes and the number of
available recharge stations is still low. In addition,
the time needed to fulfil a complete recharge is very
high and the stops during the routes increase the total
time spent to complete a trip. Fortunately, the auto-
motive sector is interested in BEVs development and
invests in this research area, hence, the innovation is
fast and continuous. For this reason, considering the
introduction of BEVs in transportation is a strategic
investment.
The introduction of BEVs in the classical Vehi-
cle Routing Problem (VRP), requires the modeling
and insertion of some specific constraints. Hence,
several restrictions must be taken into account, es-
pecially the limited battery capacity of the vehicles
and the possibility to recharge the batteries at the Al-
ternative Fuel Stations (AFSs). (Erdo
˘
gan and Miller-
72
Di Puglia Pugliese, L., Guerriero, F. and Macrina, G.
An Adjustable Robust Formulation and a Decomposition Approach for the Green Vehicle Routing Problem with Uncertain Waiting Time at Recharge Stations.
DOI: 10.5220/0010256500720081
In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 72-81
ISBN: 978-989-758-485-5
Copyright
c
 2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

Hooks, 2012) introduced the first routing model that
considers AFSs. In particular, the authors modeled
a Mixed Integer Linear Programming (MILP) of a
VRP in which a fleet is composed of Alternative Fu-
elled Vehicles (AFVs) with a limited fuel capacity.
They considered the possibility to stop the vehicles
and recharge their batteries at the available AFSs and
named this problem Green VRP (GVRP). In that pa-
per, the vehicles are uncapacitated and time window
constraints are not included. Several techniques are
developed to find a solution that minimizes the to-
tal distance traveled. (Schneider et al., 2014) and
(Felipe et al., 2014) extended the model presented
in (Erdo
˘
gan and Miller-Hooks, 2012) introducing the
Electric-VRP (E-VRP), in which the fleet is com-
posed of BEVs. In particular, (Schneider et al., 2014)
studied the E-VRP with time windows and recharge
stations, which considers capacity constraints on ve-
hicles and time windows on customers. The vehicles
can stop at any available AFS for fulfilling a com-
plete battery recharge. The time spent to recharge
the battery depends on the battery charge of the ve-
hicle on arrival at the AFS. The authors proposed a
meta-heuristic that combines variable neighbourhood
search and tabu search to solve their problem. (Felipe
et al., 2014) introduced more realistic elements to the
E-VRP. In particular, the authors considered the pos-
sibility of partial recharging at the stations. Further-
more, a battery recharge can be done with different
technologies, in this case different recharging times
and costs should be taken into account. For exam-
ple the overnight depot recharging is the cheapest and
slowest technology. The authors propose a construc-
tive algorithm based on a greedy generation method,
a deterministic local search, and a Simulated Anneal-
ing algorithm.
The E-VRP has been widely studied and, start-
ing from these works, many authors proposed sev-
eral extensions and variants and developed solutions
methods, see, e.g., (Ding et al., 2015; Keskin and
C¸ atay, 2016; Desaulniers et al., 2016; Goeke, 2019;
L
¨
offler et al., 2020) for variants which consider par-
tial recharges, and (Lin et al., 2009; Hiermann et al.,
2016; Hiermann et al., 2016; Joo and Lim, 2018;
Basso et al., 2019) for versions of the problem which
consider full recharge only. A particular variant of
the E-VRP is the mixed-fleet GVRP, in which the
fleet is composed of both BEVs and ICEVs, see, e.g.
(Gonc¸alves et al., 2011; Sassi et al., 2014; Macrina
et al., 2019a; Macrina et al., 2019b; Hiermann et al.,
2019). Since the battery charge level is a non-linear
function, recently several authors focused on this spe-
cific feature and proposed mathematical formulations
which consider this issue (see, e.g., (Montoya et al.,
2017; Froger et al., 2019)).
Another important aspect to take into account is
the limited number of recharge stations, located in
the urban area. Hence, on the one hand, combin-
ing the optimization of recharge stations location and
BEVs routing could be a leading strategy (see, e.g.,
(Yang and Sun, 2015; Paz et al., 2018; Zhang et al.,
2019)), on the other one, it is necessary to take into
account the possibility that a charger is not available
when a vehicle stops at a recharge station. Focus-
ing on the latter problem, we propose a variant of
the E-VRP with time windows and partial recharges,
which considers an uncertain waiting time at the
recharge stations. A similar framework has been stud-
ied by (Keskin et al., 2019) and (Keskin et al., 2021).
In particular, (Keskin et al., 2019) considered time-
dependent queuing times at the stations, they split the
time horizon into predetermined number of time in-
tervals and assigned an average queue length to each
recharge station, for different time intervals. They
formulated the problem as a MILP and proposed a
math-heuristic based on Adaptive Large Neighbor-
hood Search (ALNS) to solve it. (Keskin et al., 2021)
studied an E-VRP with time windows and stochastic
waiting times at recharge stations. They proposed a
two-stage simulation-based heuristic using ALNS to
solve their problem. In particular, in the first stage
they used the expected waiting time values at the sta-
tions for determining the routes. After the arrival
of the vehicles at the recharge station, their queuing
times are revealed, hence, the second stage corrects
the infeasible solutions in case the actual waiting time
at a station exceeds its expected value.
In our work, we propose a mathematical formu-
lation based on (Schneider et al., 2014), but we ex-
tend it by introducing the possibility to perform par-
tial recharges and waiting time at recharge stations.
We assume the waiting time uncertain and suppose
to not know the distribution of the realization of the
uncertain parameters. Thus, we address the problem
under the robust optimization framework. We pro-
pose a two-stage robust formulation defining routing
first stage variables and scheduling second stage ones
where the latter depend on the realization of the un-
certain waiting time. We define a decomposition ap-
proach based on a row-and-column generation strat-
egy in which second stage variables and the related
constraints are added on the fly within an iterative pro-
cedure.
The paper is organized as follows. Section 2
presents the problem and the mathematical formula-
tions. Section 3 defines the proposed solution strat-
egy based on a decomposition approach. Section 4
shows the computational results. Section 5 concludes
An Adjustable Robust Formulation and a Decomposition Approach for the Green Vehicle Routing Problem with Uncertain Waiting Time at
Recharge Stations
73

