Last Mile Delivery with Lockers: Formulation and Heuristic
Willian Jorge Pereira Oliveira
and Andr
e Gustavo dos Santos
Universidade Federal de Vic¸osa, Vic¸osa, Brazil
Combinatorial Optimization, Vehicle Routing, Facility Location, Location Routing Problem, Last Mile
The creation of efficient routes is essential for different areas having several practical applications mainly in
the transport of goods. With the growth of e-commerce and consequently the increase in demand for delivery
to end users, minimizing costs in the delivery process has gained more importance, especially the last mile
stage. It is in this context that the use of lockers emerges to optimize last mile deliveries. Lockers have
compartments of different sizes, with self-service interface and they can be positioned in supermarkets, parks
and other areas that are of interest to customers. The problem addressed in this work is to determine the
positioning of the lockers and the necessary routes to supply them and to serve the remaining customers. We
present a mathematical model to define the problem, but due to the complexity of the problem obtaining a
solution can be very expensive and require a lot of computational effort, therefore we present a heuristic,
based on Variable Neighborhood Descent (VND), using a greedy construct inspired by the Clark & Wright
savings method. By comparing the results of the heuristic with the Gurobi optimizer, we conclude that the
heuristic is capable of obtaining competitive solutions in less time than the exact methods.
The creation of routes and facilities allocation have
been the focus of studies for a long time due to its
applicability in different areas. Several characteris-
tics can be explored in these problems, such as the
homogeneity of the vehicle fleet to be used, deliv-
ery time constraints, among others. The growth of
e-commerce in recent years and, consequently, the de-
mand for delivery to customers, turned the efficient
routes planning an increasingly important problem.
Due to the worldwide urbanization trend, traffic prob-
lems have increased. There are several challenges
to deal with the delivery of goods, among them the
last mile problems have gained more attention, mainly
due to the traffic of large centers.
The last mile step consists of distributing products
from the distribution center to final consumers. This
step may have different characteristics from the rest
of the delivery process and that is why it is necessary
to study it separately.
The problem addressed in this work assumes the
inclusion of lockers as an alternative to reduce costs
in last mile deliveries. Lockers are kiosks with com-
partments accessed by self-service interface that can
be used to store the goods of customers until they can
collect them, reducing completely the fail to deliver-
ing caused by the absence of the customer. The model
proposed defines the problem mathematically and a
heuristic is suggested to obtain satisfactory solutions
to the problem. In the next section we detail the use of
lockers with real-world examples and related works in
the literature.
There are several challenges in last mile deliveries.
Among them, we can highlight the increasing in con-
gestion in large centers, the increasing in pollution,
the demand of the consumer in relation to services,
in addition to restrictions, such as the time window
available for delivery. The use of lockers has been
shown as an alternative to deal with such challenges
due to the flexibility of the collection and supply
schedule. The use of several shifts of the same vehi-
cle can help to reduce the size of the fleet. Song et al.
(2009) shows that the use of lockers generates savings
in the last mile stage, mainly due to the repeated de-
liveries that need to be made when the consumer is
Oliveira, W. and Santos, A.
Last Mile Delivery with Lockers: Formulation and Heuristic.
DOI: 10.5220/0009580004600467
In Proceedings of the 22nd International Conference on Enterprise Information Systems (ICEIS 2020) - Volume 1, pages 460-467
ISBN: 978-989-758-423-7
2020 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
not at the address to receive your order.
Lockers are collection points, where goods are
available for a certain period of time and users can
collect them on an appropriate time. User interaction
with the locker is done through a self-service interface
that requires user authentication to ensure the security
of orders.
Lockers can be located anywhere that facilitates
their access, such as shopping malls, restaurants, gas
stations and also in residential areas as long as there
is demand from customers.
The installation of the locker in places that have
parking, toilets, among other facilities, helps in pop-
ularizing the locker. It can also bring advantages to
existing businesses close to the locker, as the circula-
tion of customers can generate consumption and con-
sequently profit for traders.
The location is the most important and chal-
lenging feature of the implementation of the self-
collection, because, in addition to the investment in
the construction of the locker, it is essential for its
popularization. In this work, we consider that each
locker has a coverage distance, that is, we consider
that users with addresses in the coverage area would
accept that their orders were delivered to the locker.
