Simultaneous Simulated Annealing-Based Crossover Within a

Multi-Agent Model for Solving the Green Share-a-Ride Problem

Elhem Elkout

1,4 a

, Houssem Eddine Nouri

2,4 b

and Olfa Belkahla Driss

3,4 c

Ecole Nationale des Sciences de l’Informatique, University of Manouba, Tunisia

Institut Sup

erieur d’Informatique et de Gestion de Kairouan, University of Kairouan, Tunisia

Ecole Sup

erieure de Commerce de Tunis, University of Manouba, Tunisia

LARIA UR22ES01, Ecole Nationale des Sciences de l’Informatique, University of Manouba, Tunisia

Keywords:

Green Transport, Green Share-A-Ride Problem, Simulated Annealing, Crossover Operator, Multi-Agent.

Abstract:

This research addresses the Green share-a-ride problem (Green-SARP), which is an extension of the share-a-

ride problem (SARP) by considering a limited driving range of vehicles in combination with limited refueling

infrastructure. The goal of Green-SARP is to remove the possibility of a vehicle running out of fuel during

a route by allowing refueling at any alternative fuel station. In this work, we present a new simultaneous

Simulated Annealing-based Crossover within a Multi-Agent model (SAC-MA) to solve Green-SARP. In fact,

adding to the neighbor operators, the crossover operator is integrated to diversify the search allowing to explore

new areas in the search space. Experimental studies are carried out in order to evaluate the performance of

our approach, based on new generated data instances, allowing to show its efﬁciency compared to a Simulated

Annealing algorithm (SA).

1 INTRODUCTION

In recent years, numerous studies have focused on

the Share-a-Ride Problem (SARP) and its various ex-

tensions. First proposed by (Li et al., 2014), SARP

involves using taxis to simultaneously transport pas-

sengers and parcels, inspired by the well-established

Dial-a-Ride Problem (DARP) (Cordeau and Laporte:,

2007). While DARP aims to minimize total trans-

portation costs and enhance passenger service, SARP

introduces the dual objective of managing both pas-

senger transport and parcel delivery. The key assump-

tion in SARP is that passenger service takes prece-

dence over parcel delivery to ensure high-quality ser-

vice (Li et al., 2014). Initial solutions to the dynamic

SARP were developed using heuristic approaches, in-

cluding the insertion of unfulﬁlled requests into exist-

ing routes, followed by optimization through neigh-

borhood search.

Subsequent research on SARP has introduced en-

hancements and variants tailored to real-life urban

transportation scenarios. For example, (Nguyen et al.,

2015) proposed a hybrid transportation model for

https://orcid.org/0000-0001-7422-7951

https://orcid.org/0000-0003-0901-1278

https://orcid.org/0000-0003-3077-6240

Tokyo, enhancing realism by incorporating additional

constraints and a time slack strategy for route plan-

ning. This approach utilized the Adaptive Large

Neighborhood Search (ALNS) heuristic, which be-

gins with a basic greedy insertion of requests and ap-

plies simulated annealing as a local search method.

Several notable variants of SARP have also been

proposed. The General Share-a-Ride Problem (G-

SARP) (Yu et al., 2018) extends SARP by relax-

ing certain constraints, such as allowing vehicles to

serve multiple requests simultaneously without pri-

oritizing passenger or parcel service. To solve G-

SARP, a simulated annealing algorithm and a tabu

search were developed to evaluate its performance.

Similarly, (Beirigo et al., 2018) introduced the Share-

a-Ride with Parcel Lockers Problem (SARPLP), fo-

cusing on shared autonomous vehicles equipped with

ﬂexible compartments for passengers and parcels.

Further developments include a time-dependent

model for SARP (Do et al., 2017), incorporating

speed windows to reﬂect realistic urban conditions.

Another extension, the cooperative SARP (coop-

SARP) (Cavagnini and Morandi, 2021), examines

collaboration between different service providers.