the paper.
2 PROBLEM DEFINITION
Let G(V , A) be a directed graph, where V is the set
of nodes and A = {(i, j), ∀i, j ∈ V } is the set of arcs.
The set of nodes V contains customers, recharge
stations, and nodes s, which is the depot, and t, a copy
of s, where vehicles routes start and end, respectively.
Hence, V = N ∪ R ∪ {s, t}, where N and R are the
set of customers and recharge stations, respectively.
A demand q
i
and a service time s
i
are associated with
each customer i ∈ N , expressed in kg and hours,
respectively. Each customer must be visited by a
single vehicle. Each node i ∈ V is characterized by a
time window [e
i
, l
i
]. Furthermore, the vehicles have
limited transportation capacity and battery capacity;
thus, let Q be the maximal capacity of the vehicle
(expressed in tonne), and B be the maximal battery
capacity (expressed in kWh). For each arc (i, j) ∈ A,
d
i j
refers to the distance from i to j (expressed in
km), t
i j
denotes the travel time (expressed in hours),
and c
i j
the cost per unit of distance. It is assumed that
d
i j
≤ d
ih
+ d
h j
and t
i j
≤ t
ih
+ t
h j
for all i, h, j ∈ V ,
hence the triangle inequality holds for both the
distance and the time. Each recharge station i ∈ R
is characterized by a recharge speed ρ
i
(expressed in
kWh per hour), a waiting time w
i
, and a recharge cost
¯c
i
per unit of kWh. In particular, w
i
is the time that
a vehicle must wait before starting the recharge at
station i ∈ R . It is worth observing that if the vehicle
does not recharge, then no waiting time is consid-
ered. The value π denotes the coefficient of energy
consumption, that is assumed to be proportional to
the distance travelled (expressed in
KW h
Km
). We assume
to have a limited number of available vehicles to
perform the deliveries, equal to E. In the following,
we describe the variables used to model the problem:
• x
i j
is equal to 1 if the vehicles travel from i to j,
zero otherwise, ∀(i, j) ∈ A;
• y
j
is equal to 1 if the vehicle recharge at station j,
zero otherwise, ∀ j ∈ R ;
• z
i j
amount of energy available when arriving at
node j from the node i [kWh], ∀(i, j) ∈ A;
• g
i j
amount of energy recharged by the vehi-
cle at the node i for travelling to j [kWh],
∀i ∈ R , ∀ j ∈ V ;
• τ
j
arrival time of the vehicle to the node j [h],
∀ j ∈ V ;
• u
i
amount of load available at node i [kg], ∀i ∈ V .
The problem is formulated as follows:
min
∑
i∈R
¯c
i
∑
j∈V
g
i j
+
∑
(i, j)∈A
c
i j
x
i j
(1)
s.t.
∑
j∈V
x
i j
= 1, ∀i ∈ N (2)
∑
j∈V
x
i j
≤ 1, ∀i ∈ R (3)
∑
j∈V \
{
s
}
x
i j
−
∑
j∈V \
{
s,t
}
x
ji
= 0, ∀i ∈ V \
{
t
}
(4)
∑
j∈V \
{
s
}
x
s j
≤ E, (5)
∑
i∈V \
{
t
}
x
it
≥ 1, (6)
∑
i∈V \
{
s
}
x
si
−
∑
j∈V \
{
t
}
x
jt
= 0 (7)
u
j
≥ u
i
+ q
j
x
i j
− Q(1 − x
i j
),
∀i ∈ V \
{
s,t
}
, j ∈ V \
{
s
}
(8)
u
s
= 0 (9)
y
i
≥
∑
j∈V
g
i j
B
, ∀i ∈ R (10)
τ
j
≥ τ
i
+ (t
i j
+ s
i
)x
i j
− M1(1 −x
i j
),
∀i ∈ N ∪ {s}, ∀ j ∈ V (11)
τ
j
≥ τ
i
+t
i j
x
i j
+
1
ρ
i
g
i j
+ w
i
y
i
− M1(1 −x
i j
),
∀i ∈ R , ∀ j ∈ V (12)
e
j
≤ τ
j
≤ l
j
, ∀ j ∈ N (13)
z
i j
≤ (z
hi
+ g
i j
) − πd
i j
x
i j
+ M
1
(1 − x
i j
) +
+M
1
(1 − x
hi
),
∀h ∈ V , ∀i ∈ V \
{
s
}
, ∀ j ∈ V ,
i 6= j, i 6= h, j 6= h (14)
z
s j
≤ B −πd
s j
x
s j
+ M
1
(1 − x
s j
), ∀ j ∈ V (15)
g
i j
≤ B −z
hi
+ M
1
(1 − x
i j
) + M
1
(1 − x
hi
),
i ∈ R , h ∈ V , j ∈ V (16)
x
i j
∈ {1, 0}, ∀i ∈ V , ∀ j ∈ V ;
y
i
∈ {1, 0}, ∀i ∈ R ; u
i
≥ 0, ∀i ∈ V ;
z
i j
≥ 0, ∀i ∈ V , ∀ j ∈ V ;
g
i j
≥ 0, ∀i ∈ R , ∀ j ∈ V ;τ
i
≥ 0, ∀i ∈ V (17)
The objective function minimizes the sum of the to-
tal recharge cost and the total travel cost, expressed
ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems
74

