A Bee Colony Optimization Algorithm for the Long-Term Car Pooling

Problem

Mouna Bouzid, Ines Alaya and Moncef Tagina

COSMOS-ENSI, National School of Computer Sciences, University of Manouba, Manouba 2010, Tunisia

Keywords:

Metaheuristic, Bee Colony Optimization, The Long-Term Car Pooling Problem.

Abstract:

Recently, the big number of vehicles on roadways and the increase in the rising use of private cars have made

serious and signiﬁcant trafﬁc congestion problems in large cities around the world. Severe trafﬁc congestion

can have many detrimental effects, such as time loss, air pollution, increased fuel consumption and energy

waste. Public transportation systems have the capacity to decrease trafﬁc congestion and be an answer to this

increasing transport demand. However, it cannot be the only solution. Another recommended solution for

reducing the harmful factors leading to such problems is car pooling. It is a collective transportation system

based on the idea that a person shares his private vehicle with one or more people that have the same travel

destination.

In this paper, a Bee Colony Optimization (BCO) metaheuristic is used to solve the Car Pooling Problem. The

BCO model is based on the collective intelligence shown in bee foraging behavior. The proposed algorithm is

experimentally tested on benchmark instances of different sizes. Computational results show the effectiveness

of our proposed algorithm when compared to several state of the art algorithms.

1 INTRODUCTION

Swarm intelligence is a ﬁeld of nature inspired meta-

heuristics that was successfully applied to many op-

timization problems. We can cite the Ant colony op-

timization (Dorigo et al., 2006; Zouari et al., 2017),

the Particle swarm intelligence metaheuristic (Huang,

2015; Wu et al., 2011; Liu and Qin, 2014; Nouiri

et al., 2018; Karaboga, 2005), and the swarm intel-

ligence based on bees (Karaboga and Basturk, 2008;

Karaboga and Akay, 2011; Bacanin et al., 2010; Sub-

otic et al., 2010; Xue et al., 2018; Jadon et al., 2018).

The Bee Colony Optimization (BCO) is a meta-

heuristic in which the principal idea is taken from the

analogy between the natural behavior of bees search-

ing for food, and the behavior of optimization algo-

rithms searching for an optimum of combinatorial op-

timization problems. Based on exploration and ex-

ploitation ability, bees are able to intensify and to di-

versify the search at the same time.

In fact, the BCO metaheuristic is a competitive

approach for solving hard combinatorial optimiza-

tion problems, such as the stochastic Vehicle Routing

Problem (Lu

c and Teodorovi

c, 2003b; Teodorovi

2008), Job Shop Scheduling Problem (Chong et al.,

2006; Chong et al., 2007; Wong et al., 2010b), p-

Median Problem (Teodorovic and

Selmic, 2007) and

Traveling Salesman Problem (Wong et al., 2010a).

In this paper, our research is focused in another

problem named car pooling. This problem is a collec-

tive transportation system based on the idea that sets

of car owners having the same travel destination share

their vehicles. It has emerged to be a viable possibil-

ity for reducing private car usage around the world.

Car pooling has already been considered as an im-

portant alternative transportation service throughout

the world. In fact, many organizations such as large

companies or public administrations encourage their

citizens or employees to pick up or take back col-

leagues while driving to or from a common site. The

beneﬁts which can be obtained are particularly rele-

vant both in terms of reduction of the use of private

cars and of the parking space required.

Thanks to massive use of internet, car pooling be-

comes more and more popular. Therefore, the num-

ber of applications proposing car pool increase day

after day, but they are not optimized, and use gener-

ally the FiFo ”First in First out” principle. In addition,

after bibliographical studies, it appears that the BCO

algorithm has never been used for solving the Long-

term Car Pooling Problem (LTCPP). The aim of this

work is to propose an intelligent approach based on

Bouzid, M., Alaya, I. and Tagina, M.

A Bee Colony Optimization Algorithm for the Long-Term Car Pooling Problem.

DOI: 10.5220/0009831803190327

In Proceedings of the 15th International Conference on Software Technologies (ICSOFT 2020), pages 319-327

ISBN: 978-989-758-443-5

319

the BCO algorithm to solve the Long-term Car Pool-

ing Problem (LTCPP), which can be deployed in such

application. In order to test the proposed algorithm

we used various benchmark instances.

The different parts of this paper are organized as

follows. In the next section, we deﬁne the Long-term

Car Pooling Problem. We recall in section 3 the basic

idea of BCO. In section 4, we describe the proposed

BCO algorithm for solving our problem. Experiments

and results are shown in section 5. Finally, conclu-

sions and perspectives are presented.

2 THE LONG-TERM CAR

POOLING PROBLEM

In this section, we give an overview of the Car Pool-

ing Problem and we present a mathematical formula-

tion for the LTCPP.

