A Real-World Multi-Depot, Multi-Period, and Multi-Trip Vehicle

Routing Problem with Time Windows

Mirko Cavecchia

1 a

, Thiago Alves de Queiroz

2 b

, Riccardo Lancellotti

3 c

, Giorgio Zucchi

4 d

and

Manuel Iori

1 e

Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia,

42122, Reggio Emilia, Italy

Institute of Mathematics and Technology, Federal University of Catal ˜ao, 75704-020, Catal ˜ao-GO, Brazil

Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, 41125, Modena, Italy

R&D Department, Coopservice S.coop.p.a, 42122, Reggio Emilia, Italy

Keywords:

Vehicle Routing Problem, Multiple Depots, Multiple Periods, Multiple Trips, Variable Neighbourhood

Search, Mixed Integer Linear Programming.

Abstract:

Rich vehicle routing problems frequently arise in real-world applications. The related literature is trying to ef-

fectively address many of them, considering all their constraints and the need to solve large practical instances.

In this paper, we handle a rich vehicle routing problem that emerges from an Italian service provider company

operating in the delivery of pharmaceutical products. This problem is characterized by constraints that include

multiple depots, periods, and trips, as well as speciﬁc time windows, customer-vehicle incompatibilities, and

restrictions on vehicle working hours. To solve this problem, we propose a two-phase decomposition approach.

In the ﬁrst phase, we generate a set of trips that satisfy all problem constraints for a single period. Then, in

the second phase, three algorithms based, respectively, on a greedy heuristic, a variable neighbourhood search

metaheuristic, and a mixed integer linear programming model are tested to combine the generated trips into

routes for the overall set of periods. Computational results on real-world instances show that the model hits

the one-hour time limit for cases with more than 150 trips. The metaheuristic algorithm performs better in

terms of objective function value. The greedy algorithm is faster but is outperformed by the other methods in

terms of solution quality.

1 INTRODUCTION

The logistics industry has experienced rapid growth

in recent years, with the COVID-19 pandemic fur-

ther highlighting the need for accurate and timely

decision-making. In 2020, the global logistics indus-

try was valued at approximately USD 5.73 trillion,

and this ﬁgure is projected to nearly triple by 2028

(Gosain, 2023). Today, one of the main challenges

concerns the development of solution methods that

closely reﬂect real-world scenarios, in which different

factors, such as constraints related to daily operations

https://orcid.org/0000-0002-4129-9876

https://orcid.org/0000-0003-2674-3366

https://orcid.org/0000-0002-9470-8784

https://orcid.org/0000-0002-5459-7290

https://orcid.org/0000-0003-2097-6572

and unforeseen events, must be considered. The goals

are to reduce operational costs, address environmental

concerns, and improve labour efﬁciency.

This study addresses a real-world vehicle routing

problem (VRP) faced by an Italian service provider

company that delivers pharmaceutical products to

hospitals and medical facilities. The problem involves

a set of customers, each with a speciﬁc time window,

who must be served by a heterogeneous ﬂeet of vehi-

cles. Each customer can demand products in differ-

ent periods, and the products have to be delivered in

the periods in which they have been demanded. Vehi-

cles are allowed to perform multiple trips within each

given period. The vehicles are associated with differ-

ent depots, and each vehicle is required to begin and

conclude its route at the same depot. The objective

is to minimize the overall cost associated with the ve-

hicle usage. This is computed by considering a ﬁrst

112

Cavecchia, M., Alves de Queiroz, T., Lancellotti, R., Zucchi, G. and Iori, M.

A Real-World Multi-Depot, Multi-Period, and Multi-Trip Vehicle Routing Problem with Time Windows.

DOI: 10.5220/0013153100003893

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 14th International Conference on Operations Research and Enterprise Systems (ICORES 2025), pages 112-122

ISBN: 978-989-758-732-0; ISSN: 2184-4372

cost if a vehicle is used at least once in the set of pe-

riods, and a second cost incurred each time a vehicle

is used. Due to geographical constraints, some cus-

tomers cannot be served by larger vehicles, thus im-

posing incompatibilities between customers and ve-

hicles. The resulting problem is a challenging variant

of a multi-trip vehicle routing problem with time win-

dows (MTVRPTW).

The main contributions of this work are (i) the

study of a challenging MTVRPTW originating from a

real-world company, (ii) the proposal of a number of

solution algorithms, among which a variable neigh-

bourhood search (VNS) based heuristic (Hansen and

Mladenovi

c, 2001), and (iii) the execution of exten-

sive computational tests on real-world instances.

To solve the problem, we generalize the two-phase

decomposition approach by (Cavecchia et al., 2024),

originally developed to solve the single-period vari-

ant of the problem. In this paper, we extend it to

deal with a multi-depot, multi-period, and multi-trip

vehicle routing problem with time windows (MM-

MVRPTW). In the ﬁrst phase of the decomposition

approach, we generate a set of trips that start and end

at the same depot. In the second phase, we assign

these trips to the set of available vehicles, creating

routes with multiple trips. The aim is to minimize the

costs associated with the usage of the vehicles. We

use this decomposition approach in multiple ways.