by the equation (1). Constraints (2) impose that each
customer is visited exactly once, while (3) guarantee
that each recharge station can be visited at most once.
Flow conservation is given by constraints (4) and they
ensure that for each vertex, the number of incoming
arcs is equal to the number of outgoing arcs. Con-
straints (5) guarantee that the number of routes does
not exceed the number of available vehicles. Con-
straints (6) and (7) model the number of vehicles that
incoming and outgoing to/from the depot s and its
copy t. Constraints (8) ensure that the capacity of
each vehicle is not exceeded and constraint (9) en-
sures that the vehicle is empty at the depot. Con-
straints (10) define the values of variables y. In partic-
ular, y
i
is forced to assume value equal to 1 if g
i j
> 0,
for some j ∈ V . Constraints (11) and (12) link the ar-
rival times to the routing variables for the customers
and recharge stations, respectively. Time windows
are given by constraints (13). Constraints (14) and
(15) model the battery capacity, while constraints (16)
ensure that the energy recharged at the available sta-
tion does not exceed the battery capacity. Constraints
(17) define the domain of the decision variables. It is
worth observing that since we assumed the triangle in-
equality holds for the time, a vehicle visits a recharge
station only if it need to recharge its battery. Hence,
we can exclude from the model variable y. It follows
that constraints (10) are removed and variable y
i
is
dropped from constraints (12).
2.1 Robust Formulation
In our setting, we suppose to have not precise infor-
mation about the waiting time w ∈ R
|R |
, and we as-
sume to know a mean value ¯w
i
and a deviation ˆw
i
at
each recharge station i ∈ R . Hence, ¯w
i
−δ
−
i
ˆw
i
≤ w
i
≤
¯w
i
+ δ
+
i
ˆw
i
, ∀i ∈ R , with δ
−
i
, δ
+
i
∈ (0, 1), ∀i ∈ R .
The waiting time w and the recharge time g play
a key role in the delivery process. In fact, recharg-
ing operations are necessary to ensure the completion
of the tours, avoiding the out-of-battery, however, the
recharge time and the waiting time have to be care-
fully managed. Recharge and waiting time too long
can delay the delivery to the customers causing the
infeasibility due to the time window constraints.
Since all customers have to be served within their
own time windows, the solution must remain feasible
by satisfying the time windows requirements for any
realization of w. For this purpose we consider a ro-
bust formulation of the problem where the problem
is optimized under a worst case scenario. Hence, the
waiting time is assumed to be w
i
≤ ¯w
i
+δ
+
i
ˆw
i
, ∀i ∈ R .
We model the realization of the uncertain param-
eter w with a polytope. In particular, we consider the
budgeted uncertainty set from (Bertsimas and Sim,
2003), which has attracted the attention of many re-
searchers to handle uncertainty in several contexts,
see, e.g. (Poss, 2014; Lu and Gzara, 2015; Pessoa
et al., 2015; Bruni et al., 2017; Di Puglia Pugliese
et al., 2019). For the problem at hand, the budgeted
uncertainty polytope is defined as U
Γ
= {w ∈ R
|R |
:
w
i
= ¯w
i
+ δ
+
i
ˆw
i
, δ
+
i
∈ (0, 1), ∀i ∈ R ,
∑
i∈R
δ
+
i
≤ Γ}.
In order to take into account the uncertain wait-
ing time, we formulate a route-based two-stage robust
model.
Let p =
h
i
0
= s, i
1
, . . . , i
n−1
, i
n
= t
i
be a route de-
fined as an ordered sequence of nodes, starting and
ending from/to node s and t, respectively. Let N(p)
be the set of customers served by route p, R(p)
be the set of recharge stations visited by route p,
and A(p) be the set of arcs A(p) = {(i
h
, i
h+1
), h =
0, . . . , n − 1} traversed by route p. Let P be a feasi-
ble solution to the problem, i.e., P is a set of feasible
routes p with respect to the capacity and the avoiding-
out-of-battery constraints, such that
S
p∈P
N(p) = N
and
T
p∈P
N(p) =
/
0 to guarantee constraints (2), and
S
p∈P
R(p) ⊆ R and
T
p∈P
R(p) =
/
0 to guarantee con-
straints (3). In other words, P represents a solu-
tion to the routing problem (1)–(9), (14)–(16) de-
fined by the first stage variables x, u, z, g. Given a
route p, c(p) defines the traveling cost, whereas g(p)
represents the recharge cost.The traveling cost c(P )
and the recharge cost g(P ) of a set P are defined
as the sum of the traveling costs and the sum of
the recharge costs of all routes p ∈ P, respectively.
Let
¯
P be the set of all first stage feasible solutions
P , i.e.,
¯
P =

(x
P
, u
P
, z
P
, g
P
) : (1) −(9), (14) − (16)

.
The two-stage robust formulation is given below:
min c(P )+ g(P ) (18)
s.t.
P ∈
¯
P , (19)
τ(w, P ) ∈ T (P ), ∀w ∈ U
Γ
, (20)
τ(w, P ) ≤ l, ∀w ∈ U
Γ
, (21)
where τ(w, P ) is the two-stage variable representing
the arrival times at nodes starting from the depot at
time zero and using the set of routes P. Time τ(w, P)
depends on the specific value of w since the vehi-
cles may have to recharge their battery along their
routes. T (P ) is a polytope defining variable τ(w, P ),
described in what follows:
An Adjustable Robust Formulation and a Decomposition Approach for the Green Vehicle Routing Problem with Uncertain Waiting Time at
Recharge Stations
75

T (P ) =



















τ
j
(w, P ) ≥
τ
i
(w, P ) + (t
i j
+ s
i
)x
P
i j
− M1(1 − x
P
i j
),
∀i ∈ N ∪ {s}, j ∈ V
P
\ {s},
τ
j
(w, P ) ≥
τ
i
(w, P ) +t
i j
x
P
i j
+
1
ρ
i
g
P
i j
+ w
i
− M1(1 − x
P
i j
),
∀i ∈ R
P
, j ∈ V
P
\ {s},
τ
j
(w, P ) ≥ e
j
, ∀ j ∈ N .
(22)
The set V
P
= N ∪ R
P
is the set of nodes included
in solution P , i.e., it contains all nodes i ∈ N and
possibly some nodes associated with recharge stations
i ∈ R
P
=
S
p∈P
R(p) ⊆ R .
Given a solution P , the value of the arrival time
τ(w, P ) is adjusted based on the realization of w.
The route-based two-stage model presents an infinite
number of constraints (20)–(21) being U
Γ
a polytope.
However, one can readily see that, since τ(w, P ) is de-
fined via convex functions, actually linear (see (22)),
we can consider the convex hull of U
Γ
, hence we
can restrict our attention to extreme points of U
Γ
,
i.e., ext