2.1 An Overview of Car Pooling

Problem

Car pooling (Bruglieri et al., 2011; Manzini et al.,

2012; Correia and Viegas, 2011; Yan et al., 2011;

Vargas et al., 2008) is a collective transportation sys-

tem. In this transport service, several people share

a common vehicle simultaneously, in order to reach

common destinations. The principal objective of this

mobility service is to decrease the number of private

vehicles.

By joining the car pooling system, many advan-

tages are offered. In fact, it saves time, parking

spaces, and decreases the number of accidents. Fur-

thermore, it encourages sociability between people

since it creates social interaction between colleagues,

friends and neighbors. By sharing journey expenses,

such as, tolls fuel, costs, and road stress, travel costs

can be reduced. Car pooling can make our environ-

ment more friendly, ecological and sustainable way

to travel as sharing journeys reduces carbon emissions

and trafﬁc congestion on the roads.

Car pooling problem can be operated in two dif-

ferent ways. It can be either a Daily Car Pooling

Problem (DCPP) or a Long-term Car Pooling Prob-

lem (LTCPP).

In the case of DCPP (Baldacci et al., 2004; Calvo

et al., 2004; Swan et al., 2013), on each day a number

of users, considered as servers, are available for pick-

ing up and later bringing back colleagues (clients) on

that particular day. Then, the problem asks to assign

clients to servers and to identify which routes should

be driven by the servers in order to minimize the num-

ber of unassigned clients, subject to time window and

car capacity constraints.

However, in the case of LTCPP, each user is avail-

able to act both as a server and as a client. The LTCPP

requires to ﬁnd user pools or crews where each user

will in turn, on different days, picks up the remaining

pool members and to identify the routes to be driven

by each member in the car pool. The main objective

of this problem is to minimize the number of vehicles

and the length of the path traveled by all users when

acting as servers subject to car capacity and time win-

dow constraints (Guo, 2012).

The Long-term Car Pooling Problem can be seen

as a compound problem. It is a combination of a clus-

tering problem and a routing problem. This model

is frequently used by universities and large compa-

Figure 1: An example of the LTCPP.

ICSOFT 2020 - 15th International Conference on Software Technologies

320

nies which provide long-term carpool service for their

students or employees, since it is the most stable

car pooling model. Indeed, in the LTCPP users will

not usually be changing (in a relatively long period).

Therefore, the maximum number of users in a pool

is equal to the capacity of the smallest car among

those owned by all pool members, because ﬁnally

each member should pick up all the other ones. Figure

1 presents an example of the LTCPP. In this example

we suppose that we have 7 users and the maximum

car capacity is 4. In the ﬁrst day, users 1, 2 and 3 are

pooled together where 1 is considered as server. Users

4, 5, 6 and 7 construct another pool and the user 5 is

considered as server. In the other days, each member

could be considered as server.

In this article, our study focuses on the Long-

term Car Pooling Problem, which is an NP-complete

problem (Varrentrapp et al., 2002). Due to its hard-

ness and signiﬁcance it has constantly been stud-

ied by researchers who approached it from vari-

ous perspectives. After bibliographical study, we

noticed that different heuristics and metaheuristics

have been developed by different authors. We can

cite the Saving Functions Based Algorithm (Ferrari

et al., 2003), an Approximated Non-deterministic

Tree Search (ANTS) algorithm (Maniezzo et al.,

2004), a Simulation Based Approach (SB) (Correia

and Viegas, 2008), a Multi-Matching System (Yan

et al., 2011), a Clustering Ant Colony Algorithm

(CAC) (Guo et al., 2012), a Guided Genetic Algo-

rithm (GGA) (Guo et al., 2011).

2.2 LTCPP Formulation

The LTCPP can be modeled, by a direct graph G =

{U ∪ {0},A}, where U is the set of users, and A =

{arc(i, j)/i ∈ U, j ∈ U ∪ {0}} is the set of directed

weighted arcs where each arc(i, j) ∈ A is associated

with a non-negative travel cost d

i j

and a travel time

i j

. Every user i ∈ U for the Long-term Car Pooling

is characterized by: the origin (home), the node 0 that

corresponds to the destination, the earlier time e

for

leaving home; the acceptable time r

for arriving at

destination; the capacity Q

of his car; furthermore the

maximum time T

that the user will accept to drive.

The LTCPP can be seen as a multi-objective prob-

lem, looking for minimizing the total number of ve-

hicles traveling to the common destination and the

distance to be driven by each user, when acting as a

driver.

However, based on the research of Varrentrapp,

Maniezzo and St

utzle (Varrentrapp et al., 2002), we

can combine these two objectives in a single objec-

tive function. Then, the LTCPP can be considered as

a single-objective problem.

Each user of pool k, will use on different days his

car to pick up his pool mates and drive to the common

destination. Therefore, each driver must know his

Hamiltonian path starting from his origin (his home),

then linking all other nodes corresponding to other

pool members’ homes exactly once, until reaching the

common destination.