Wang et al. (2017) highlight the closure of sev-
eral lockers in Singapore between 2015 and 2017, the
main indication being that the location of the lock-
ers did not favor its popularization. The Locker Al-
liance (LA) was proposed by the government of Sin-
gapore in 2019 and consists of a network of lockers
distributed across residential areas and metro stations
to help to optimize last mile delivery. Lyu and Teo
(2019) propose methods to define the location of the
network lockers and the demand to be aimed at them.
In door-to-door delivery, it is necessary to con-
sider additional time for interaction with the con-
sumer and the possibility that the consumer is not
available to receive his goods. Building lockers, we
can reduce operating costs due to the condensation of
several deliveries at the same point, reducing the dis-
tance covered, eliminating the interaction time with
the consumer and consequently reducing the delivery
time. The supply of the lockers can also be done at pe-
riods with less traffic, using the same vehicle for dif-
ferent work shifts. With the decrease in delivery ve-
hicles circulating during heavy traffic hours, we have
a small reduction in the emission of gases harmful to
the environment (Edwards et al., 2009).
In this work, the objective is to determine which
lockers will be built, the routes to supply them and the
routes to meet the demand of customers who have not
been allocated to lockers.
We can consider this problem as a variation of the
location-routing problem since it is necessary to de-
termine the location of the lockers and create routes
to meet the demands. Customers’ demands can be
met in two different ways: by door-to-door delivery
or through the collection from lockers. Given a set
of potential locations for building lockers and a set of
customers with their respective demands, it is neces-
sary to create two groups of routes: the delivery routes
and supply routes, and the associations between cus-
tomers and lockers. The built-in lockers must also be
served through supply routes.
The last mile problem with the use of lockers is dis-
cussed by Wen and Li (2016). The paper presents
a model using lockers in vehicle routing, consider-
ing relevant aspects such as CO
emission, customer
time window and congestion. Actual data from Ming-
guangcun in Beijing is used to apply the proposed
model. In this case study, it was possible to conclude
that the use of lockers contributes to a better solution,
because, relaxes the restriction of time window and
reduces traffic congestion.
Faugere and Montreuil (2016) analyzed the busi-
ness model of twelve companies using lockers and/or
access points in order to identify different characteris-
tics in their services. The data were obtained through
the companies’ websites and online press. The enti-
ties involved in the process, and the impacts generated
by them, are observed: customers, salespeople, deliv-
ery personnel, cities, and the environment. The re-
sults consist of the classification of the business mod-
els used by the companies and their general charac-
Through the observed points, it was possible to
conclude that the use of lockers generates a decrease
in the delivery time. However, it is a good solution
only if they are conveniently located for customers.
For logistics providers, they may have a reduced fleet
since delivery points are condensed in the lockers,
also eliminating multiple trips due to the absence of
the customer at the time of delivery. For sellers, the
price of deliveries is expected to be reduced in the
long term. Finally, reducing gas emissions and con-
gestion is an advantageous consequence for cities and
the environment.
Veenstra et al. (2018) proposed to integrate the fa-
cility location problem with the vehicle routing prob-
lem and lockers service. This integrated problem was
applied to the delivery of medicines in the Nether-
lands. This problem differs from other classic prob-
lems due to the possibility of serving a customer in
two ways, through the locker or with door-to-door
Last Mile Delivery with Lockers: Formulation and Heuristic
service. A mathematical model and a heuristic are
defined to solve the problem. The objective is to min-
imize total costs, considering the routes and the cost
of opening the lockers. Computational tests were per-
formed with two sets of instances: random and based
on real data. The branch-and-bound method was ap-
plied to the proposed mathematical model, obtaining
results for instances with 100 patients and 50 poten-
tial lockers with a time limit of 7200 seconds. The
proposed heuristic achieves solutions that surpass the
optimization software, CPLEX, with only a fraction
of the time (4.18% better for the set of random in-
stances and 3.64% better for the instances based on
real data). The heuristic proved to be consistent in
obtaining the same results considering 10 executions
for each instance.