U
Γ

= {w ∈ R
|R |
: w
i
= ¯w
i
+ δ
+
i
ˆw
i
, δ
+
i
∈
{0, 1}, ∀i ∈ R ,
∑
i∈R
δ
+
i
≤ Γ} which contains a finite
number of elements, named scenarios, since U
Γ
is a
polytope. However, even if there exist a finite number
of scenarios, they grow exponentially with the dimen-
sion of the problem, i.e.,


ext

U
Γ



=
∑
Γ
γ=1

|R |
γ

. It
can be computationally intractable enumerate all sce-
narios, thus a decomposition approach, where primal
cuts are derived iteratively, is defined in the next sec-
tion.
3 SOLUTION APPROACH
Model (18)–(21) induces a decomposition of the
problem into two subproblems. The first one, re-
ferred to as the master problem, determines the routes
P ∈
¯
P , i.e., (18) and (19); the second subproblem, re-
ferred to as separation problem, defines the schedule
τ(w, P ) and checks the feasibility, i.e., (20) and (21).
During the iterations of the decomposition ap-
proach, once a solution P
k
is obtained at a given it-
eration k, the separation problem is solved. The latter,
given the first stage solution, defines the second stage
variables based on the realization of the uncertain pa-
rameter w. Hence, we obtain a scenario w
k
where
some recharge stations i ∈ R
P
k
experience a delay. If
τ(w
k
, P
k
) ≤ l, then the first and second stage variables
design an optimal solution to (18)–(21). Otherwise,
solution P
k
must be excluded from
¯
P .
We define a row-and-column generation algorithm
(R&C) (Agra et al., 2013) where constraints (20) and
(21) are iteratively included. The idea is to relax all
constraints (20) and (21) but those in a finite subset
U
∗
. Then, the variables and constraints that corre-
spond to scenarios in ext

U
Γ

\ U
∗
are added on the
fly. The new scenario w
k
to be included into U
∗
at
a given iteration k is determined by solving the sep-
aration problem (20)–(21) with T (P
k
). In particular,
the scheduling is determined by solving problem (20)
and then the feasibility of constraints (21) is checked.
In the case problem (20)–(21) is infeasible, then the
scenario w
k
is included into U
∗
, otherwise, P
k
is an
optimal solution. This type of strategy can be viewed
as a Benders decomposition approach with primal
cuts (Bruni et al., 2018). Indeed, introducing a new
scenario into U
∗
means to introduce into the master
problem second-stage variables and constraints. The
steps of the proposed R&C approach are depicted in
Algorithm 1.
Algorithm 1: R&C.
1: k = 0, U
∗
= {w
k
}, w
k
i
= ¯w
i
, ∀i ∈ R , stop=false
2: while !stop do
3: k + +
4: stop=true
5: Solve problem (18)–(21) with U
∗
obtaining P
k
6: Solve problem (20) with T (P
k
) obtaining a
scenario w
k
7: if τ
j
(w
k
, P
k
) > l
j
for some j ∈ N then
8: U
∗
← U
∗
∪

w
k

9: stop=false
10: end if
11: end while
12: P
k
is an optimal solution.
3.1 Solving the Separation Problem
Solving the separation problem (20)–(21) at iteration
k means to determine the realization of w consider-
ing ext(U
Γ
), i.e., the scenario w
k
and consequently
τ(w
k
, P
k
). We can rewrite constraints (20) in the form
max
w∈ext(U
Γ
)
T (P ) (23)
Dualizing (23) and letting f
1
∈ R
N ∪{s}×V
P
+
, f
2
∈
R
R
P
×V
P
+
, and f
3
∈ R
N
+
be the dual variables associ-
ated with the three inequalities, we obtain the follow-
ing problem
max
w∈ext(U
Γ
)
∑
i∈N ∪{s}
∑
j∈V
P
(t
i j
+ s
i
) f
1
i j
+ (24)
+
∑
i∈R
P
∑
j∈V
P

t
i j
+
1
ρ
i
g
p
i j
+ w
i

f
2
i j
+ (25)
ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems
76

+
∑
i∈N
e
i
f
3
i
, (26)
subject to classical flow-conservation constraints de-
fined over variable f
1
and f
2
, and f
3
≥ 0. The details
of the dualization are reported in the Appendix. We
remark that P
k
is a set of routes p. Without loss of
generality, we can duplicate nodes s and t for each
route p ∈ P
k
. Hence, T (P
k
) can be decomposed into
|P
k
| disjoint sets T (p), each associated with a route
p ∈ P
k
. It follows that variables f
1
and f
2
assume
value equal to 1 for each (i, j) ∈ A(p), p ∈ P
k
and zero
otherwise. We can rewrite (23) in the form
max
w∈ext(U
Γ
)
[
p∈P
k
T (p). (27)
and the associated dual becomes:
max
w∈ext(U
Γ
)
∑
p∈P
k
∑
(i, j)∈A(p):i∈N(p)∪{s}
(t
i j
+ s
i
) + (28)
+
∑
p∈P
k
∑
(i, j)∈A(p):i∈R(p)

t
i j
+
1
ρ
i
g
p
i j
+ w
i

+ (29)
+
∑
p∈P
k
∑
i∈N(p)
e
i
f
3
i
, (30)
with f
3
i
≥ 0, ∀i ∈ N . We note that only the term (29)
depends on the uncertain parameter w. Thus, we can
rewrite (28)–(30) as
∑
p∈P
k
∑
(i, j)∈A(p):i∈N(p)∪{s}
(t
i j
+ s
i
) + (31)
+ max
w∈ext(U
Γ
)
∑
p∈P
k
∑
i∈R(p)
w
i
+
+
∑
p∈P
k
∑
(i, j)∈A(p):i∈R(p)