Let path(i, k) be a feasible Hamiltonian path,

starting from i ∈ k, linking all j ∈ k\{i} and ending

in 0. Suppose |k| ≤ Q

, where |k| is the size of this

pool, Q

is the minimum car capacity of pool k and

all constraints are satisﬁed. d

denote the travel cost

from user’s home directly to the destination, while ρ

is a penalty imposed for a client driving alone.

The cost of a pool k is then computed as follows:

Cost(k) =

(

∑

i∈k

cost(path(i,k))

|k|

,i f |k| > 1,

∑

i∈k

+ ρ

,otherwise.

The total cost of a complete solution to the LTCPP

is then equal to the sum of the costs of the pools in

it. By this view, both objective functions can be op-

timized at the same time. In fact, provided that the

penalty of a client driving alone is sufﬁciently greater

than zero, it is better to pool clients together than to

leave them alone.

In order to formulate this problem, we present the

following notations (Guo et al., 2012):

• K : Set of indices of all pools;

• U : Set of indices of all users;

• A : Set of indices of all arcs;

• X

i j

: Binary variable equals to 1 if arc(i,j) is trav-

eled by a server h of a pool k;

• Y

: Binary variable equals to 1 if user i is in pool

• S

: Positive variable indicating the pick-up time

of user i by server h;

• F

: Positive variable for denoting the arrival time

of user i at destination when traveling with server

• d

i j

: Positive value indicating the travel cost be-

tween users i and j;

• t

i j

: Positive value indicating the travel time be-

tween users i and j;

• ρ

: Positive value for denoting the penalty for user

i when he travels alone;

• ϕ

: Binary variable equals to 1 if user i is not

pooled with any other user;

• e

: Positive value denoting the earlier time for

leaving home of user i;

A Bee Colony Optimization Algorithm for the Long-Term Car Pooling Problem

321

• r

: Positive value indicating the acceptable time

for arriving at work of user i;

• Q

: Positive value denoting the minimum car ca-

pacity of pool k;

• T

: Positive value for denoting the extra driving

time speciﬁed by server h;

Objective Function:

F= min(

∑

k∈K

∑

h∈U

∑

(i, j)∈A

i j

∑

i∈U

∑

i∈U

)

Constraints:

∑

j∈U\{h}

i j

= Y

i,h ∈ U;k ∈ K; (1)

∑

j∈U

= Y

i,h ∈ U; k ∈ K; (2)

∑

j∈U

i j

∑

j∈U

i,h ∈ U;k ∈ K; (3)

∑

k∈K

+ϕ

= 1 i ∈U; (4)

∑

(i, j)∈A

i j

≤ Q

h ∈ U; k ∈ K; (5)

∑

(i, j)∈A

i j

≤ T

h ∈ U ; k ∈ K; (6)

≥ e

i,h ∈ U; (7)

−S

≥ t

i j

−M ∗(1−

∑

k∈K

i j

) (i, j) ∈ A; h ∈U; (8)

≥ S

−M ∗(1 −

∑

k∈K

) i,h ∈ U; (9)

≤ r

+M ∗(1−

∑

k∈K

∑

j∈U

i j

) i,h ∈U; (10)

i j

∈ {0,1} h ∈ U ; k ∈ K;(i, j) ∈ A; (11)

∈ {0,1} i ∈ U; k ∈ K; (12)

∈ {0,1} i ∈ U; (13)

≥ 0 i,h ∈ U; (14)

≥ 0 i,h ∈ U; (15)

If there is an arc-connecting user i and user j or

user j and user i, equations (1) and (2) force a user i

to be added in a pool k. Constraint (3) makes sure the

continuity of the path. Equation (4) shows that each

user must be assigned to a pool or be penalized. The

car capacity and the extra driving time constraints are

translated respectively in (5) and (6). Equations (7)

and (8), where M is a big constant, collectively set

feasible pick-up times, while (9) and (10) are mini-

mum and maximum values of feasible arrival times,

respectively. Constraints (11) and (12) and (13) are

binary constraints while (14) and (15) are positivity

constraints.

3 THE BEE COLONY

OPTIMIZATION ALGORITHM

The BCO is a population-based algorithm inspired by

bees’ behavior in the nature. It was proposed for deal-

ing with hard combinatorial optimization problems.

The main idea of this approach is to build the multi

agent system (colony of artiﬁcial bees) where each ar-

tiﬁcial bee called agent, can generate one solution to

the problem. Looking for the best feasible solutions,

artiﬁcial bees investigate through the search space.

After that, they cooperate and exchange information.