Wang et al. (2017) discussed relevant questions
about the use of lockers. The paper uses Singapore’s
case as an example, where the lockers from POPSta-
tion, a leader in this market, are positioned at 2500m
from each other. From 2015 to 2017 it was observed
that several lockers were closed permanently. Ac-
cording to the authors, the location of the lockers is
crucial for its popularization, so they must be posi-
tioned in an attractive way for the customer. The com-
petition between various companies and their lock-
ers should also be considered. The article also con-
tributed to the literature because it was the first work
related to lockers and your positions, which is based
on real and public data. Real distances between loca-
tions were also used instead of the Euclidean distance,
commonly used in the literature.
Huang et al. (2019) present the problem of vehicle
routing using electric cars and stations. The stations
can be of three types, the first one, only to recharge
cars used on delivery of goods, the second one, only
to store goods of customers, such as locker, the third
one, a hybrid station, which can be used for both pur-
poses. In the work, two integer programming models
are proposed: in the first one, the routes for supply-
ing the lockers and the routes for customer deliveries
are separated; in the second one, a hybrid route is al-
lowed. A hybrid heuristic is proposed which, com-
pared to CPLEX, presents good results efficiently and
effectively. The results obtained show that the use
of lockers combined with door-to-door deliveries can
help to reduce the cost of deliveries.
Relevant reasons are presented for calculating the
routes separately. Firstly, it is not possible to measure
the time that would be spent on each customer. There-
fore, on a hybrid route, if a delivery to a customer is
delayed, one is forced to delay the delivery to a locker,
which will affect multiple users. Second, the hours of
interest for deliveries are different: lockers may be lo-
cated at points with congestion during business hours,
so it is interesting to deliver at less busy times, while
for door-to-door delivery, deliveries are usually dur-
ing business hours. Third, the required skills of the
driver are different depending on the type of delivery
he/she will make: for door-to-door deliveries, the de-
livery person must have interpersonal skills while for
the supply to lockers the driver must have availability
in times less conventional.
In this paper, a variation is also proposed that al-
lows for more work shifts. The same vehicle can be
used during the day for door-to-door deliveries and,
at night, for supplying lockers. In this variation, the
vehicle may have more than one route associated with
it and the fixed cost of the vehicle is considered only
once. In all of the proposed models, the objective is to
minimize the total costs including expenses with the
construction of the stations and the fixed costs of the
vehicles, considered heterogeneous. We do not con-
sider the construction costs, as lockers are built only
once and used for years. Considering these costs on
a daily delivery would highly overestimate the cost.
The instances used to test the methods were created
by the authors by combining several instances of the
literature. They consider electrical vehicles but char-
acteristics about charging electric cars at stations are
not considered, such as charging waiting time, the
possibility of changing batteries or charging queue.
As part of The Federated Lockers and Collection
Points program, the government of Singapore has pro-
posed the creation of the Locker Alliance (LA) which
consists of a network of lockers to complement exist-
ing lockers and to improve the performance of deliv-
eries to consumers. Lyu and Teo (2019) aim to deter-
mine the best design of the locker network in order to
increase its coverage and use. With the increasing in
coverage, it is expected that the concentration of ex-
isting deliveries in the central business district (CBD)
would be reduced by 7.5% due to the possibility of
collecting items in locations away from the CBD.
Unlike the current work, the solutions obtained by
them are evaluated on the perspective of the attrac-
tiveness of the use of lockers, that is, the objective
is to optimize the location of the lockers so that the
volume of users who choose this type of delivery is
maximized. The data collected in the study are from
a time when LA did not exist, and therefore, the work
takes into account that after the implementation of the
network, certain changes will occur in the choice of
users. Using real data, a model was developed and
calibrated to calculate whether the user would be sub-
ject to using the locker, thus, it was possible to mea-
sure the effectiveness of the network design concern-
ing the popularity of locker. The supply and delivery
ICEIS 2020 - 22nd International Conference on Enterprise Information Systems
routes of users who have not opted for the locker are
not addressed in their work.
We propose here an Integer Linear Programming for-
mulation to formally define the problem.
As input we have the location of the depot, the
customers and the candidate positions for lockers. We
consider a complete graph, i.e., we know the distance
between any pair of locations.
