A Posteriori Approach of Real-time Ridesharing Problem with

Intermediate Locations

Kamel Aissat

and Ammar Oulmara

University of Lorraine - LORIA, Nancy, France

University of Lorraine, Ile de Saulcy, Metz, France

Keywords:

Operations Research, Transport, Ridesharing, Shortest Path Problem, Geographical Maps.

Abstract:

Ridesharing is a travel mode that provides several beneﬁts and solutions, such as the reduction of travel cost,

the reduction of the trafﬁc congestion and the provision of travel options. In the classical ridesharing approach,

the driver makes a detour to the rider’s origin in order to pick-up the rider, then drives him to his destination

and ﬁnally the driver goes to his own destination. This implies that the driver endures the whole detour and

may not accept such matching if the detour is too long. However, the matching could be accepted if the

rider meets the driver at an intermediate location. In this paper, we present a general ridesharing approach in

which a driver and a rider accept to meet each other at an intermediate pick-up location and to separate at an

intermediate drop-off location not necessarily their origins and destinations locations, respectively.

Thus, for a given rider, we propose an exact and heuristic methods to determine the best driver and the best

meeting locations that minimize a total travel cost. Finally, we perform a numerical study using a real road

network and a real dataset. Our experimental analysis shows that our heuristics provide efﬁcient performance

within short CPU times and improve participants cost-savings and matching rate compared to the classical

ridesharing.

1 INTRODUCTION

Several worldwide urban cities face mobility prob-

lems among which the insufﬁciency and the satura-

tion of the current offer of public transport, trafﬁc

congestion and carbon emissions. Those problems

impact the environment and the quality of life of cit-

izens. The transportation activity is responsible for

a great part of carbon emissions. At least half of to-

tal carbon emissions is due to private vehicles (Metz

et al., 2005). Nowadays, governments begins to be

conscious of the urgency to conserve the environ-

ment by investing in more environmentally friendly

and safe modes of transportation like Electric vehicles

(Sassi and Oulamara, 2014) and ridesharing systems.

Ridesharing helps reducing the number of vehicles

since most private vehicle trips are made with only

one passenger occupancy. The total cost of single pas-

senger trips is quantiﬁed in (Hartwig and Buchmann,

2006). This study estimates that the cost of “empty

seats” in the world would be about 500 billion dollars

per year. Thus, a signiﬁcant scope for reduction of

cost and the number of cars on the road requires in-

telligent usage of private vehicles. In that case, a ve-

hicle is considered as a means of mobility rather than

an ownership. In order to avoid empty seats during

trips, recent initiatives such as ridesharing services,

have been deployed with success.

A ridesharing service brings together users with

similar itineraries and time schedules. More pre-

cisely, a driver with his vehicle and a rider, traveling

from their individual starting locations to their desti-

nations, share a common part of their itinerary using

the driver’s vehicle as well as a part of vehicle-related

expenses.

The considerable progress in embedded and mo-

bile computing technologies such as smartphones,

along with the pervasive diffusion of online social net-

working tools and geolocation devices (Global Posi-

tioning System - GPS), facilitate the use of rideshar-

ing services. Thus, a new and innovative solution of

ridesharing service has emerged. It consists in an au-

tomatic and instant matching of riders through a net-

work service by using smartphones as both geolo-

cation and communication devices. This service is

called real-time ridesharing.

Several methods have been proposed in the liter-

ature to solve the simplest form of ridesharing, also

Aissat K. and Oulmara A..

A Posteriori Approach of Real-time Ridesharing Problem with Intermediate Locations.

DOI: 10.5220/0005256100630074

In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 63-74

ISBN: 978-989-758-075-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

called classical ridesharing, see (Agatz et al., 2012).

In this simplest form, the driver picks up the rider at

his starting location, drops him off at his ending lo-

cation and then continues to his target location. This

method is the most implemented in ridesharing ser-

vices. Thus, users (drivers and riders) send their re-

quests (demands and offers of ridesharing) and the

system proposes the best matchings of offers and de-

mands regarding the itinerary of each user. However,

the matching opportunities of drivers and riders in the

classical ridesharing decrease when itineraries of rid-

ers and drivers are slightly different. In order to over-

come this lack of ﬂexibilty, instead of allowing the

driver to go to a rider’s origin to pick him up and drop

him off at his destination, in this paper, we allow the

driver and the rider to meet and to separate at inter-

mediate locations, not necessarily at the rider’s origin

and rider’s destination (see Figure 1).