Additionally, (Yu et al., 2021) introduced the

Share-a-Ride Problem with Flexible Compartments

(SARPFC), which allows the passenger compartment

600

Elkout, E., Nouri, H. E. and Driss, O. B.

Simultaneous Simulated Annealing-Based Crossover Within a Multi-Agent Model for Solving the Green Share-a-Ride Problem.

DOI: 10.5220/0013314300003890

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

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 1, pages 600-607

ISBN: 978-989-758-737-5; ISSN: 2184-433X

to be used for parcel storage, aiming to maximize rev-

enue.

Building on these advancements, the Green Share-

a-Ride Problem (Green-SARP) was proposed by

(Elkout and Belkahla, 2022) to address environmen-

tal concerns. Green-SARP combines the principles of

SARP with those of the Green Vehicle Routing Prob-

lem (GVRP), incorporating Alternative Fuel Vehicles

(AFVs) and the need for refueling at Alternative Fuel

Stations (AFS). This formulation aims to reduce the

environmental impact of shared mobility while main-

taining operational efﬁciency.

In 2023, (Elkout et al., 2023) introduced an en-

hanced Simulated Annealing algorithm with a Cor-

rection Mechanism (SA-CM) to solve Green-SARP.

Their experimental results demonstrated that SA-CM

could produce high-quality solutions close to the

optimal results obtained by CPLEX, outperforming

CPLEX for instances with more than 10 requests.

These ﬁndings highlight the potential of heuristic-

based methods to solve complex urban transportation

problems efﬁciently.

This innovative problem Green-SARP was ﬁrst in-

troduced by (Elkout and Belkahla, 2022), by leverag-

ing the ﬂexibility of AFVs and integrating AFS nodes

into the route design, the Green-SARP ensures the

sustainability of urban transport systems while meet-

ing the demands of modern shared mobility.

In this paper, we introduce a simultaneous Sim-

ulated Annealing-based Crossover within a Multi-

Agent model (SAC-MA) to solve Green-SARP. We

conduct computational experiments to assess the per-

formance of our approach, utilizing newly modiﬁed

data instances, and demonstrate its efﬁciency com-

pared to a Simulated Annealing algorithm (SA).

2 SIMULTANEOUS SIMULATED

ANNEALING-BASED

CROSSOVER WITHIN A

MULTI-AGENT MODEL

Simulated Annealing is a probabilistic meta-heuristic

algorithm, it is based on a natural technique that sim-

ulates the cooling of a group of heated atoms us-

ing an analogy to thermodynamics, a process known

as annealing (Kirkpatrick et al., 1983). Simulated

Annealing accepts search movements that temporar-

ily produce degradation of an existing solution to a

problem (Kirkpatrick et al., 1983). Simulated An-

nealing (SA) is a local neighborhood search that re-

quires exploring the search space and accepting so-

lutions with some probabilities (Davtyan and Khcha-

tryan, 2020). An artiﬁcial system composed of a

population of autonomous agents is called a multi-

agent system, in order to achieve their shared goals

the agents work cooperatively. Additionally, a multi-

agent system is a computational system in which two

or more agents cooperate, compete or combine their

efforts to accomplish some individual or group objec-

tives (Ferber, 1999). In this work, we propose a new

simultaneous Simulated Annealing-based Crossover

within a Multi-Agent model (SAC-MA) for solving

Green Share-A-Ride Problem. In addition, the fact

that Green-SARP is an NP-Hard problem, so the use

of a multi-agent system allows distributed and simul-

taneous processing, which are very complementary.

The multi-agent system is composed entirely of in-

teracting agents. Each agent can communicate, coor-

dinate and cooperate with other agents to complete a

common aim. It consists of two classes of agents: A

master-Agent (MA) and a set of Simulated-annealing

Agent (SimA) , where the MA agent is the Master of

its society and the SimA agents are its Workers/Sub-

agents. The ﬁgure 1 represents the proposed SAC-

MA approach.