As output we have the the lockers chosen to be
used, which customers are associated to each of those
lockers, and the routes to delivery items to the lockers
and to the customers not served by any locker. All
the output is represented by binary variables, but we
use also integer variable to keep track of the load in
each vehicle, as a way to guarantee that the capacity
of lockers and vehicles are satisfied and also avoid
isolated sub-tours along the routes.
3.1 Input and Decision Variables
The parameters and the decision variables of the pro-
posed model are presented below.
dep: depot
I: set of customers
: set of nodes, I
= {dep} I
J: set of candidate location for lockers
: set of nodes, J
= {dep} J
n: number of customers, n = |I|
m: number of locker candidate locations, m = |J|
: demand of customer i
Q: capacity of vehicles for door-to-door delivery
QL: capacity of vehicles that supply lockers
capacity of locker j
i j
: binary, 1 if arc (i, j ) used, i, j I
, 0 otherwise
i j
: binary, 1 if arc (i, j) used, i, j J
, 0 other-
i j
: integer, flow on arc (i, j) on customers’ route
f L
i j
: integer, flow on arc (i, j) on lockers’ route
: binary, 1 if customer i is served by a locker, 0
i j
: binary, 1 if i is served by locker j, 0 otherwise
: binary, 1 if locker j is used, 0 otherwise
i j
: distance between nodes i and j, i, j I
i j
: distance between nodes i and j, i, j J
i j
: distance between customer i and locker j
: cover distance of locker j
3.2 Objective Function
The objective function is defined below.
i j
i j
+ (
i j
i j
) 0.8 (1)
The aim is to minimize the total cost of the routes.
We consider the cost proportional to the distance, then
the cost of the customers’ routes is simply the sum of
the distances of the arcs used (first term of (1)). For
the lockers supply routes (second term), we consider
the cost as 80% of the normal cost, as these routes are
favourable due to several conditions: higher flexibil-
ity in the period of attendance; no temporary absent
and no need to return,, which may happen in the cus-
tomers’ route; possibility of night attendance, in pe-
riod of no traffic congestion; possibility to reuse the
same vehicle used in normal routes during the day, so
as to reduce fixed costs of vehicles.
3.3 Constraints
i j
) + l
= 1, i I (2)
) + l
= 1, i I (3)
) (
i j
) + (l
) = q
, i I (5)
i j
Q x
i j
, i, j I
i j
, j J (7)
i j
= l
, i I (8)
i j
i j
, i I, j J (9)
i j
, i I, j J (10)
= c
, j J (11)
i j
= c
, j J (12)
dep, j
f L
i j
i j
, i, j J
f L
i j
f L
k j
, j J (15)
Last Mile Delivery with Lockers: Formulation and Heuristic
i j
{0, 1}, i, j I
i j
{0, 1}, i, j J
i j
0, i, j I
f L
i j
0, i, j J
{0, 1}, i I (20)
{0, 1}, i I, j J (21)
{0, 1}, j J (22)
Routes are modeled using flow variables to con-
trol nodes demand service and avoid sub-tours. There
are two set of routes, one set for in-door service (con-
straints (2)- (6)) and one set for lockers supply (con-
straints (11)- (15)). The remaining constraints (7)-
(10) define the assignment of customers to lockers
and, finally, (16)-(22) the range of variables.
Constraint (2) and (3) define that must be a chosen
arc leaving and respectively reaching each customer
or else the customer must be served by a locker. To-
gether, they define that each customer must be served
by a locker or by exactly one in-door route.
Constraint (4) balance the number of arcs leaving
and reaching the depot, so that every route starting at
the depot must come back to it. Notice that, although
a customer may not be served by more than one route,
the depot may be in as many routes as needed.
The demand of each customer is guaranteed by
constraint (5), either by the flow of a route or by as-
signing it to a locker. The following two constraints
establish that the capacity of vehicles may not be ex-
ceeded in any point of the route neither the capacity
of a locker may be surpassed.
Constraints (8)-(10) assure that a customer may be
assigned only to lockers nearby (within their covering
distance) and at most to one locker. If customer i is
assigned to a locker j, the corresponding variables l
(customer served by locker) and c
(locker used) are
set to 1.