First of all, we present a mixed integer linear pro-

gramming (MILP) model that attempts to optimally

combine, in phase two, the trips created in phase one.

We then propose a constructive heuristic, which we

call bounded starting time greedy (BSTG) algorithm,

which assigns the generated trips to the available ve-

hicles according to increasing values of their starting

times. We ﬁnally develop a novel VNS-based heuris-

tic that attempts several assignments of the generated

trips by invoking mechanisms of perturbation and lo-

cal search.

The remainder of this paper is organized as fol-

lows. Section 2 reviews recent contributions related

to the problem that we tackle, including surveys and

seminal studies. Section 3 provides a formal prob-

lem description, outlining its objective function and

constraints. Section 4 presents the details of the two-

phase approach, detailing each developed algorithm.

Section 5 describes the instances that we solved and

the computational results that we obtained from all

the algorithms, contrasting them in terms of objec-

tive function values, computing times, and numbers

of vehicles used. Section 6 offers some concluding

remarks and potential directions for future research.

2 LITERATURE REVIEW

The VRP is a well-known NP-hard problem that

arises in various logistics scenarios. Researchers have

progressively incorporated additional constraints to

better reﬂect real-world conditions, such as time win-

dows, multiple depots, heterogeneous vehicle ﬂeets,

multiple trips, and multi-period planning, among oth-

ers (Toth and Vigo, 2014). When a VRP includes

multiple constraints, it is usually referred to as a rich

VRP (Lahyani et al., 2015). The literature has de-

veloped a large variety of algorithms to address rich

VRPs, ranging from exact methods to advanced meta-

heuristics (Elshaer and Awad, 2020).

Time windows are a common constraint in VRPs,

restricting vehicles to serve each customer within a

speciﬁed time period. Time windows can be either

soft or hard. The former ones are ﬂexible and may

be violated, although this causes a penalty cost. The

latter ones must be strictly observed and a viola-

tion makes the problem solution infeasible. Surveys

on the VRP with time windows (VRPTW) can be

found in (Br

aysy and Gendreau, 2005a) and (Br

aysy

and Gendreau, 2005b). These authors reviewed key

problem components and various heuristic and meta-

heuristic solution methods developed in the litera-

ture. Recently, (Pan et al., 2021) addressed a vari-

ant of the VRPTW involving time-dependent travel

times. They proposed a hybrid heuristic using an

adaptive large neighbourhood search (ALNS) and a

tabu search, successfully improving results from pre-

vious studies. Similarly, (Zhou et al., 2022) devel-

oped a MILP formulation and a hybrid heuristic com-

bining VNS with tabu search to solve a variant of

the VRPTW, which incorporates two-echelon logis-

tics and pickup and delivery constraints. Other stud-

ies involving the VRPTW were recently surveyed by

(Zhang et al., 2022). Besides deﬁning and presenting

the problem mathematical formulation, the authors

discussed the recent progress made by the literature

on this problem.

The VRPTW has also been investigated with ad-

ditional features, such as the presence of multiple de-

pots and periods, besides the opportunity to use mul-

tiple trips per period. Concerning the variant with

multiple trips, (Brand

ao and Mercer, 1998) handled a

real-world problem related to the food industry. They

proposed a tabu search algorithm where the initial so-

lution is generated by combining nearest-neighbour

and insertion movements. The tabu search was com-

pared with prior literature and produced solutions

that balanced daily tours more effectively. (Despaux

and Basterrech, 2016) introduced a hybrid algorithm

merging simulated annealing and local search tech-

A Real-World Multi-Depot, Multi-Period, and Multi-Trip Vehicle Routing Problem with Time Windows

113

niques, generating new solutions by moving cus-

tomers from their routes as well as assigning vehi-

cles to different routes. (Cattaruzza et al., 2016) intro-

duced a new VRPTW variant arising from a real-case

scenario concerning a two-level mutualized distribu-

tion in cities. This variant requires that the vehicle

routes satisfy multiple trips and release dates. The au-

thors proposed a hybrid genetic algorithm incorporat-

ing a local search procedure to reﬁne solutions after

the crossover operation. (Huang et al., 2024) tack-

led a problem originating from waste collection oper-

ations. The authors proposed an arc ﬂow model and

a branch-price-and-cut algorithm based on a set parti-

tioning model. A column generation algorithm and a

surrogate relaxation strategy were combined together

to generate trips. Their algorithm proved to be efﬁ-

cient in comparison with the previous literature.