2.1 Master Agent

It is the one who interacts with the user, the

master-agent (MA) receives the number of Simulated-

annealing Agents to create all parameters for the Sim-

ulated Annealing-based Crossover algorithm. It is

responsible of creating Simulated-annealing Agents

(SimA) based on the input number given by the user.

Then the MA generates an initial population with

different dynamic lists based only on pickup nodes.

Noting that the number of dynamic lists depend of

the number of the Simulated-annealing Agent given

by the user. So, the MA agent provides for each

worker-agent its necessary information such as the

agent identiﬁcation (as an autonomous agent and also

as a system member), its dynamic list solution from

the initial population and the parameters for the Sim-

ulated Annealing-based Crossover algorithm. If the

stopping criterion is attained, the MA agent chooses

the best solution from the last solutions received from

Simulated-annealing Agents and displays it as the

global solution for the problem.

2.1.1 Solution Representation

The Green-SARP involves a depot, a set of requests,

and a set of AFS nodes. The answer to this prob-

lem provides 3 travels with 3 AFV each having a 60-

gallon fuel capacity and a consumption rate equal to

0.2.

Simultaneous Simulated Annealing-Based Crossover Within a Multi-Agent Model for Solving the Green Share-a-Ride Problem

601

Figure 1: Simultaneous Simulated Annealing-based Crossover within a Multi-Agent Model.

Figure 2 illustrates a sample solution represen-

tation and its interpretation into vehicle routes. As

shown in Figure 2, the ﬁrst vehicle is assigned to visit

both passengers 1 and 3 and one parcel 7. The vehi-

cle requires more than 60 gallons to serve all affected

requests; thus, it decides to visit the AFS node. The

remaining vehicles are assigned similarly to how the

ﬁrst vehicle. The solution representation for Green-

SARP consists of k dynamic lists. Each list comprises

numbers including a depot node, a set of pick-up and

drop-off nodes of the passenger and parcel demands,

and AFS nodes. The ﬁrst and the last position in each

dynamic list must be respectively an origin and desti-

nation depot designate by 0. Each dynamic list repre-

sents a vehicle that must serve the different nodes one-

by-one from left to right. If the next node is a negative

number then the vehicle must visit an AFS node and

then continue its travel. To explain the Green-SARP, a

sample problem of 8 and 3 vehicles requests is shown

in ﬁgure 3, in table 3 we present all the necessary pa-

rameters:

In Figure 2, the ﬁrst dynamic list illustrates the

route of the ﬁrst vehicle. It starts at the depot and

Table 1: A simple instance of the Green-SARP.

Parameters Value

number of requests σ = 8

number of parcel re-

quests

m = 3 Vp,0=

{1,2,3,4,5} Vp,d=

{9,10,11,12,13}

number of passenger

requests +

n = 5 Vf,0 = {6, 7 ,8}

Vf,d = {14, 15 ,16}

number of vehicles K = 3

number of AFS f = 4

then visits some pick-up and delivery nodes when the

fuel remaining is not sufﬁcient the vehicle visit AFS

node denote (-1) then continues its travel and returns

to the depot.

2.1.2 Population Initialization

The initial population is generated randomly to in-

crease the diversity and distribution of the individual

solutions in the search space. In fact, each new dy-

namic list solution is based only on pickup nodes and

should have a predeﬁned distance from all the other

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602

Figure 2: Solution representation for Green-SARP.

dynamic lists to be considered as a new member of the

initial population. The dissimilarity distance is cal-

culated by verifying the difference between two dy-

namic lists in terms of the pickup node order. Noting

that a new solution is accepted only if its greater than

a ﬁxed threshold of difference equals to 50% percent-

age.

2.2 Simulated-Annealing Agent

Simulated-annealing Agent (SimA) are created by the

Master-Agent, where each one has its own search

space area. After generating its initial solution,

each SimA agent uses a Simulated Annealing-based

Crossover algorithm to explore the search space. To

improve the search technique, the SimA agents co-

operate and communicate by messages allowing to

share their best current solutions among them in order

to avoid in each case the reuse of the same research

point. SimA agents complete the process by sending

their last best solutions to the Master-Agent, which

considers the most dominant of them as the global so-

lution for the Green-SARP.