Constraints (11)-(15) control the routes to serve
lockers in the same way constraints (2)- (6) control
the door-to-door routes. Particularly, (11) and (12)
define that, if a locker is used, it must be served by
a route (arc arriving and leaving the node); (13) bal-
ances the number of arcs leaving and reaching the de-
pot (the number of routes to supply lockers); (14) as-
sure that the capacity of the vehicle is respected and
(15) that when visiting a locker the demand of all cus-
tomers assigned to it must be unload there.
Finally, constraints (16)-(22) define the domain of
the decision variables.
As detailed later in the experiments, optimal solutions
using the proposed formulation are reached only for
small instances, with up to 75 customers. Heuristic
method is needed to handle larger instances.
The proposed heuristic follows the standard Vari-
able Neighborhood Descent (VND) metaheuristic as
proposed by Mladenovi
c and Hansen (1997), using a
local search with three neighborhood. To provide an
initial solution, a random greedy constructive method
was used. The construction is followed by a local
search and the two steps are performed 200 times, af-
ter which the best solution found is returned.
A solution consists of two parts, a set of routes and
a set of assimilations. Each route can be of delivery
or supply. Both are represented by a vector of identi-
fiers, of customers or lockers. Where n is the number
of lockers, the assimilations consist of n lines, starts
with locker identifier followed by the customers as-
similates to each locker.
An initial solution is generated by a greedy ran-
domized solution: a subset of candidate lockers is
chosen at random, and the customers within the cover-
age distance of these lockers are sorted by distance to
the depot. The further away the customer the sooner
it is associated to a locker.
Customers may end up not associated to any
locker, some because they are not within the coverage
radius of any locker and others because when they are
considered the possible lockers cannot serve their de-
mand anymore due to previously assignments. Routes
are built to serve those customers using the Clare and
Wright saving heuristic (Clarke and Wright, 1964),
satisfying vehicle capacities. The lockers are served
by separated routes, created with the same method.
The routes of the constructed solution is then im-
proved by a local search using three different neigh-
borhoods: 2-swap, 2-opt and exchange. They are used
sequentially, as in the VND metaheuristic.
The 2-swap neighborhood consists in swapping
two customers inside a route. A best improvement
strategy is used in this stage: the best swap is chosen
and performed until there is no possible improvement.
For the 2-opt neighborhood, two nodes are ran-
domly chosen and the visit sequence between those
nodes is reversed. Successive improvements are made
until a stop criteria is reached. We used as stop con-
dition 500 iteration without improvement.
The third and last neighborhood is the exchange.
For each node a, the nearest node b in a different route
is determined. If moving node a to the route of b just
before or after b improves the cost, it is considered for
exchange. Then, the best improvement is made. The
ICEIS 2020 - 22nd International Conference on Enterprise Information Systems
process continues until no improvement can be made.
The algorithm of VND-based heuristic can be
seen below:
for(int i=0;i<ITERATIONS;i++){
s = constructiveMethod();
int k = 1;
s’ = bestOfNeighborhood(k);
if (f(s’) < f(s)){
s = s’;
k = 0;
if (f(s) < f(s*)){
s* = s;
return s*;
We adapt a set of classic instances from the Vehicle
Routing Problem (VRP) (Rochat and Taillard, 1995).
We chose 8 instances, with 75, 100 and 150 cus-
tomers, and adapt them by inserting 10 candidates for
lockers. The position of these candidates were manu-
ally chosen in areas with many customers. The capac-
ity of each locker was defined randomly in the range
of 10 to 20 times the average of customers demand,
so a locker may serve around 10 to 20 customers.
Table 1 lists some characteristics of each instance:
the number of customers (n) and the capacity of the
vehicles (Q). We do not impose a limit on the ca-
pacity of the vehicles that supply lockers (QL), as the
capacity of lockers in the instances used already limit
a reasonable capacity.
Table 1: Instances.
ID #Customers #Vehicle Capacity
c tai75a 75 1445
c tai75b 75 1679
c tai75c 75 1122
c tai75d 75 1699
c tai100a 100 1409
c tai100b 100 1842
c tai100c 100 2043
c tai150a 150 1544
The machine used was an i5-7400 CPU @ 3.00GHz
with 8GB RAM. The Integer Linear Programming
(ILP) model was implemented in Julia, using JuMP
as modelling language, and run on Gurobi solver, with
time limit of 1 hour. The heuristic was implemented
in C++ and run 3 times for each instance. The average
value is reported for analysis.