t
i j
+
1
ρ
i
g
p
i j

+ (32)
+
∑
p∈P
k
∑
i∈N(p)
e
i
f
3
i
. (33)
It follows that the scenario associated with solution
P
k
is determined by solving the following problem
max
w∈ext(U
Γ
)
∑
i∈R
P
k
w
i
. (34)
Problem (34) can be easily solved by determining the
set R
max
containing max{|R
P
k
|, Γ} recharge stations
associated with the highest value of ¯w + ˆw. Hence,
the scenario w
k
is determined by setting w
k
i
= ¯w
i
+
ˆw
i
, ∀i ∈ R
max
and w
k
i
= ¯w
i
, ∀i ∈ R
P
k
\ R
max
.
Given that each node i ∈ N ∪ R
P
k
belongs to
a single route p ∈ P
k
, we can define τ
i
(w
k
, p), ∀i ∈
N(p) ∪ R(p), p ∈ P
k
as the arrival time to node i in
route p. Thus, once w
k
is available, τ(w
k
, P
k
) =
S
p∈P
k
τ(w
k
, p) are computed as follows
τ
s
(w
k
, p) = 0, ∀p ∈ P
k
, (35)
τ
j
(w
k
, p) = max{e
j
, τ
i
(w
k
, p) +t
i j
+ s
i
},
∀(i, j) ∈ A(p) : i ∈ N(p) ∪ {s}, p ∈ P
k
, (36)
τ
j
(w
k
, p) = max{e
j
, τ
i
(w
k
, p) +t
i j
+
+
1
ρ
i
g
i j
+ w
k
i
},
∀(i, j) ∈ A(p) : i ∈ R(p), p ∈ P
k
. (37)
4 COMPUTATIONAL RESULTS
The aim of this Section is to evaluate the effect of
the uncertain waiting time on the optimal solution.
The decomposition approach has been implemented
in Java and the tests have been carrier out on an In-
tel(R) Core(TM) i7-4720HQ CPU, 2.60 GHz, 8GB
RAM machine under Microsoft Windows 10 operat-
ing system. In the next Section we describe the con-
sidered instances, whereas we present the numerical
results in Section 4.2.
4.1 Instances Generation
We consider the benchmark instances proposed in
(Schneider et al., 2014) for the E-VRP with time win-
dows with 5 and 10 customers, i.e., N ∈ {5, 10}.
These instances are created based on the benchmark
instances for the VRP with time windows proposed
by (Solomon, 1987) where recharge stations are in-
cluded at random. For more details on the charac-
teristics of the considered instances, the reader is re-
ferred to (Schneider et al., 2014). We modify such
instances by introducing the parameters w and Γ. In
particular, we generate the mean value of the waiting
time w as ¯w
i
= ˜w ∗(1 +α), ∀i ∈ R and α chosen in the
uniform interval (0, 1). Whereas, the pick value ˆw is
computed as ˆw
i
= ¯w
i
∗(1 +β), ∀i ∈ R with β ∈ (0, 1).
Starting from the original instances, we generate a set
of test problems by considering ˜w ∈ {10, 15, 30, 45}
and β ∈ {0.5, 0.7,0.9}. We consider several degree
of the risk aversion of the decision maker, by letting
Γ =
d
ρ ∗ |R |
e
with ρ ∈ {0.25, 0.50, 0.75}.
4.2 Experimental Results
Before showing the numerical results, we analyze
the characteristics of the generated instances starting
from the benchmarks (Schneider et al., 2014).
An Adjustable Robust Formulation and a Decomposition Approach for the Green Vehicle Routing Problem with Uncertain Waiting Time at
Recharge Stations
77

4.2.1 Instances Analysis
The instances generated are not all feasible. Table 1
reports the percentage of feasible instances.
Table 1: Percentage of feasible instances.
|N | = 5 |N | = 10
ρ ρ
˜w β 0.25 0.50 0.75 0.25 0.50 0.75
10
0.5 100% 100% 100% 100% 100% 100%
0.7 100% 100% 100% 100% 100% 92%
0.9 100% 100% 100% 100% 92% 92%
15
0.5 100% 100% 100% 83% 83% 83%
0.7 100% 100% 100% 83% 83% 83%
0.9 100% 100% 100% 83% 83% 83%
30
0.5 100% 83% 83% 83% 83% 83%
0.7 83% 83% 83% 83% 83% 83%
0.9 83% 83% 83% 83% 83% 83%
45
0.5 75% 75% 75% 67% 67% 67%
0.7 75% 75% 75% 67% 67% 67%
0.9 75% 75% 75% 67% 67% 67%
The results reported in Table 1 highlight that adding
the waiting time to the recharge stations make some
benchmark instances infeasible. In particular, the
higher the values of ˜w, β, and ρ, the lower the per-
centage of feasible instances. In the sequel, we refer
only to the instances that are feasible for each value
of ˜w, β, and ρ.
Referring to the risk aversion of the decision
maker, Table 2 reports the average value of Γ and the
average number of recharge stations (#RS) included
in the solution by varying the value of ρ.
Table 2: Average value of Γ and average number of recharge
stations included in the optimal solution at varying ρ.
|N | = 5 |N | = 10
ρ ρ
0.25 0.50 0.75 0.25 0.50 0.75
Γ 1.00 2.44 3.33 1.38 2.88 4.25
#RS 2.06 2.04 2.04 2.68 2.70 2.70
Analyzing the results reported in Table 2, we observe
that Γ computed with ρ = 0.75 presents the same risk
aversion of Γ computed by considering ρ = 0.50. In-
deed, #RS is lower than Γ when ρ = 0.50. It means
that, on average, in all recharge stations the vehicles
experience a delay. Thus, increasing the value of ρ to
0.75 does not produce any modification on the real-
ization of the uncertain parameter w. Hence, on av-
erage, the solutions obtained with Γ computed by set-
ting ρ = 0.50 are the same as those determined by
considering ρ = 0.75.
4.2.2 Numerical Results
The aim of this Section is twofold. Firstly, we ana-
lyze the transportation system behaviour by varying
the values of the parameters ˜w, β, and ρ, and we high-
light some insight on how the uncertain waiting time
influences the optimal solutions. Secondly, we ana-
lyze the characteristics of the robust solutions in terms
of both cost and reliability in comparison with the de-
terministic ones.
Behavior of the System under Uncertainty. Table
3 reports the average results in term of objective func-
tion under column obj, number of recharge stations
included in the solutions under column #RS, number
of iteration under column #iter, and execution time (in
seconds) under column time, at varying the values of
the parameters ˜w, β, and ρ.
Table 3: Average numerical results.
obj #RS #iter time
|N | = 5
˜w
10 216.99 2.00 0.11 0.32
15 220.44 2.06 0.12 0.36
30 222.04 2.00 0.02 0.38
45 231.56 2.11 0.07 0.35
β
0.5 222.34 2.02 0.06 0.36
0.7 222.82 2.06 0.08 0.34
0.9 223.11 2.06 0.10 0.36
ρ
0.25 221.21 2.06 0.03 0.33
0.50 223.53 2.04 0.11 0.36
0.75 223.53 2.04 0.11 0.37
|N | = 10
˜w
10 302.91 2.63 0.00 7.86
15 309.27 2.75 0.13 14.40
30 311.63 2.63 0.00 25.63
45 313.89 2.76 0.31 35.49
β
0.5 309.29 2.70 0.10 20.62
0.7 309.29 2.68 0.10 20.33
0.9 309.69 2.70 0.11 21.59
ρ
0.25 308.89 2.68 0.07 20.49
0.50 309.69 2.70 0.13 21.18
0.75 309.69 2.70 0.13 20.87
The results collected in Table 3 highlight that obj in-
creases by increasing the value of all parameters, i.e.,
˜w, β, and ρ. This is an expected behaviour. Indeed,
when the mean value of the waiting time ¯w increases,
the number of feasible solutions, with respect to the
time windows, decreases. We observe a similar trend
for variations of the pick value ˆw expressed in terms
of β. However, in this case, the variations of obj are
less evident than that observed when the mean waiting
time ¯w changes. In addition, the higher the risk aver-
sion of the decision maker, i.e., the higher the value
of Γ, the higher the value of obj.
The number of recharge stations in the optimal so-
lution remains almost the same by varying the param-
eters values (see columns #RS). In addition, a clearly
trend is not observed. These results suggest that the
uncertain parameter w does not influence the vehicles
ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems
78