Using collective knowledge and information sharing,

artiﬁcial bees update their current solution and then,

concentrate on the more promising areas and incre-

mentally abandon solutions from the less promising

ones. Step by step, this population of agents collec-

tively generate and/or yield their solutions. The BCO

algorithm runs iteratively until a stopping condition is

met.

The basic principles of collective bee intelligence

in solving combinatorial optimization problems were

proposed for the ﬁrst time by Lucic and Teodorovic

(Lu

c and Teodorovi

c, 2003b; Lu

c and Teodorovi

2003a).

The algorithm is composed of two alternating

phases: forward pass and backward pass. During ev-

ery forward pass, each artiﬁcial bee is exploring the

search space. It makes a predeﬁned number of local

moves, which gradually construct and/or improve the

solution, yielding to a new solution. Having created

various partial solutions, they return to the nest and

perform the second phase, called backward pass. In

the hive, the artiﬁcial bees share information about

their solutions.

In nature, bees communicate through a waggle

dance which would notify the other bees about the

quantity and the quality of nectar they have discov-

ered, and the proximity of the path to the hive. In the

BCO algorithm, the artiﬁcial bees publicize the qual-

ity of the solution created, i.e. the objective function

value. During the backward pass, bees compare all

partial generated solutions. Based on a certain proba-

bility, each bee decides whether to stay faithful to its

created partial solution or not. Note that, bees with

higher objective function value have greater chance

to continue their own exploration. Hence, if bees

dance and thus recruit the nestmates before return-

ing to the created partial solution, they are considered

as recruiters. Moreover, if a bee chooses to become

uncommitted follower, it must select a new solution

from recruiters by the roulette wheel (better solutions

have higher opportunities of being chosen for explo-

ration).

ICSOFT 2020 - 15th International Conference on Software Technologies

322

The two phases (forward and backward pass) al-

ternate, until a stopping condition is met. There are

different possible stopping conditions, like, the maxi-

mum total number of forward/backward passes with-

out the improvement of the objective function, the

maximum total number of forward/backward passes,

etc.

In the ﬁrst stage of the search, all the bees are lo-

cated in the hive.

4 THE PROPOSED BCO

ALGORITHM FOR SOLVING

THE LTCPP

The basic idea of the Bee Colony Optimization (BCO)

is to model the problem to solve as the search for

a minimum cost path in a graph, and to use artiﬁ-

cial bees to search for good paths. Let’s represent

each user for the Long-term Car Pooling Problem by a

node. In our proposed algorithm, each bee is allowed

to explore and search for a car pool. The bee starts

to build a car pool by randomly determining the ﬁrst

user to be added to its car pool initially empty. We

decompose our problem into stages. Each stage is as-

sociated to an unpooled user. In fact, the ﬁrst user in

the car (driver) represents the ﬁrst stage; the second

user to join the pool represents the second stage, etc.

In every stage, the bee chooses to visit one node.

Thus during the forward pass, every bee travels

to a node. It behaves according to a selection pro-

cess named the roulette wheel selection. Based on the

probability values, this model of choice is employed

to aid the bee in its decision making on which user

to be pooled. The probability gives to the bee the

likelihood to move from user i to user j. In fact, if

constraints presented previously (in section 2.1) are

satisﬁed, the probability value between two different

users i and j, P

i j

, is deﬁned as follows:

i j

∑

k∈U

(16)

Where d

i j

represents the travel cost from user i to user

j. Note that the travel cost is inversely proportional to

i j

. In other words, the shorter the distance, the higher

the likelihood of that user to be chosen.

After that, the bee returns to the hive (in the back-

ward pass), where it participates in a decision making

process. Then, every bee decides whether to aban-

don their created partial solution and become again

uncommitted follower, or to continue its own explo-

ration and become recruiter.

During the second forward pass, each bee visits a

new node, creates a partial solution, and after that per-

forms again a backward pass and returns to the hive.

Then, it performs a third forward pass, etc.The

two phases of the search algorithm, forward and back-

ward pass, are alternated iteratively, until the total

number of forward/backward passes reaches the car

pool size. Then, all generated pools are evaluated and

the best one is added to the current solution.

One iteration is ﬁnished, when all the users are

clustered by bees. The algorithm terminates when a

maximal number of iterations is reached.

The overall structure of the BCO-LTCPP is out-

lined in Algorithm 1.

Algorithm 1: BCO-LTCPP Algorithm.