The results of the ILP model are reported in Table
2: solution value, runtime and the linear gap reported
by Gurobi. Optimal results were found for only 3 in-
stances, among those with less customers. However,
the difficult is not related only to the number of cus-
tomers. Instance c tai75a has the same number of
customers and lockers but the optimal solution was
not guaranteed in the time available, finishing with a
gap of 3.6%. This may be due to the relative loca-
tion of lockers and customers. For example, a cus-
tomer may be located in an area covered by two or
more lockers, so may be served by any of them either
by a door-to-door route. The higher the alternatives
the greater the solution space, which may increase the
running time to reach an optimal solution.
Table 2: Results of the ILP formulation.
Instance Solution Runtime gap
c tai75a 803.613 3600 3.63
c tai75b 537.113 25 -
c tai75c 804.118 194 -
c tai75d 420.707 55 -
c tai100a 971.408 3600 3.13
c tai100b 982.592 3600 4.93
c tai100c 870.561 3600 2.59
c tai150a 1097.150 3600 16.01
Table 3 show the results of the proposed heuris-
tic. Column gap
represents the percentual differ-
ence between the solution of the heuristic and the one
reached by the ILP in 1 hour. One may notice that
the heuristic found the same result for 4 instances.
Column gap
show the percentual difference to the
lowew bound found the ILP model, which is the best
known still possible result. Besides the two optimal
solutions, for three other the gap was less than 4%.
The time to reach the optimal solution is substan-
tially less in comparison to the ILP model, and for the
others the gap was less than 5%.
In Figure 1 we can see the comparison between
the lower bound and the two approaches proposed
in this work. The results obtained by ILP and VND
are close for most instances, increasing for larger in-
stances, however, as previously mentioned, the time
spent by VND to obtain solutions is significantly less.
Last Mile Delivery with Lockers: Formulation and Heuristic
Table 3: Results of the heuristic.
Instance VND time gap
c tai75a 803.613 9 - 3.63
c tai75b 537.113 6 - -
c tai75c 813.119 9 1.11 1.11
c tai75d 420.707 6 - -
c tai100a 971.408 22 - 3.13
c tai100b 1002.820 19 2.02 6.85
c tai100c 908.177 17 4.14 6.62
c tai150a 1136.280 54 3.44 18.91
Figure 1: Comparison of the Lower Bound reported by
CPLEX and the feasible solutions from ILP and VND for
each instance.
The instances used in this work, proposed by
Rochat and Taillard (1995), are commonly used for
the VRP and some of its variations. Table 4 show
on column VRP the best known results for the clas-
sic VRP, which may be used to study and foresee
the saving that the use of lockers can bring. Solu-
tions using lockers save in average 52% of the trans-
portation costs. The real saving is not that amount,
of course, because we are considering that routes to
supply locker are 20% cheaper and we are not consid-
ering the costs of building and maintaining the lock-
ers. However, the saving is clear, as the fixed costs of
locker are spread over years and the costs of the table
are a daily cost.
Table 4: Impact of the use of lockers.
Instance VRP ILP VND
c tai75a 1618.36 803.613 803.613
c tai75b 1344.64 537.113 537.113
c tai75c 1291.01 804.118 813.119
c tai75d 1365.42 420.707 420.707
c tai100a 2041.34 971.408 971.408
c tai100b 1940.61 982.592 1002.820
c tai100c 1406.21 870.561 908.177
c tai150a 3055.23 1097.150 1136.280
Table 5 shows, for each instance, the number of
customers that are served by lockers in the best so-
lution found by each method. One might intuitively
think that the optimal solution would include the max-
imum possible number of customers to lockers. How-
ever, this is not always the case, as shown by the re-
sults for the instance c tai100c: the heuristic solu-
tion is 4% costly, despite having more customers as-
sociated to lockers. This happens because, in some
scenarios, the cost to visit a locker may be higher
than the cost to include the costumers associated to
it in other route. Hence, even disregarding installa-
tion costs, lockers should not be build deliberately, as
there would be still supply costs.