recharging. Indeed, they have to serve all customers
avoiding the out-of-battery. Thus, the solutions adjust
themselves in terms of sequence of visiting of both
customers and recharge stations, in order to guarantee
the satisfaction of the time windows.
As expected, the higher the deviation ˆw, the higher
the #iter. Hence, a higher number of scenarios have to
be included in the master problem in order to obtain a
feasible solution. The same trend is observed at vary-
ing the value of ρ. This behaviour is due to the fact
that the higher the risk aversion of the decision maker,
the lower the solution space. Thus, since our approach
starts with a relaxation of the polytope defining the re-
alization of w, a higher number of scenarios have to
be included in order to construct a feasible solution
space.
The execution time is limited for the instances
with N = 5. For these problems, the solution ap-
proach requires the same computational time, by
varying all the parameters values (see column time).
For the instances with 10 customers, an increase in
the computational overhead is observed by increasing
the value of ˜w. On the other hand, the execution time
remains almost unchanged by varying the values of
both β and ρ.
Effect of the Robustness. In order to show how
the robust solution behaves under uncertain waiting
time, we perform a sampling analysis. In particular,
we build 1 hundred samples generated as described in
what follows. The mean waiting time of sample s is
calculate as ¯w
s
i
= ˜w ∗ (1 + r
1
) with r
1
randomly cho-
sen in the interval (0, 1), the deviation is calculated
as ˆw
s
i
= ¯w
s
i
∗ (1 + r
2
β), with r
2
randomly chosen in
the interval (0, 1). Hence, we set either w
s
i
= ¯w
s
i
+ ˆw
s
i
or w
s
i
= max{0, ¯w
s
i
− ˆw
s
i
}, ∀i ∈ R . It follows that
w
s
i
∈ (0, 2 ˜w(1 + β)), ∀i ∈ R , for each sample s. Thus,
we recalculate the scheduling considering all gener-
ated samples and we check the feasibility with respect
to the time window constraints for each optimal solu-
tion, both robust and deterministic.
Table 4 reports the average percentage of infeasi-
ble solutions when considering the robust (see column
%inf R) and the deterministic ones (see column %inf
D). Column PoR reports the price of robustness cal-
culated as
obj−obj
D
obj
D
× 100, where obj is the cost of the
robust solution, whereas obj
D
is the cost of the deter-
ministic one.
As expected, the robust solutions are more reliable
than the deterministic ones. Indeed, the average per-
centage of infeasibility over the generated samples is
0.93% and 1.29% for the robust solutions against the
3.06% and the 3.39% observed for the deterministic
ones, considering N = 5 and N = 10, respectively.
Table 4: Average results showing the effect of the robust-
ness.
|N | = 5 |N | = 10
PoR %inf R %inf D PoR %inf R %inf D
˜w
10 1.22% 0.06% 2.78% 0.00% 0.36% 0.93%
15 1.59% 1.58% 3.93% 0.73% 0.11% 1.19%
30 0.02% 0.40% 0.59% 0.00% 0.22% 0.22%
45 0.51% 1.69% 4.41% 0.73% 4.46% 9.56%
β
0.5 0.61% 0.73% 1.72% 0.32% 0.70% 2.00%
0.7 0.82% 0.84% 2.81% 0.32% 1.39% 2.89%
0.9 0.96% 1.22% 4.25% 0.45% 1.77% 4.03%
ρ
0.25 0.10% 2.28% 2.93% 0.19% 2.79% 2.97%
0.50 1.15% 0.26% 2.93% 0.45% 0.54% 2.97%
0.75 1.15% 0.26% 2.93% 0.45% 0.54% 2.97%
AVG 0.80% 0.93% 3.06% 0.37% 1.29% 3.39%
In addition, the price of robustness is quite limited
with an average cost for the robust solutions 0.80%
and 0.37% higher than that of the deterministic ones,
on average, for the instances with 5 and 10 customers,
respectively. As expected, the higher the risk aver-
sion of the decision maker, the higher the PoR and
the lower the %inf R for all the considered instances.
5 CONCLUSIONS
We have studied a vehicle routing problem variants
in which a fleet of electrical vehicles must serve a set
of customers within their own time windows. Par-
tial battery recharging of the vehicles to any available
recharge station is allowed. We consider a particular
case in which a charger can be busy when a driver
visits a station, thus he must wait. We address the
problem under the robust optimization framework, by
modelling the realization of the uncertain parameter
with the budget of uncertainty polytope. We propose
a robust formulation and a row-and-column genera-
tion method, based on a two-stage reformulation, to
solve our problem.
The proposed approach is tested on benchmark in-
stances for the E-VRP with time windows. We con-
sider only small-size instances. The collected com-
putational results highlight that the robust solutions
are more reliable than the deterministic ones with a
limited increase of the total cost. Solving to opti-
mality large-size instances is not viable. We remark
that the proposed decomposition approach solves in-
stances of the E-VRP with time windows and partial
recharges, at each iteration. The corresponding MILP
cannot be solved with commercial solver to optimal-
ity, within reasonable computational time for larger
instances. Future work should consider heuristic ap-
proaches for solving larger problems. It is possible to
derive a heuristic based on the proposed decomposi-
tion approach by imposing a time limit to the solver
An Adjustable Robust Formulation and a Decomposition Approach for the Green Vehicle Routing Problem with Uncertain Waiting Time at
Recharge Stations
79