Regarding the multiple depot VRPTW variants,

(Li et al., 2016) studied the case in which vehicles

can start and end their routes in different depots. They

proposed a hybrid genetic algorithm with an adaptive

local search mechanism. The computational results

they obtained showed that changing depot may lead

to an interesting 2% reduction in the traveling costs.

(Bae and Moon, 2016) examined the multiple depot

VRPTW with additional constraints involving the de-

livery and installation of electronic components. They

developed a MILP model and two heuristics. The

ﬁrst heuristic assigns customers to their nearest de-

pot and builds routes using a nearest-neighbor-based

algorithm. The second heuristic is a genetic algo-

rithm making use of penalty and ﬁtness functions,

as well as crossover operators. Recently, (Su et al.,

2024) combined a genetic algorithm with a VNS to

address a multiple depot VRPTW variant involving

green requirements and customer satisfaction, with

the genetic algorithm serving as the upper layer and

the VNS as the local search mechanism. Computa-

tional results on instances of different sizes demon-

strated the good performance of their approach.

With respect to VRP variants with multiple peri-

ods, (Wen et al., 2010) solved a dynamic, real-world

problem for a large Swedish distributor. They pro-

posed a MILP model and a three-phase rolling hori-

zon heuristic. In the ﬁrst phase, customers are se-

lected to be visited. In the next phase, the selected

customers are arranged in routes by observing a plan-

ning problem, and, in the last phase, a tabu search is

used to improve the routes. (Mancini, 2016) intro-

duced a MILP model and an ALNS to solve a rich

VRP with multiple periods. Their ALNS considered

four destroy operators based on randomly selecting

customers and reallocating them to different routes.

More recently, (Zhang et al., 2024) tackled a real-

world application in the drug distribution system for

epidemic situations. To solve large instances, the au-

thors proposed a hybrid tabu search algorithm. It in-

cludes intra- and inter-route operators, as well as de-

stroy and repair mechanisms to improve efﬁciency.

In this work, we address a very general VRPTW

variant that combines multiple periods, multiple de-

pots, and multiple trips. To the best of our knowl-

edge, the literature on this rich problem is scarce.

As cited above, most of the works considered these

constraints separately, but none with all of them to-

gether. When solving similar problems, the literature

has mostly focused on developing metaheuristic algo-

rithms, especially when dealing with large and real-

world instances. Similar to the literature, we also aim

to propose an effective metaheuristic. Our aim is in-

deed to tackle a real-world problem and solve real-

world instances using such an approach.

3 PROBLEM DESCRIPTION

The optimization problem that we face is deﬁned on

a directed graph G = (N, A), where N represents the

set of nodes, consisting of a set D of depots and a

set C of customers. The set A of arcs is deﬁned as

{(i, j) : i, j ∈ N, i ̸= j}, where each arc (i, j) is associ-

ated with a non-negative travel time t

i j

. The problem

spans over a time horizon divided into a set P of peri-

ods. In each period, each vehicle is constrained by a

limit T on the maximum number of hours it can work.

For each customer i ∈ C and for each period p ∈ P, we

are given a demand q

, a service time s

, and a hard

time window [e

, l

], where e

is the earliest pos-

sible arrival time and l

is the latest possible arrival

time. A vehicle is allowed to arrive in i before e

but in such a case it will have to wait until e

. Arriv-

ing after l

is, instead, not allowed. A heterogeneous

vehicle ﬂeet, divided into a set V of vehicle types, is

available at each single depot. Each vehicle type v ∈V

consists of a set K

of different vehicles. Each vehi-

cle k ∈ K

, for v ∈ V , has a loading capacity Q

. The

sum of the loaded demands cannot exceed the vehi-

cle capacity at any moment during the execution of a

route.

The MMMVRPTW is subject to the following

constraints: each route can consist of one or more

trips, and each trip must start and end at the same de-

pot; the load on each trip must not exceed the vehi-

cle capacity; and, each route may not last longer than

the maximum number T of working hours. When

computing the total time of a route, a ﬁxed time ∆

for loading/unloading operations is imposed between

two consecutive trips performed by the same vehi-

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

114

cle. Each customer is assigned to a unique trip and

must be visited exactly once. Each visit should be

performed during the customer time window, and the

customer’s demand must be entirely fulﬁlled during

that visit. Furthermore, the problem incorporates in-

compatibilities between vehicles and customers. In

other words, only speciﬁc vehicle types can serve cer-

tain customers due to geographical location and de-

mand (e.g. on a mountainous territory only a small

vehicle can perform the delivery).

The objective of the problem is to ﬁnd an optimal

solution that satisﬁes all the problem constraints while

minimizing the total cost of vehicle usage. Speciﬁ-

cally, the overall cost includes two components. The

ﬁrst is a ﬁxed cost, A

, which depends on the vehicle

type v ∈ V and is incurred if a vehicle of this type is

used at least once across the set of periods. The sec-

ond is a variable cost, B

, which applies each time a

vehicle of type v is used for a route in a period. The

resulting problem is not only NP-hard (as it general-

izes the VRP) but also very difﬁcult in practice.