Table 5: Customers served by lockers.
Instance Customers ILP VND
c tai75a 75 62 62
c tai75b 75 69 69
c tai75c 75 58 57
c tai75d 75 72 72
c tai100a 100 82 82
c tai100b 100 83 83
c tai100c 100 59 61
c tai150a 150 125 121
Due to the increasing popularity of e-commerce the
last mile step in the delivery of goods has become
more important and relevant in the delivery planning.
The use of lockers has gained attention as an alterna-
tive to reduce costs, to optimize the delivery time and
is also beneficial to the environment, as it contribute
to reduce gas emission.
In this work we study methods for a vehicle rout-
ing combined with a facility problem for delivering
goods using lockers. To define and solve the prob-
lem we propose an integer linear programming model,
implemented and solved by Gurobi. Due to the com-
plexity of the formulation, a heuristic is also proposed
to handle larger instances. The heuristic is based on
the VND metaheuristic and uses 3 neighborhoods to
improve a greedy initial solution.
Experimental tests were made with 8 cases
adapted from VRP classical instances. The ILP for-
mulation found the optimal solution for three in-
stances and a linear gap below 5% for other four in-
stances within 1 hour of execution time. The heuristic
found solutions close to the ones found by the ILP for
most of the instances in few seconds.
ICEIS 2020 - 22nd International Conference on Enterprise Information Systems
We show the impact of the use of the lockers and
as future works we plan to incorporate time windows
and traffic conditions, as the problem in the real con-
text has such characteristics. We also plan to improve
the heuristic by adding more neighborhoods and use
good solutions to instead of creating new random so-
lutions on each iteration.
The authors thank Coordenac¸
ao de Aperfeic¸oamento
de Pessoal de N
ıvel Superior (CAPES) and Fundac¸
de Amparo
a Pesquisa do Estado de Minas Gerais
(FAPEMIG) for the financial support of this project.
Clarke, G. and Wright, J. W. (1964). Scheduling of vehicles
from a central depot to a number of delivery points.
Operations research, 12(4):568–581.
Edwards, J., McKinnon, A., Cherrett, T., McLeod, F., and
Song, L. (2009). The impact of failed home deliveries
on carbon emissions: Are collection/delivery points
environmentally-friendly alternatives. In 14th Annual
Logistics Research Network Conference, page M117.
Faugere, L. and Montreuil, B. (2016). Hyperconnected city
logistics: Smart lockers terminals & last mile deliv-
ery networks. In Proceedings of the 3rd International
Physical Internet Conference, Atlanta, GA, USA, vol-
ume 29.
Huang, Z., Huang, W., and Guo, F. (2019). Integrated sus-
tainable planning of self-pickup and door-to-door de-
livery service with multi-type stations. Computers &
Industrial Engineering.
Lyu, G. and Teo, C.-P. (2019). Last mile innovation: The
case of the locker alliance network. Available at SSRN
c, N. and Hansen, P. (1997). Variable neigh-
borhood search. Computers & operations research,
Rochat, Y. and Taillard,
E. D. (1995). Probabilistic diversi-
fication and intensification in local search for vehicle
routing. Journal of heuristics, 1(1):147–167.
Song, L., Cherrett, T., McLeod, F., and Guan, W. (2009).
Addressing the last mile problem: transport impacts
of collection and delivery points. Transportation Re-
search Record, 2097(1):9–18.
Veenstra, M., Roodbergen, K. J., Coelho, L. C., and Zhu,
S. X. (2018). A simultaneous facility location and ve-
hicle routing problem arising in health care logistics
in the netherlands. European Journal of Operational
Research, 268(2):703–715.
Wang, Y., Ong, T., Lee, L. H., and Chew, E. P. (2017).
Capacitated competitive facility location problem of
self-collection lockers by using public big data. In
2017 13th IEEE Conference on Automation Science
and Engineering (CASE), pages 1344–1344. IEEE.
Wen, J. and Li, Y. (2016). Vehicle routing optimization
of urban distribution with self-pick-up lockers. In
2016 International Conference on Logistics, Infor-
matics and Service Sciences (LISS), pages 1–6. IEEE.
Last Mile Delivery with Lockers: Formulation and Heuristic