2.2.1 Individual’s Solution

The initial solution is created using a random method.

Step 1. We create an empty list then we add all pickup

nodes in random order.

Step 2. We create K empty dynamic lists, we divide

randomly the ﬁrst list for K parts, and each part is af-

fected in one vehicle.

Step 3. we insert after each pickup node her delivery

node in each vehicle.

Step 4. We apply the corrector mechanism to correct

the route for each vehicle and have a feasible solution,

and we insert the origin and the destination depot at

ﬁrst and at the end for each dynamic list. Finally, we

insert the AFS nodes consider the tank fuel capacity

remaining and we select the nearest AFS node. The

ﬁgure 3 presents the ﬁrst three steps for generating the

initial solution. After creating the K dynamic lists and

applying the correct mechanism in each vehicle, we

found a solution we called it ”deactivated solution”.

Figure 3: The ﬁrst steps for generating initial solution for

Green-SARP.

Finally, where we insert the depot and the AFS nodes

if necessary we ﬁnd the complete solution named ”ac-

tivated solution”. The ﬁgure 4 illustrates the initial

solution after checking all the constraints.

Figure 4: The last step for generating initial solution for

Green-SARP.

2.2.2 Neighborhood

The neighborhood move begins with a feasible solu-

tion and progresses to the construction of new neigh-

boring solutions of this solution. At each iteration,

one of the four neighborhood moves is chosen ran-

domly. In fact, the choice of this neighborhood move

is justiﬁed by applying the intensiﬁcation technique

to exploit the search space. Each neighborhood move

relaunches the search from a current solution to attain

an elite solution of the search space. These moves are

called: SWAP, Permutation, Insertion, and Reverse.

Each Neighborhood moves in the dynamic lists be-

fore inserting the AFS nodes and the origin and des-

tination depots, i.e these moves and operators are ap-

plied to the ”deactivated solution”. In the following,

we present these four moves.

(1) Permutation. The permutation move is per-

Simultaneous Simulated Annealing-Based Crossover Within a Multi-Agent Model for Solving the Green Share-a-Ride Problem

603

formed by randomly selecting two positions, say p1

and p2 (p1 ̸=p2) in the same dynamic list i.e in the

same vehicle, and then exchange the two nodes in the

two selected positions.

(2) SWAP. The SWAP move is performed by ran-

domly selecting many positions, in the same dynamic

list and then exchanging the nodes in the positions se-

lected.

(3) Reverse. Reversing the sequence of the elements

included among two randomly chosen positions p1

and p2 (p1 ̸=p2) in the same dynamic list.

(4) Insertion. in the ﬁrst time selecting randomly two

positions from the same dynamic list symbolized by

p1 and p2 (p1 ̸=p2), and the node of the position p2

is inserted immediately before position p1.

2.2.3 Crossover Operator

The crossover operator is necessary for the entire pro-

cedure, enabling the creation of new offspring vehi-

cles by combining two parent vehicles in order to

reach new promising areas in the search space. To

apply this operator, ﬁrst of all a selection operator

is used to randomly select two different parent ve-

hicles V1 and V2. On the other hand, by counting

the same length equals exactly to two pickup trav-

els the one point crossover is applied else it is the

two point crossover. On the other hand, by having

different length, two sub-methods are used, cutFis-

rtPart and cutRemainPart, to divide the long list of

the ﬁrst parent vehicle into two parts based on the

length of the second parent vehicle. Then, the one

point crossover is applied if the length of the second

vehicle is equal exactly to two pickup travels else it

is the two point crossover. Finlay, the remaining el-

ements, obtained by the cutRemainPart method, are

distributed randomly between the two new offspring

vehicles.

One Point Crossover: a point on both selected ve-

hicles is ﬁxed randomly, and designated a ’crossover

point’. The nodes to the Right of that point are per-

muted between the two vehicles, which results two

offspring. the ﬁrst child gets the current value of the

ﬁrst vehicle and the second child takes the value of

the second vehicle.