4 PROPOSED APPROACH

As anticipated, in this paper, we propose a two-phase

approach. In the ﬁrst phase, we generate a set of trips

that respect all the problem constraints. However,

since a vehicle can perform multiple trips within a sin-

gle period (multi-trip) and be reused across different

periods (multi-period), the second phase focuses on

combining these trips to minimize the cost associated

with the usage of the vehicles for all periods.

Figure 1: Trips calculated with the ILS metaheuristic in

(Kramer et al., 2019).

In the ﬁrst phase, we apply the metaheuristic pro-

posed by (Kramer et al., 2019). The goal is to gen-

erate a set of trips while respecting the problem con-

straints outlined in Section 3. Speciﬁcally, the authors

developed an iterated local search (ILS) metaheuris-

tic, which includes a restart mechanism after a speci-

ﬁed number of iterations. The ILS consists of a con-

structive heuristic, a local search, and a perturbation

procedure. The local search uses seven neighbour-

hood structures based on simple movements (e.g.,

swaps, relocations, and 2-opt exchanges). The per-

turbation step introduces randomness by performing

swaps and relocations. We refer the reader to the

original publication for further details. Figure 1 illus-

trates an example of the trips obtained with the ILS, in

which the blue points are customers, the orange points

are depots, and trips are depicted in red.

The trips obtained in the ﬁrst phase are then used

and combined in the second phase to create the ﬁ-

nal routes. In order to do that, we propose a VNS-

based heuristic, where the initial solution is generated

using a constructive heuristic. This solution is sub-

sequently reﬁned through a shaking procedure and

a local search. In the proposed algorithm, we de-

ﬁne a stopping criterion based on two conditions: the

number of consecutive iterations without improve-

ment and a maximum time limit. Further details are

provided in the following section. Additionally, we

adapt the MILP model and the BSTG heuristic pro-

posed by (Cavecchia et al., 2024) in order to accom-

modate multiple periods and the more general cost

function. All these algorithms are applied to solve

a number of real-world instances, and computational

results demonstrate their competitiveness.

4.1 Adapted Methods

In (Cavecchia et al., 2024), a single period VRPTW

variant with multiple depots and multiple trips was

solved by means of a MILP model, two greedy

heuristics and an ILS. The authors employed the

forward-time slack procedure proposed by (Savels-

bergh, 1992) in order to deﬁne the earliest and lat-

est starting times of each trip. The idea behind this

procedure is to allow a vehicle to depart as early as

possible without generating infeasible trips. It calcu-

lates the forward time slack, which indicates how far

the departure time of each customer can be shifted

forward.

As in (Cavecchia et al., 2024), we ﬁrst generate

the trips R

for each period p ∈ P using the meta-

heuristic proposed by (Kramer et al., 2019). Next, we

apply the forward-time slack procedure to determine

the earliest starting time ¯e

and the latest starting time

A Real-World Multi-Depot, Multi-Period, and Multi-Trip Vehicle Routing Problem with Time Windows

115

for each trip r ∈ R

. These parameters are then used

by the MILP model described below. The model also

receives in input the duration T

of each generated trip

r ∈ R

As in (Cavecchia et al., 2024), the model includes

four sets of binary decision variables and three sets

of continuous decision variables. However, now the

decision variables have an additional index p repre-

senting the period they refer to. To correctly assess

the new cost function, we introduce an additional bi-

nary decision variable w

• x

rkv

: binary variable that takes the value 1 if trip

r ∈ R

is assigned to vehicle k ∈ K

of type v ∈ V

on period p ∈ P; otherwise it takes the value 0;

• y

: binary variable that takes the value 1 if vehi-

cle k ∈ K

of type v ∈ V is used; otherwise it takes

the value 0;

• w

: binary variable that takes the value 1 if vehi-

cle k ∈ K

of type v ∈ V is used on period p ∈ P;

otherwise it takes the value 0;

• z

rskv

: binary variable that takes the value 1 if trip

r ∈ R

precedes trip s ∈ R

, both assigned to vehi-

cle k ∈ K

of type v ∈ V on period p ∈ P; otherwise

it takes the value 0;

• α

: continuous variable that represents the start-

ing time of vehicle k ∈ K

of type v ∈ V on period

p ∈ P;

• β

: continuous variable that represents the end-

ing time of vehicle k ∈ K

of type v ∈ V on period

p ∈ P;

• t

rkv

: continuous variable that represents the start-

ing time of trip r ∈ R

assigned to vehicle k ∈ K

of type v ∈ V on period p ∈ P.