Initialize parameters: Number of bees n and Maxi-

mum Iteration;

While Maximum Iteration is not met do

Initialize the car pool of every bee b G

←

Initialize the current solution S

cur

←

Initialize the current set of users not yet pooled

cur

← U ;

Repeat

Select randomly a new user u;

←the car capacity of user u;

For every bee b do{

Insert u into G

and eliminate it from U

cur

;}

j ← 1;

While ( j < NC

)do{

(a) The forward pass

For every bee b do{

Evaluate all unserved users in U

cur

;

Choose an unserved user i who satisﬁes

constraints using the roulette wheel

selection based on probability P

i j

(16);

Insert user i into G

and eliminate it

from U

cur

;}

(b)The backward pass

For every bee b do{

Every bee decides randomly whether to

continue its own exploration and

becomes recruiter, or to become

uncommitted;

If (b is uncommitted) then

Randomly chooses a recruiter to

follow;}

j ← j + 1;}

Find the best pool G − best

from all pools G

;

cur

← S

cur

G − best

;

Until all users are clustered;

Update the best solution;

End

Output the best solution;

A Bee Colony Optimization Algorithm for the Long-Term Car Pooling Problem

323

5 EXPERIMENTAL STUDY AND

DISCUSSION

This section describes, ﬁrst, the benchmark problems

used in our experiments. To test the efﬁciency and

the performance of our algorithm, we have chosen

to compare our experimental results with four ap-

proaches for solving the LTCPP: the GGA, the Ants,

the CAC and the simulation based approach (SB),

since we haven’t found any other BCO in the liter-

ature for the LTCPP and because all these approaches

have been proven to have the ability to solve our prob-

lem and provide high solution quality. Since the CAC

is one of the most recent swarm intelligence meta-

heuristic for solving the Long-term Car Pooling Prob-

lem, and, in order to compare both of the algorithms

in the same environment, we have implemented the

CAC algorithm and tested it exactly as it was de-

scribed in (Guo et al., 2012), and we have maintained

the same parameter values. The results of GGA, Ants

and SB are obtained directly from (Guo, 2012).

5.1 Benchmark Problems

Since there are no benchmarks for the LTCPP in the

literature, we have chosen to use in our experimen-

tal study, the benchmarks developed in (Guo, 2012),

which are originally derived from the Pickup and

Delivery Problems with Time Windows (PDPTW)

benchmarks by Li and Lim (Li and Lim, 2003).

These benchmarks include two sets of instances

where each one is composed of 9 instances with the

number of users ranging from 100 to 400. The ﬁrst set

has the users clustered distributed, therefore is named

with C. The users in the second set are allocated ran-

domly, so the set is named with R.

For all instances, we considered the depot in the

original instances as the destination, while we re-

tained the coordinates of the customers, who become

the users. The cost d

i j

was assumed to be equal to the

Euclidean distance between user i and j. Travel times

i j

were set equal to the distances divided by 50 km/h

(average travel speed). For each user, the penalty ρ

was calculated as ρ

= 2d

, where we denoted by d

the travel cost from user i’s home directly to the desti-

nation, and the car capacity Q

was set to 4. The max-

imum ride time T

was computed as T

= 1.5t

, where

represents the time needed to travel from the user

i’s home to the destination. The latest arrival times

is an integer value randomly selected in the interval

[510,540], and the earliest departure time of user i was

estimated to be equal to e

= r

− max(t

+ 30,2t

5.2 Computational Results

For both algorithms, there are some control parame-

ters, which are used for its efﬁcient performance. Af-

ter some times of test runs, the parameters settings of

our algorithm are speciﬁed as follows:

-Number of iterations: IT = 1000,

-Number of bees: n = number of users in the instance.

The BCO-LTCPP algorithm was implemented in

JAVA, and all the tests were performed on computer

with Intel core i5-3317U, 1.70 GHz and 6 Go of

RAM.

5.2.1 Comparison of BCO-LTCPP with

State-of-the Art Algorithms

We have performed 10 runs for each instance. The

results presented in this study are the averages of

ten runs (AVG). We also report the best result (Best)

among the ten replications. In addition, the results are

also analyzed through the non-parametric Wilcoxon

rank-signed test (Sheskin, 2003) (W-test) in order to

verify if the difference between the compared algo-

rithms is statistically signiﬁcant. Note that for W-

Test, the level of signiﬁcance considered is 0.05. We

use (+) and (-) to denote if the BCO-LTCPP result

is, respectively, signiﬁcantly or not signiﬁcantly bet-

ter than the other algorithms.

Table 1 summarizes the results of the C set in-

stances. On this set of instances, the BCO-LTCPP

outperforms the GGA, Ants, SB and CAC on all in-

stances in best found solution. Considering the av-

erage solution quality, our algorithm outperforms the

GGA and Ants on 8 instances, and outperforms the

CAC and SB on all instances. Besides, the superior-

ity of our algorithm is statistically clear. In fact, the

W- test shows that statistically our algorithm outper-

forms widely the other algorithms (GGA, Ants and

SB). Also, statistically, the BCO-LTCPP is signiﬁ-

cantly better than the CAC in seven instances among

the nine instances.