Two Point Crossover: two crossover points are gen-

erated randomly from the selected vehicles. The

pickup nodes between the two points are permuted

between these vehicles.

2.2.4 Simulated Annealing-based Crossover

The SA procedure starts by initializing the current

temperature T0 and generating a one random initial

solution X, and Sets the current best solution to de-

note Xbest, and the current best objective function

of X denotes Fbest = F(X). To improve the current

solution X, the corrector mechanism is performed to

obtain a feasible solution. In each iteration, a new

solution Y is generated from the exploitation of the

space of search using the neighborhood move based

on a random method or from the exploration of search

space using the Crossover operator. To select which

neighborhood move or crossover operator to use, the

algorithm ﬁrst generated a randomly p value. After

that, the objective function values of X and Y are eval-

uated using the formula, ∆= F(Y)-F(X). Y is better

than X if ∆ is positive, in that case Y will replace X.

Otherwise, Y will replace X with a probability equal

to exp (∆/T). So the accepted solution Y is compared

to Xbest, if F(Y) is better than Fbest consequently re-

place Fbest with F(Y). The current temperature T0 is

decreased after Iter iterations according to the formula

T = α*T0. If one of the termination criteria is sat-

isﬁed, the algorithm will end. There is one termina-

tion condition in this algorithm: The ﬁnal temperature

TF is attained TF =T, this stopping criteria consider

the condition of Non-improve: No improvement is

reached in succession temperature reductions. Figure

5 shows the ﬂow chart of the proposed SAC heuristic.

3 EXPERIMENTAL STUDIES

To evaluate our model and demonstrate the beneﬁts

of using multi-agent systems, we developed an en-

hanced Simulated Annealing called Improved Sim-

ulated Annealing-based Crossover (SAC), which is

based on an initial population of solutions. Both the

Improved Simulated Annealing (SA) and Simultane-

ous Simulated Annealing-based Crossover within a

Multi-Agent Mode (SAC-MA) were implemented in

Java on a PC with Intel processor core i7 vPro and 32

GB of RAM, using the Eclipse IDE to code these ap-

proaches and the Jade platform to create the SAC-MA

multi-agent system.

3.1 Test Instances

To evaluate and compare the efﬁciency of these ap-

proaches, numerical tests were conducted using the

data set generated by Masmoudi et al (Masmoudi

et al., 2019) from the literature on the green dial-a-

ride problem , and the parameters provided by Li et

al. for SARP (Li et al., 2014).

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

604

Figure 5: A ﬂow chart of Simulated Annealing-based

Crossover.

3.2 Parameter Tuning

The SA metaheuristic uses ﬁve parameters: Iter, T0,

TF, Non-improving and α.

• Iter: indicates the number of iterations for the

search to be proceeded at a speciﬁc temperature,

• T0: indicates the initial temperature,

• TF: describes the ﬁnal temperature when the SA

procedure stops,

• Non-improving: is the number of temperature re-

ductions during which the value of the objective

function is not improved,

• α: is the factor in charge of the cooling schedule

proposed by (Kirkpatrick et al., 1983).

Following an experimental study, the ‘popsize‘

parameter was set to 10 agents for small instances, 50

for medium instances (fewer than 40 requests), and

100 for large instances (more than 40 requests).

The Non-improving parameter was set to 100 for

small instances, 70 for medium instance, and 50 for

large instance. The other parameter values are deﬁned

as follows:

• Iter = 1000,

• T0 = 12,

• TF = 0,001.

Therefore, we chose for the rest of the parameters the

best values used in the literature of SARP variants (Yu

et al., 2018), (Yu et al., 2021) α = 0.9.

3.3 Computational Results

To evaluate the performance of our SAC-MA ap-

proach, we established a rigorous comparative

methodology. Our results are compared to those ob-

tained by the SA algorithm developed by (Elkout

et al., 2023). Tables 2 present the comparison of SAC-

MA with the SA algorithm (Elkout et al., 2023) for

small and large instances. They display the optimal

value obtained by SA, followed by the best value, the

CPU time, and the average value obtained by SAC-

MA.