The MILP model has the objective function (1)

that accounts for minimizing the total ﬁxed cost asso-

ciated with the different vehicles used across all pe-

riods (A

), as well as the daily cost encountered by

considering the vehicles used in every single period

). The model constraints are given by equations

(2)-(21) and are direct extensions of those in (Cavec-

chia et al., 2024). The model is solved independently

for each depot d ∈ D, as the trips in R

have already

been associated with a depot d and are independent of

one another. In other words, each depot corresponds

to an instance of the problem.

Constraints (2)-(4) ensure that if a vehicle k is

used during period p, then it must be considered in

the solution cost through variables y and w. Con-

straints (5) impose that each trip r on period p must

be assigned to exactly one vehicle k. Precedence con-

straints are ensured in (6) and (7) for any two trips

assigned to the same vehicle. Constraints (8) ensure

that trips assigned to the same vehicle have different

starting times, i.e., one must start after the other ﬁn-

ishes, considering the total duration T

and the ﬁxed

loading time ∆ between the consecutive trips r and s.

Constraints (9) require each trip to respect its earliest

¯e

and latest

starting times. Constraints (10), (11),

and (12) guarantee that each vehicle performs a set of

trips that respects the working hours limit T . In (13),

valid inequalities are applied to ensure that the work-

ing hours of each vehicle are respected. Constraints

(14) stipulate that vehicles can only perform trips ac-

cording to the customer’s geographical location and

demand, thus taking care of the vehicle-customer in-

compatibilities. Formally, set I contains those gener-

ated trips r ∈ R

that a vehicle of type v cannot per-

form due to geographical restrictions. Finally, con-

straints (15)-(21) deﬁne the domain of the variables.

In addition to the MILP model, we adapt the

bounded starting time greedy (i.e., BSTG) algorithm

developed by (Cavecchia et al., 2024). This heuris-

tic considers a variable σ to track the vehicles from

each depot and type used in each period. First, the

trips r ∈ R

are sorted in a non-decreasing order of

the earliest starting time ¯e

. In case of ties, trips with

longer duration are given priority. Trips are then as-

signed to vehicles, considering their working hours

and type. Algorithm 1 in (Cavecchia et al., 2024) pro-

vides the pseudocode for the BSTG algorithm. Differ-

ently from the original algorithm, we give preference

to vehicles in σ that have already been used in pre-

vious periods. If no such vehicle is available, a new

one is created and added to σ. Priority is given to ve-

hicle types with the lowest associated costs. We take

advantage of previously used vehicles and try to ﬁrst

assign trips to used vehicles of minimum cost, follow-

ing their costs.

4.2 VNS-Based Heuristic

The VNS is a metaheuristic that has been applied

to solve many hard combinatorial optimization prob-

lems. We can ﬁnd relevant VNS applications in

problems like vehicle routing, cutting, packing, fa-

cility location, scheduling, lot sizing, and many oth-

ers (Hansen and Mladenovi

c, 2001). Recently, (Lan

et al., 2021) surveyed healthcare problems that were

solved by VNS-based heuristics. Besides classify-

ing the works in the literature into ﬁve classes, the

authors detailed the main VNS phases, e.g., shaking

and local search, developed by the literature to obtain

high-quality solutions. They also commented on the

efﬁciency of classical operators based on swap, ex-

change, insertion, and move, among others, to deﬁne

neighbourhood structures.

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

116

min Z =

∑

v∈V

∑

k∈K

∑

p∈P

∑

v∈V

∑

k∈K

(1)

s.t. x

rkv

≤ w

, ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(2)

≥ w

, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

(3)

rkv

≤ y

, ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(4)

∑

v∈V

∑

k∈K

rkv

= 1, ∀p ∈ P, ∀r ∈ R

(5)

rskv

+ z

srkv

≥ x

rkv

+ x

skv

− 1, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

, ∀r, s ∈ R

: r ̸= s (6)

rskv

+ z

srkv

≤ 1, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

, ∀r, s ∈ R

: r ̸= s (7)

rkv

+ (T

+ ∆) ≤ t

skv

+ M(1 − z

rskv

), ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

, ∀r, s ∈ R

: r ̸= s (8)

¯e

≤ t

rkv

≤

, ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(9)

≤ t

rkv

+ M(1 − x

rkv

), ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(10)

≥ t

rkv

+ (T

+ ∆) − M(1 − x

rkv

), ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(11)

− α

≤ (T + ∆)y

, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

(12)

∑

r∈R

+ ∆)x

rkv

≤ (T + ∆)y

, ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(13)

rkv

= 0, ∀p ∈ P, ∀(r, v) ∈ I , ∀k ∈ K

: q

> Q

(14)

rkv

∈ {0, 1}, ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(15)

∈ {0, 1}, ∀v ∈ V, ∀k ∈ K

(16)

∈ {0, 1}, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

(17)

rskv

∈ {0, 1}, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

, ∀r, s ∈ R

: r ̸= s (18)

rkv

≥ 0, ∀p ∈ P, ∀r ∈ R

, ∀v ∈ V, ∀k ∈ K

(19)