for solving the master problems. Hence, a feasible
rather than an optimal solution is considered, at each
iteration.
It is worth noting that we consider a simplified dis-
charging model of the battery, in which the discharg-
ing is a linear function depending on the distance trav-
elled. Actually, the discharge is influenced by sev-
eral factors, such as the speed, the load of the vehicle,
the gradient, and so on. Hence, the discharging func-
tion is non-linear. As future work, investigating how
a more realistic discharging function influences the
transportation system under uncertain waiting time,
should be worthy.
An interesting version of the problem is to con-
sider uncertain discharge rate. This assumption al-
lows to avoid to take into account complicating non-
linear discharging function and to prevent energy dis-
ruption.
REFERENCES
Agra, A., Christiansen, M., Figueiredo, R., Hvattum, L. M.,
Poss, M., and Requejo, C. (2013). The robust vehicle
routing problem with time windows. Computers &
operations research, 40(3):856–866.
Basso, R., Kulcs
´
ar, B., Egardt, B., Lindroth, P., and
Sanchez-Diaz, I. (2019). Energy consumption estima-
tion integrated into the electric vehicle routing prob-
lem. Transportation Research Part D: Transport and
Environment, 69:141–167.
Bertsimas, D. and Sim, M. (2003). Robust discrete opti-
mization and network flows. Mathematical program-
ming, 98(1-3):49–71.
Bruni, M., Di Puglia Pugliese, L., Beraldi, P., and Guer-
riero, F. (2017). An adjustable robust optimization
model for the resource-constrained project schedul-
ing problem with uncertain activity durations. Omega,
71:66–84.
Bruni, M., Di Puglia Pugliese, L., Beraldi, P., and Guer-
riero, F. (2018). A computational study of ex-
act approaches for the adjustable robust resource-
constrained project scheduling problem. Computers
& Operations Research, 99:178–190.
Desaulniers, G., Errico, F., Irnich, S., and Schneider, M.
(2016). Exact algorithms for electric vehicle-routing
problems with time windows. Operations Research,
64:1388–1405.
Di Puglia Pugliese, L., Guerriero, F., and Poss, M. (2019).
The resource constrained shortest path problem with
uncertain data: A robust formulation and optimal so-
lution approach. Computers & Operations Research,
107:140–155.
Ding, N., Battay, R., and Kwon, C. (2015). Conflict-
free electric vehicle routing problem with ca-
pacitated charging stations and partial recharge.
https://www.chkwon.net/papers.
Erdo
˘
gan, S. and Miller-Hooks, E. (2012). A green vehi-
cle routing problem. Transportation Research Part E:
Logistics and Trasportation Review, 48(1):100–114.
Felipe, A., Ortu
˜
no, M. T., Righini, G., and Tirado, G.
(2014). A heuristic approach for the green vehi-
cle routing problem with multiple technologies and
partial recharges. Transportation Research Part E,
71:111–128.
Froger, A., Mendoza, J., Jabali, O., and Laporte, G. (2019).
Improved formulations and algorithmic components
for the electric vehicle routing problem with nonlin-
ear charging functions. Computers & Operations Re-
search, 104:256–294.
Goeke, D. (2019). Granular tabu search for the pickup and
delivery problem with time windows and electric ve-
hicles. European Journal of Operational Research,
278:821–836.
Gonc¸alves, F., Cardoso, S. R., Relvas, S., and Barbosa-
P
´
ovoa, A. P. F. D. (2011). Optimization of a distribu-
tion network using electric vehicles: A vrp problem.
15ˆCongresso Nacional da Associac¸
˜
ao Portuguesa de
Investigac¸
˜
ao Operacional.
Hiermann, G., Hartl, R. F. Puchinger, J., and Vidal, T.
(2019). Routing a mix of conventional, plug-in hy-
brid, and electric vehicles. European Journal of Op-
erational Research, 272:235–248.
Hiermann, G., Puchinger, J., Ropke, S., and Hartl, R. F.
(2016). The electric fleet size and mix vehicle routing
problem with time windows and recharging stations.
European Journal of Operational Research, 252:995–
1018.
Joo, H. and Lim, Y. (2018). Ant colony optimized rout-
ing strategy for electric vehicles. Journal of Advanced
Transportation, 2018:9.
Keskin, M. and C¸ atay, B. (2016). Partial recharge strategies
for the electric vehicle routing problem with time win-
dows. Transportation Research Part C, 65:111–127.
Keskin, M., C¸ atay, B., and Laporte, G. (2021). A
simulation-based heuristic for the electric vehicle
routing problem with time windows and stochastic
waiting times at recharging stations. Computers and
Operations Research, 125:105060.
Keskin, M., Laporte, G., and C¸ atay, B. (2019). Electric
vehicle routing problem with time-dependent waiting
times at recharging stations. Computers & Operations
Research, 107:77–94.
L
¨
offler, M., Desaulniers, G., Irnich, S., and Schneider,
M. (2020). Routing electric vehicles with a single
recharge per route. Networks, 76(2):187–205.
Lin, J., Zhou, W., and Wolfson, O. (2009). Electric vehicle
routing problem. In Transportation Research Proce-
dia, pages 508–521, Tenerife, Canary Islands (Spain).
The 9th International Conference on City Logistics.
Lu, D. and Gzara, F. (2015). The robust crew pairing prob-
lem: model and solution methodology. Journal of
Global Optimization, 62(1):29–54.
Macrina, G., Di Puglia Pugliese, L., Guerriero, F., and
Laporte, G. (2019a). The green mixed fleet vehicle
routing problem with partial battery recharging and
time windows. Computers & Operations Research,
101:183–199.
ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems
80