Table 2 shows the experimental results of set R in-

stances. In fact, we can draw that the BCO-LTCPP

outperforms the GGA, Ants, SB and CAC on all in-

stances in best-found solution. Also, the BCO-LTCPP

outperforms the GGA and Ants on 8 instances and the

CAC and SB on all instances, considering the average

solution. Furthermore, our algorithm is statistically

a clear winner when compared to the GGA, the Ants

and the SB. In addition, the W-test shows that the re-

sults of the BCO-LTCPP algorithm are signiﬁcantly

better than the CAC algorithm on eight instances.

ICSOFT 2020 - 15th International Conference on Software Technologies

324

Table 1: Experimental results of set C instances for BCO, GGA, Ants, SB and CAC algorithms.

Instance Size BCO-LTCPP GGA Ants SB CAC

Best AVG Best AVG Best AVG Best AVG Best AVG W-test

C101 100 1491 1576.1 1585,5 1599,3 1585.5 1592,9 1647,4 1669,2 1523 1595.4 (+)

C102 100 1617 1696 1701,9 1712 1711.4 1748,5 1717,5 1724,8 1652 1722.9 (-)

C103 100 1503 1545 1513,7 1543,9 1512.6 1535,1 1532,2 1599,4 1524 1559.8 (+)

C201 200 2600 2653.5 2672,2 2749,4 2784.4 2854,2 2761,7 2868,6 2660 2754.3 (+)

C202 200 2765 2817.7 2836,7 2876,5 2936.1 3004,5 3081,7 3114,1 2816 2912.6 (+)

C203 200 2709 2785 2716 2891,8 2845.9 3003,5 2975,1 3182,4 2789 2874.9 (+)

C401 400 5308 5496.2 5489,4 5690,6 5833.5 6281,4 6174,2 6860,3 5661 5727 (+)

C402 400 4350 4586.2 4548,3 4786,4 4893.5 5153,2 5383,7 5524,5 4552 4590.9 (-)

C403 400 5776 5917.7 5909,6 6085,2 6125.6 6742,1 6675,2 6994,5 5803 5974.5 (+)

W-test (+) (+) (+) (-) (+) (+)

Table 2: Experimental results of set R instances for BCO, GGA, Ants, SB and CAC algorithms.

Instance Size BCO-LTCPP GGA Ants SB CAC

Best AVG Best AVG Best AVG Best AVG Best AVG W-test

R101 100 2136 2197,2 2207,1 2235,9 2207.1 2281,5 2235,1 2265,4 2199 2254.4 (+)

R102 100 1801 1898,2 1824,5 1867,5 1834.6 1864,2 1832,8 2091,7 1906 1976.2 (+)

R103 100 2098 2146,7 2209,1 2286 2299.2 2438,7 2204,7 2418,5 2102 2177.9 (+)

R201 200 3933 4054,4 4034,8 4188,3 4101.5 4253,5 4425 4567,1 4190 4272.2 (+)

R202 200 3598 3655,3 3646,8 3751,7 3772.2 4071,9 3952,4 4283,3 3698 3771.2 (+)

R203 200 3852 4017,8 3923,2 4158,4 4368.5 4541,5 4092,4 4257,5 4242 4372.9 (+)

R401 400 7441 7659,7 7514,9 7799,5 8396.1 8580,4 8787,8 8993,8 7777 7894.6 (+)

R402 400 6023 6233,6 6172,7 6254 6512.7 6893,3 7258,7 7417,5 6045 6186.5 (-)

R403 400 7403 7617,5 7670,2 7872,9 8113.1 8338,9 8841,9 8933,5 7885 7978.8 (+)

W-test (+) (+) (+) (+) (+) (+)

Table 3: CPU time of the different algorithms (in seconds).

Instances BCO GGA Ants SB CAC

C101 2 13 17 91 8

C102 3 9 14 94 7

C103 2 12 18 85 8

C201 14 31 57 329 39

C202 14 28 64 473 43

C203 14 42 58 394 39

C401 46 248 424 934 231

C402 42 203 357 683 243

C403 46 295 511 1257 263

R101 2 18 18 100 6

R102 2 15 17 97 7

R103 3 17 21 80 7

R201 15 43 108 430 35

R202 16 41 84 231 33

R203 12 38 116 540 34

R401 59 392 581 1106 217

R402 49 277 479 896 231

R403 55 311 631 1037 237

Computer \ Langage Intel core i5-3317U

1.70GHz (JAVA)

Intel core i7 740QM

2.9GHz (JAVA)

Intel core i7 740QM

2.9GHz (JAVA)

Intel core

i7 740QM

2.9GHz

(JAVA)

Intel core i5-3317U

1.70GHz (JAVA)

A Bee Colony Optimization Algorithm for the Long-Term Car Pooling Problem

325

5.2.2 Comparison of CPU Time of the Different

Algorithms

Table 3 presents the elapsed CPU time to obtain the

best solution. In this table, we can see that, the com-

puting time of the BCO-LTCPP is much lower than

other approaches. We note that these values are given

as an indication since the different algorithms are

tested on different computers. However, we remark

that the proposed BCO-LTCPP algorithm, despite it

was run on the less powerful processor, takes much

less CPU time.