≥ 0, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

(20)

≥ 0, ∀p ∈ P, ∀v ∈ V, ∀k ∈ K

(21)

According to (Hansen et al., 2019), the VNS starts

with an initial (random) solution and iterates over a

given set of neighbourhood structures in order to ob-

tain a solution that is globally optimal concerning all

structures. At each iteration, a new neighbour solu-

tion is generated in the shaking phase. Depending on

the VNS version, this neighbour solution is improved

through a local search phase. The resulting solution is

then evaluated and accepted if it is better than the pre-

vious one. In such a case, the search restarts from the

ﬁrst neighbourhood structure. Otherwise, the search

continues over the next neighbourhood.

In this work, we propose a heuristic based on the

VNS framework, as outlined in Algorithm 1. The al-

gorithm receives in input the routes R

generated in

the ﬁrst phase and the set of periods P. As output, it

returns the best solution found, x, which consists of

the routes performed by each vehicle over P. In line

1, we apply the adapted BSTG algorithm to obtain an

initial solution. The BSTG algorithm assigns vehicles

to trips following the order of the periods given in P.

This means that a different order of the periods may

imply a different (and hopefully better) solution. In

the ﬁrst while loop (line 2), we use as stopping crite-

ria the maximum number of iterations without an im-

provement or a maximum time limit. This ﬁrst loop

starts by imposing the search over the ﬁrst neighbour-

hood. Next, we enter a second inner loop that iterates

through all the neighbourhood structures (line 4). We

develop K

max

= 3 structures that all work on the set

of periods, namely: (i) swap two periods in the order,

(ii) insert one period before another one in the order,

and (iii) swap three periods. The idea is to generate

different orders for the periods, thus leading to good

diversiﬁcation of the search.

The shake phase in line 5 uses the neighbourhood

structures to generate a new order P

′

to the periods in

P. Next, we modiﬁed the adapted BSTG algorithm

in order to receive the current solution x and P

′

. This

modiﬁcation consists of using the trips as they have

A Real-World Multi-Depot, Multi-Period, and Multi-Trip Vehicle Routing Problem with Time Windows

117

Algorithm 1: VNS-based heuristic.

Data: R

, P

Result: x

1 x ← BSTG(R

, P);

2 while not reaching one of the stopping

criteria do

3 k ← 1;

4 while k ≤ K

max

5 P

′

← SHAKE(x, P, k);

6 x

′

← BSTG(x, P

′

);

7 x

′′

← LOCALSEARCH(x

′

);

8 if COST(x

′′

) < COST(x) then

9 x ← x

′′

;

10 k ← 1;

11 else

12 k ← k + 1;

13 end

14 end

15 end

been assigned in the solution. Then, these trips are

reassigned in accordance with the order of the periods

in P

′

. After that, we try to improve the neighbour

solution x

′

with the local search phase, resulting in

a solution x

′′

. In line 8, we compare the costs of x

′

and x

′′

, updating x in case x

′′

is better and making the

search restart from the ﬁrst neighbourhood. In case

no improvement is obtained, we continue the search

with the next neighbourhood (line 12).

Our implementation of the local search consists of

three steps. In the ﬁrst step, we perform all possible

swaps of two routes, stopping when a movement can

improve the solution (i.e., we use a truncated local

search, with the aim of keeping a low computational

effort). This improved solution is then used in the sec-

ond step, where all possible insertions between two

routes are performed. Similarly, we stop in the ﬁrst

insertion movement able to improve the input solu-

tion. Finally, in the third step, we select one vehicle

from a period and a type in order to reassign all its

trips to other vehicles in that period that are able to

accommodate them. The vehicle we attempt to empty

is taken from the period and vehicle type that is most

used among all ones. In fact, we try to empty all ve-

hicles of that type, until reaching one in which this is

possible. If no vehicle can be emptied, we return the

solution given as output by the previous local search

operator.

5 COMPUTATIONAL RESULTS

The two-phase approach was implemented in the fol-

lowing way: for the ﬁrst phase we invoked the algo-

rithm by (Kramer et al., 2019), which was coded in

C++; we then coded all the algorithms used in the

second phase in Python. The computer used to run

the experiments is an Intel(R) Xeon(R) E3-1245 3.50

GHz CPU, 32 GB of RAM, running under Ubuntu

22.04 LTS.

The MILP model (1)-(21) was solved with Gurobi

Optimizer version 11. We set the maximum number

of consecutive iterations without improvement of the

VNS-based heuristic equal to the number of neigh-

bourhood structures (i.e., equal to three). In compli-

ance with the real-world case study, we set the ﬁxed

loading/unloading time to ∆ = 30 minutes, and the

limit on the maximum number of working hours to

T = 480 minutes. Each algorithm of the second phase

was executed ﬁve times. Each execution was limited

to 3600 seconds. In the objective function, the val-

ues of A

= {160, 140, 120, 100, 80, 60} are assigned

to vehicle types v ∈ V , ordered in descending capac-

ity. Additionally, B

is set to 1 uniformly across all

vehicle types, regardless of their speciﬁc characteris-

tics. For each algorithm, the tables below report the

average results regarding objective function value and

computing time considering the ﬁve executions per

instance. The term TL stands for time limit and is

used when all executions reach 3600 seconds.