Macrina, G., Laporte, G., Guerriero, F., and
Di Puglia Pugliese, L. (2019b). An energy-
efficient green-vehicle routing problem with mixed
vehicle fleet, partial battery recharging and time win-
dows. European Journal of Operational Research,
276(3):971–982.
Montoya, A., Gu
´
eret, C., Mendoza, J. E., and Villegas, J. G.
(2017). The electric vehicle routing problem with non-
linear charging function. Trasportation Research Part
B, 103:87–110.
Paz, J. C., Granada-Echeverri, M., and Escobar, J. W.
(2018). The multi-depot electric vehicle location rout-
ing problem with time windows. International Jour-
nal of Industrial Engineering Computations, 9:123–
136.
Pessoa, A. A., Di Puglia Pugliese, L., Guerriero, F., and
Poss, M. (2015). Robust constrained shortest path
problems under budgeted uncertainty. Networks,
66(2):98–111.
Poss, M. (2014). Robust combinatorial optimization with
variable cost uncertainty. European Journal of Oper-
ational Research, 237(3):836–845.
Sassi, O., Cherif, W. R., and Oulamara, A. (2014). Ve-
hicle routing problem with mixed feet of conventional
and heterogenous electric vehicles and time dependent
charging costs. Technical report. https://hal.archives-
ouvertes.fr/hal-01083966.
Schneider, M., Stenger, A., and Goeke, D. (2014). The
electric vehicle routing problem with time windows
and recharging stations. Transportation Science,
48(4):500–520.
Solomon, M. M. (1987). Algorithms for the vehicle rout-
ing and scheduling problems with time window con-
straints. Operations Research, 35(2):254–265.
Yang, J. and Sun, H. (2015). Battery swap station location-
routing problem with capacitated electric vehicles.
Computers & Operations Research, 55:217–232.
Zhang, S., Chen, M., and Zhang, W. (2019). A novel
location-routing problem in electric vehicle trans-
portation with stochastic demands. Journal of Cleaner
Production, 221:567–581.
APPENDIX
The problem defined by constraints (23) can be recast
as follows:
min 0 (38)
s.t.
max
w∈ext(U
Γ
)
T (P ). (39)
The polytope T (P ) can be rewritten as:
τ
j
(w, P ) − τ
i
(w, P ) ≥ (t
i j
+ s
i
)x
P
i j
− M1(1 −x
P
i j
),
∀i ∈ N ∪ {s}, j ∈ V
P
\ {s}, (40)
τ
j
(w, P ) − τ
i
(w, P ) ≥ t
i j
x
P
i j
+
1
ρ
i
g
P
i j
+ w
i
+
−M1(1 − x
P
i j
), ∀i ∈ R
P
, j ∈ V
P
\ {s}, (41)
τ
j
(w, P ) ≥ e
j
, ∀ j ∈ N . (42)
We remark that x
P
and g
P
represent the first stage
solution. In particular, x
P
i j
assume value equal to 1
for each arc (i, j) included in some route, i.e., x
P
i j
=
1, ∀(i, j) ∈ A(P ) where A(P) = {(i, j) ∈
S
p∈P
A(p)}
is the set of all arcs included in the solution P .
Variables g
P
i j
> 0 for each i ∈ R
P
and j ∈ V
P
such that (i, j) ∈ A(P ) since a recharge station is vis-
ited only if the vehicle recharges the battery. We can
consider two disjoint subsets, i.e., A
1
(P ) = {(i, j) ∈
A(P ) : i ∈ N ∪ {s}} and A
2
(P ) = {(i, j) ∈ A(P ) : i ∈
R
P
}. Hence, constraints (40)–(42) become
τ
j
(w, P ) − τ
i
(w, P ) ≥ t
i j
+ s
i
,
∀(i, j) ∈ A
1
(P ), (43)
τ
j
(w, P ) − τ
i
(w, P ) ≥ t
i j
+
1
ρ
i
g
P
i j
+ w
i
,
∀(i, j) ∈ A
2
(P ), (44)
τ
j
(w, P ) ≥ e
j
, ∀ j ∈ N . (45)
Let A
|A
1
(P )|×|V
P
|
1
and A
|A
2
(P )|×|V
P
|
2
be the coefficient
matrix of constraints (43) and (44), respectively, and
A
|N |×|N |
3
be the coefficient matrix of constraints (45).
Let t
1
, t
2
, and e be the right-hand-side terms of con-
straints (43), (44), and (45), respectively. The dual of
problem (38), (39) is the following:
min max
w∈ext(U
Γ
)
t
1
f
1
+t
2
f
2
+ e f
3
(46)
s.t.
A
T
1
f
1
= 0, (47)
A
T
2
f
2
= 0, (48)
A
T
3
f
3
= 0, (49)
f
1
, f
2
, f
3
≥ 0, (50)
where f
1
∈ R
|A
1
(P )|
+
, f
2
∈ R
|A
2
(P )|
+
and f
3
∈ R
|N |
+
are the
variables dual to constraints (43), (44), and (45), re-
spectively. We note that A
T
= A
T
1
∪A
T
2
represents the
incident matrix related to the first stage routing solu-
tion. Hence, f = f
1
∪ f
2
are flow variables and con-
straints (47) and (48) represent the flow-conservation
constraints. The constraints A
T
f = 0 is a circulation,
indeed, all nodes are of transit type. This because
each route starts and ends at the same depot.
An Adjustable Robust Formulation and a Decomposition Approach for the Green Vehicle Routing Problem with Uncertain Waiting Time at
Recharge Stations
81