Figures 6 illustrate that SAC-MA performs bet-

ter and surpasses SA-CM in most cases for both

small and large instances.These best results can be

explained by the distributed search space in our pro-

posed approach SAC-MA, and the cooperative aspect

between the different agents.

Finally, ﬁgures 7 shows that SAC-MA is faster in

terms of CPU time compared to SA-CM. These ex-

cellent results conﬁrm the efﬁciency of using a multi-

agent system, which primarily aims to reduce execu-

tion time through the cooperation and communication

between Simulated Annealing agents, as well as the

speciﬁc stopping condition employed in SAC-MA.

Indeed, the counter for iterations without improve-

ment (Non-improving) is not incremented solely

when an agent ﬁnds a locally non-improving solution.

The search for the dominant solution by an agent also

depends on global solutions. In other words, even

solutions found by other agents are considered non-

improving if one agent reaches them. This interaction

signiﬁcantly reduces the required CPU time by opti-

mizing convergence towards a high-quality global so-

lution.

Simultaneous Simulated Annealing-Based Crossover Within a Multi-Agent Model for Solving the Green Share-a-Ride Problem

605

Table 2: Comparison of SAC-MA and SA-CM results for small instances.

Instance Nb-req Nb-AFS Nb-Veh SAC-MA SA-CM CPU SAC-MA CPU SA-CM

Best AVG Best AVG (minute) (minute)

A0-10-1 10 3 1 65,742 65,279 64,8 61,2 1,73 8,21

A0-10-1 10 3 2 102,410 102,090 98,7 94,4 4,46 16,08

A0-10-2 10 3 1 54,972 53,605 54 50,9 1,74 8,29

A0-10-2 10 3 2 64,043 63,648 62,1 58,1 4,48 16,35

A0-12-3 12 4 2 149,992 148,422 145 142 5,73 19,92

A0 20 1 20 6 4 523,238 453,204 490,07 406,3 11,23 40,79

A0 20 2 20 6 4 584,415 500,328 545,35 472,3 11,96 42,58

A0 25 4 25 8 5 750,295 652,113 668,33 572,8 13,34 47,32

A0 25 5 25 8 5 605,91 544,503 577,12 540,4 13,46 47,48

A0 30 8 30 9 6 679,382 660,314 653,75 623,8 14,92 51,86

A0 30 9 30 9 5 696,929 650,985 651,87 604,2 14,66 50,92

A0 35 11 35 11 5 765,68 658,534 670,56 635,4 15,33 52,69

A0 35 12 35 11 6 624,543 587,32 615,85 574,3 14,71 51,15

A0 40 14 40 12 7 798,553 764,772 710,52 699,9 15,85 54,21

A0 40 15 40 12 6 790,632 752,48 742,56 704,1 15,49 53,46

Figure 6: Comparison between SA-CM and the proposed

SAC-MA.

4 CONCLUSIONS

In this study, we introduce a Multi-Agent model

called simultaneous Simulated Annealing-based

Crossover within (SAC-MA) to solve th Green-

SARP. SAC-MA consist of two classes of agents:

the Master-Agent which is responsible of manag-

ing inputs and outputs of the system and a set of

Simulated-annealing Agents allowing to cooperate

and communicate between them to increase neigh-

borhood search efﬁciency and identify promising

areas and ﬁnd the best solution. The experimental

results showed that SAC-MA produced the best set

of solutions compared to the Simulated Annealing

algorithm.

Figure 7: Comparison between SAC-MA and SA-CM in

terms of CPU time.

In future work, we plan to adapt our approach to

a multi-objective version of the Green-SARP, and

compare the results to other approaches from the

literature. Furthermore, the proposed SAC-MA algo-

rithm could be used also for future SARP extensions,

such as Green general SARP or Green multi-depot

SARP and Green-SARP with ﬂexible compartments.

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