The two-phase approach was applied to solve real-

istic instances from an Italian service company. They

are based on three regions in Italy: Emilia Romagna,

Sardinia, and Tuscany. We have been provided with

three instances from Emilia Romagna, two from Sar-

dinia, and one from Tuscany. Table 1 contains some

details about these instances, including the name, the

number of depots, the number of vehicle types, the

number of customers, the number of trips obtained

with the algorithm of (Kramer et al., 2019) in the

ﬁrst phase of our decomposition approach, the aver-

age number of customers that a vehicle could service

in a single trip (#CT ), and the average number of pe-

riods in which a customer requires products (#PC).

Moreover, the maximum number of periods is six.

Table 2 presents the results obtained with all the

algorithms of the second phase, applied for each de-

pot. We present the objective function Z, computed

according to (1), and the computing time t(s) in sec-

onds, for each algorithm. For the MILP model, we

also include the lower bound Z

and the percent-

age gap, GAP(%), whose values are both directly re-

turned by the solver. For each instance, the best ob-

jective values are highlighted in bold. In general, we

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118

Table 1: Main characteristics of the instances.

Instance |D| |V | |C| |R

| #CT #PC

West Emilia

1 1 206 119 2.6 1.5

Romagna

Central Emilia

2 1 210 194 3.0 3.2

Romagna

East Emilia

1 1 201 177 3.0 3.1

Romagna

North Sardinia 3 2 152 48 8.4 1.5

South Sardinia 1 2 154 58 7.3 1.6

Tuscany 2 6 155 101 5.1 2.3

observe that the VNS-based heuristic is able to pro-

duce better objective function values, although it may

spend large computing times. The MILP model also

presents competitive results. However, it reaches the

time limit of one hour for those instances having the

largest number of trips. The adapted BSTG algorithm

is fast but does not perform so well compared to the

other approaches.

Observing the results in Table 2, the VNS-based

heuristic has the best objective value for the Central

Emilia Romagna (depot 1) instance. Moreover, in

comparison with the adapted BSTG algorithm, it is

still superior for the instances West Emilia Romagna

and Tuscany (depot 1). Similarly, it performs better

than the MILP model also for the East Emilia Ro-

magna instance. The best improvement obtained with

the VNS-based heuristic over the adapted BSTG algo-

rithm is for the Tuscany (depot 1) instance, reaching

a percentage decrease of 10.66%. With respect to the

MILP model, it reaches an improvement of 4.02% for

the East Emilia Romagna instance. Concerning the

comparison between the MILP model and the adapted

BSTG algorithm, the former is better for the instances

West Emilia Romagna and Tuscany (depot 1), while

the latter is superior for the instances Central Emilia

Romagna (depot 1) and East Emilia Romagna.

In terms of computing time, the results in Table

2 indicate the best performance of the adapted BSTG

algorithm. It presents an average computing time of

0.14 seconds. On the other hand, the VNS-based

heuristic has an average computing time of 1281.57

seconds, reaching the imposed time limit for the in-

stances Central Emilia Romagna (depot 1) and East

Emilia Romagna. The large computing times reported

by the VNS-based heuristic are justiﬁed by its local

search phase. Similarly, the MILP model reaches the

time limit for the same instances, with an overall av-

erage computing time of 725.87 seconds. For these

instances, the model GAPs are 7.63% and 15.58%, re-

spectively. From a scalability point of view, the model

struggles as the size of the instances grows, reaching

the time limit and increasing the GAP. In this way,

the BSTG algorithm and the VNS-based heuristic are

effective alternatives to the model.

In Table 3 we present the number of vehicles used

over the set of periods, considering the ﬁnal solution

obtained with the algorithms for the second phase.

The results are shown for each depot and vehicle type

that uses at least one vehicle. All the algorithms return

the same amount of vehicles, except for the instances

East Emilia Romagna and Tuscany (depot 1, vehicle

type 2). In these speciﬁc cases, the MILP model re-

quires one more vehicle and the adapted BSTG al-

gorithm requires one more vehicle with respect to the

other algorithms, respectively. Despite the same num-

ber of vehicles required in the ﬁnal solution by the al-

gorithms, the objective function values may be differ-

ent as previously shown in Table 2, because it refers

to the ﬁxed cost of the vehicles plus the number of

times that each vehicle is used over the periods.