Therefore, we can conclude that our algorithm is

able to reach signiﬁcant better solutions in a short

time.

6 CONCLUSIONS

In this paper, we proposed a Bee Colony Optimiza-

tion algorithm for the Long-term Car Pooling Prob-

lem. The proposed BCO algorithm, called BCO-

LTCPP, was tested on various benchmark instances.

Based on the experimental results, we can conclude

that the BCO-LTCPP is an efﬁcient approach to solve

the Long-term Car Pooling Problem. In fact, these re-

sults show that the proposed algorithm is able to reach

signiﬁcantly better solutions in a very short computa-

tional time when compared to other competitive ap-

proaches from literature on all instances.

The effectiveness of the developed BCO algo-

rithm encourages its application, for future works,

to other transportation problems, such as the daily

carpooling problem or the academic vehicle routing

Problem.

REFERENCES

Bacanin, N., Tuba, M., and Brajevic, I. (2010). An object-

oriented software implementation of a modiﬁed arti-

ﬁcial bee colony (abc) algorithm. In Proceedings of

the 11th WSEAS international conference on nural

networks and 11th WSEAS international conference

on evolutionary computing and 11th WSEAS interna-

tional conference on Fuzzy systems, pages 179–184.

World Scientiﬁc and Engineering Academy and Soci-

ety (WSEAS).

Baldacci, R., Maniezzo, V., and Mingozzi, A. (2004). An

exact method for the car pooling problem based on

lagrangean column generation. Operations Research,

52(3):422–439.

Bruglieri, M., Ciccarelli, D., Colorni, A., and Lu

e, A.

(2011). Poliunipool: a carpooling system for uni-

versities. Procedia-Social and Behavioral Sciences,

20:558–567.

Calvo, R. W., de Luigi, F., Haastrup, P., and Maniezzo, V.

(2004). A distributed geographic information system

for the daily car pooling problem. Computers & Op-

erations Research, 31(13):2263–2278.

Chong, C. S., Low, M. Y. H., Sivakumar, A. I., and Gay,

K. L. (2006). A bee colony optimization algorithm to

job shop scheduling. In Proceedings of the 2006 win-

ter simulation conference, pages 1954–1961. IEEE.

Chong, C. S., Low, M. Y. H., Sivakumar, A. I., and Gay,

K. L. (2007). Using a bee colony algorithm for neigh-

borhood search in job shop scheduling problems. In

21st European conference on modeling and simula-

tion (ECMS 2007).

Correia, G. and Viegas, J. M. (2011). Carpooling and car-

pool clubs: Clarifying concepts and assessing value

enhancement possibilities through a stated preference

web survey in lisbon, portugal. Transportation Re-

search Part A: Policy and Practice, 45(2):81–90.

Correia, G. H. d. A. and Viegas, J. M. (2008). Structured

simulation-based methodology for carpooling viabil-

ity assessment. Technical report.

Dorigo, M., Birattari, M., and Stutzle, T. (2006). Ant colony

optimization. IEEE computational intelligence maga-

zine, 1(4):28–39.

Ferrari, E., Manzini, R., Pareschi, A., Persona, A., and Re-

gattieri, A. (2003). The car pooling problem: Heuris-

tic algorithms based on savings functions. Journal of

Advanced Transportation, 37(3):243–272.

Guo, Y. (2012). Metaheuristics for solving large size long-

term car pooling problem and an extension. PhD the-

sis, Artois.

Guo, Y., Goncalves, G., and Hsu, T. (2011). A guided

genetic algorithm for solving the long-term car pool-

ing problem. In 2011 IEEE Workshop On Computa-

tional Intelligence In Production And Logistics Sys-

tems (CIPLS), pages 1–7. IEEE.

Guo, Y., Goncalves, G., and Hsu, T. (2012). A clustering ant

colony algorithm for the long-term car pooling prob-

lem. International Journal of Swarm Intelligence Re-

search (IJSIR), 3(2):39–62.

Huang, C.-L. (2015). A particle-based simpliﬁed swarm

optimization algorithm for reliability redundancy al-

location problems. Reliability Engineering & System

Safety, 142:221–230.

Jadon, S. S., Bansal, J. C., Tiwari, R., and Sharma,

H. (2018). Artiﬁcial bee colony algorithm with

global and local neighborhoods. International Journal

of System Assurance Engineering and Management,

9(3):589–601.

Karaboga, D. (2005). An idea based on honey bee swarm

for numerical optimization. Technical report, Tech-

nical report-tr06, Erciyes university, engineering fac-

ulty, computer . . . .