Figure 2 presents the graphical output generated

by the VNS-based heuristic for the Tuscany (depot 1)

instance. The chart displays the assignment of trips to

vehicles over the considered set of periods. Each col-

ored bar represents a trip, with bars of the same color

corresponding to trips assigned to the same vehicle.

The horizontal axis indicates the time, while the ver-

tical axis shows the vehicles and the created routes,

which consist of multiple trips.

A Real-World Multi-Depot, Multi-Period, and Multi-Trip Vehicle Routing Problem with Time Windows

119

Table 2: Computational results on the ten real-world instances.

Instance Depot

BSTG VNS MILP Model

Z t(s) Z t(s) Z Z

GAP(%) t(s)

West Emilia Romagna 1 3949 0.28 3947 3393.38 3947 3947 0.00% 7.94

Central Emilia Romagna

1 2450 0.25 2449 TL 2451 2264 7.63% TL

2 980 0.05 980 113.57 980 980 0.00% 34.56

East Emilia Romagna 1 3919 0.48 3919 TL 4083 3447 15.58% TL

North Sardinia

1 742 0.03 742 8.16 742 742 0.00% 0.15

2 430 0.01 430 0.22 430 430 0.00% 0.02

3 433 0.01 433 0.47 433 433 0.00% 0.03

South Sardinia 1 1632 0.08 1632 172.60 1632 1632 0.00% 1.29

Tuscany

1 1342 0.20 1199 1926.93 1199 1199 0.00% 14.65

2 208 0.01 208 0.39 208 208 0.00% 0.05

Table 3: Number of vehicles used in the solutions found.

Instance Depot Vehicle Type BSTG VNS MILP Model

West Emilia Romagna 1 1 24 24 24

Central Emilia Romagna

1 1 15 15 15

2 1 6 6 6

East Emilia Romagna 1 1 24 24 25

North Sardinia

1 1 1 1

2 4 4 4

2 2 3 3 3

3 2 3 3 3

South Sardinia 1

1 2 2 2

2 9 9 9

Tuscany

1 1 1 1

2 2 1 1

3 2 2 2

4 1 1 1

5 1 1 1

6 7 7 7

2 1 1 1

6 1 1 1

6 CONCLUDING REMARKS

The rich VRP handled in this work originates from a

real-world case study. It contains many different con-

straints and thus is solved by a two-phase approach.

The ﬁrst phase is concerned with obtaining a set of

trips that respect the problem’s constraints. The sec-

ond phase aggregates these trips into routes in order

to minimize the costs associated with the usage of

the vehicles and the number of times each vehicle

is used over a set of periods. The trips are calcu-

lated with a metaheuristic from the literature and the

routes may be deﬁned by three different algorithms:

an adapted BSTG algorithm, a VNS-based one, and a

MILP model.

Considering experiments on realistic instances,

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

120

Figure 2: Final solution for Tuscany instance generated by the VNS-based heuristic.

the proposed algorithms are effective in returning so-

lutions close to each other. Differences are observed

with the VNS-based heuristic, which can improve

some of the solutions computed by the others, e.g.,

reaching improvements superior to 10% compared to

the adapted BSTG algorithm. On the other hand, it

has the largest computing times. The MILP model

works well for small-size instances, but it is not able

to achieve optimal solutions for the larger instances

within 3600 seconds.

After all, we note there is room for improve-

ment. A sensitivity analysis (e.g., varying the vehi-

cle capacity and the customers’ time windows) could

help the company in deciding about the ﬂeet of ve-

hicles. On the other hand, the VNS-based heuris-

tic is time-consuming due to its local search phase.

One direction could be to explore other VNS vari-

ants that do not have the local search phase. An-

other interesting research avenue lies in the proposal

of different heuristics and metaheuristics (e.g., col-

umn generation-based heuristics, genetic algorithms,

and tabu search) to compare with the methods pro-

posed in this paper. One could extend the proposed

algorithms to handle the entire problem instead of ﬁrst

calculating the trips and then aggregating them, as

well as other rich VRP variants. We are also interested

in studying robust/stochastic problems, where the to-

tal duration of the trips is not deterministically deter-

mined but may assume uncertain values taken from

intervals (in case of robust optimization) or scenarios

(in case of stochastic programming).

ACKNOWLEDGEMENTS

This paper was partially ﬁnanced by the Na-

tional Council for Scientiﬁc and Technological De-

velopment (CNPq) [grant numbers 405369/2021-

2, 408722/2023-1, and 315555/2023-8], the State

of Goi

as Research Foundation (FAPEG), the Na-

tional Recovery and Resilience Plan (NRRP), Mis-

sion 04 Component 2 Investment 1.5–NextGenera-

tionEU, Call for tender n. 3277 dated 30/12/2021,

Award Number: 0001052 dated 23/06/2022, and

Coopservice Soc.coop.p.A. Their support is gratefully

acknowledged.

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