Karaboga, D. and Akay, B. (2011). A modiﬁed artiﬁcial

bee colony (abc) algorithm for constrained optimiza-

tion problems. Applied soft computing, 11(3):3021–

3031.

Karaboga, D. and Basturk, B. (2008). On the performance

of artiﬁcial bee colony (abc) algorithm. Applied soft

computing, 8(1):687–697.

ICSOFT 2020 - 15th International Conference on Software Technologies

326

Li, H. and Lim, A. (2003). A metaheuristic for the

pickup and delivery problem with time windows. In-

ternational Journal on Artiﬁcial Intelligence Tools,

12(02):173–186.

Liu, Y. and Qin, G. (2014). A modiﬁed particle swarm op-

timization algorithm for reliability redundancy opti-

mization problem. Journal of Computers, 9(9):2124–

2131.

c, P. and Teodorovi

c, D. (2003a). Computing with

bees: attacking complex transportation engineering

problems. International Journal on Artiﬁcial Intelli-

gence Tools, 12(03):375–394.

c, P. and Teodorovi

c, D. (2003b). Vehicle routing prob-

lem with uncertain demand at nodes: the bee system

and fuzzy logic approach. In Fuzzy sets based heuris-

tics for optimization, pages 67–82. Springer.

Maniezzo, V., Carbonaro, A., and Hildmann, H. (2004). An

ants heuristic for the long—term car pooling problem.

In New optimization techniques in engineering, pages

411–430. Springer.

Manzini, R., Pareschi, A., et al. (2012). A decision-support

system for the car pooling problem. Journal of Trans-

portation Technologies, 2(02):85.

Nouiri, M., Bekrar, A., Jemai, A., Niar, S., and Ammari,

A. C. (2018). An effective and distributed particle

swarm optimization algorithm for ﬂexible job-shop

scheduling problem. Journal of Intelligent Manufac-

turing, 29(3):603–615.

Sheskin, D. J. (2003). Handbook of parametric and non-

parametric statistical procedures. crc Press.

Subotic, M., Tuba, M., and Stanarevic, N. (2010). Paral-

lelization of the artiﬁcial bee colony (abc) algorithm.

Recent advances in neural networks, fuzzy systems &

evolutionary computing, pages 191–196.

Swan, J., Drake, J.,

Ozcan, E., Goulding, J., and Woodward,

J. (2013). A comparison of acceptance criteria for the

daily car-pooling problem. In Computer and informa-

tion sciences III, pages 477–483. Springer.

Teodorovi

c, D. (2008). Swarm intelligence systems for

transportation engineering: Principles and applica-

tions. Transportation Research Part C: Emerging

Technologies, 16(6):651–667.

Teodorovic, D. and

Selmic, M. (2007). The bco algorithm

for the p-median problem. Proceedings of the XXXIV

Serbian Operations Research Conferece, pages 417–

420.

Vargas, M. A., Sefair, J., Walteros, J. L., Medaglia, A. L.,

and Rivera, L. (2008). Car pooling optimization:

a case study in strasbourg (france). In 2008 IEEE

Systems and Information Engineering Design Sympo-

sium, pages 89–94. IEEE.

Varrentrapp, K., Maniezzo, V., and St

utzle, T. (2002).

The long term car pooling problem–on the sound-

ness of the problem formulation and proof of np-

completeness. Technische Universitat Darmstadt.

Wong, L.-P., Low, M. Y. H., and Chong, C. S. (2010a).

Bee colony optimization with local search for travel-

ing salesman problem. International Journal on Arti-

ﬁcial Intelligence Tools, 19(03):305–334.

Wong, L.-P., Puan, C. Y., Low, M. Y. H., Wong, Y. W., and

Chong, C. S. (2010b). Bee colony optimisation al-

gorithm with big valley landscape exploitation for job

shop scheduling problems. International Journal of

Bio-Inspired Computation, 2(2):85–99.

Wu, P., Gao, L., Zou, D., and Li, S. (2011). An improved

particle swarm optimization algorithm for reliability

problems. ISA transactions, 50(1):71–81.

Xue, Y., Jiang, J., Zhao, B., and Ma, T. (2018). A

self-adaptive artiﬁcial bee colony algorithm based on

global best for global optimization. Soft Computing,

22(9):2935–2952.

Yan, S., Chen, C.-Y., and Lin, Y.-F. (2011). A model with

a heuristic algorithm for solving the long-term many-

to-many car pooling problem. IEEE Transactions on

Intelligent Transportation Systems, 12(4):1362–1373.

Zouari, W., Alaya, I., and Tagina, M. (2017). A hybrid ant

colony algorithm with a local search for the strongly

correlated knapsack problem. In 2017 IEEE/ACS 14th

International Conference on Computer Systems and

Applications (AICCSA), pages 527–533. IEEE.

A Bee Colony Optimization Algorithm for the Long-Term Car Pooling Problem

327