Speciﬁcally, the rider travels on his own to the ﬁrst

intermediate location, where he is picked up by the

driver. He is then dropped off at the second interme-

diate location, where he will continue to his ending

location. Solving the problem of ridesharing with in-

termediate locations requires the development of efﬁ-

cient algorithms that determine the optimal interme-

diate locations regarding the itineraries of riders and

drivers.

2 LITERATURE REVIEW

Depending on the form of the ridesharing in practice,

we distinguish the following optimization problems:

Slugging problem, Taxi-ridesharing problem and

Ridesharing problem.

Slugging Problem. High Occupancy Vehicle

(HOV) lanes are trafﬁc lanes set aside for vehicles

with a minimum number of occupants. In order to

meet the occupancy requirements of HOV lanes,

users have developed a new way of travel, known as

slugging, that offers the beneﬁts of traveling on an

HOV lane without forming classical ridesharing. In

slugging, the driver maintains his original route and

does not make any detour to pick-up or drop-off the

rider. Thus, the rider walks to the driver’s origin,

boards at the driver’s departure time, alights at the

driver’s destination, then walks from there to his own

destination. Authors in (Ma and Wolfson, 2013)

formalize and study this slugging problem. Variants

of the slugging problem that also consider the vehicle

capacity constraint and travel time delay, has been

proven to be the NP-complete and they propose

a quadratic algorithm to solve it optimally. They

(a)

(b)

Figure 1: Figure [a] represents the shortest path of driver

(green path) and the rider (red path) before matching, while

ﬁgure [b] represents the new path of the driver and the rider

after the matching with two new intermediate locations (in-

termediate meeting locations are pointed with two blue ar-

rows and the common path is represented with blue path).

also develop heuristics that have been evaluated on

real data. Another work on slugging problem is

considered in (Mote and Whitestone, 2011).

Taxi-ridesharing Problem. The problem of Taxi-

ridesharing consists in assigning taxi vehicles to rid-

ers requests which are spread over different locations

in a given zone. Several constraints can be considered

such as the vehicle capacity and time windows of rid-

ers. Thus, this problem concerns the assignment of a

time to each pickup and delivery event, within these

time windows.

In contrast to Slugging problem, the route of driver

trip (i.e. taxi driver) change to accommodate riders,

and the pick-up and drop-off locations for the riders

correspond to their origins and destinations, respec-

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

tively. Authors in (Ma et al., 2013) propose a practi-

cal taxi ridesharing service in which an organization

operates a dynamic taxi ridesharing service. In their

system, the taxi drivers can independently determine

when they join and leave the service. Thus, riders sub-

mit ride queries in real time via a mobile device, e.g.,

a smartphone. Each query indicates the origin and

destination locations of the trip, as well as the time

windows during which the rider wants to be picked up

and dropped off. Once a new query is received, the

system determines an appropriate taxi which is able

to satisfy both the new query and the trips of existing

riders who are already assigned to that taxi. The up-

dated schedules and routes will then be given to the

corresponding taxi driver and riders. Another work

on taxi-ridesharing problem is considered in (Varone

and Janilionis, 2014).

Classical Ridesharing Problem. The principle

is rather similar to Taxi-ridesharing. The main

differences are, (i) the classical ridesharing is based

on private cars in which the rider shares his trip

with a simple driver, while in taxi ridesharing the

presence of the taxi driver is obligatory, (ii) the taxi

ridesharing usually needs appropriate pricing mech-

anisms, generally more expensive than a classical

ridesharing, to incite taxi drivers. Several research

has been reported recently in the ﬁelds of ridesharing,

see (Agatz et al., 2012) and (Furuhata et al., 2013).

In ridesharing system, when a driver’s offer or rider’s

request arrives in the system, some options should

be speciﬁed. For instance, when the driver offers

a ride, he may specify if he is willing to take a

single rider or multiple riders. Similarly, when the

rider looks for a ride, he speciﬁes if he wants to

ride with a single driver or may accept to ride with

multiple drivers and will be transferred from one

to another to reach his ﬁnal destination. Thus, we

distinguish four variants, namely, single-driver and

single-rider (Geisberger et al., 2010), (Amey, 2011),

single-driver and multiple-riders (Baldacci et al.,

2004), (Herbawi and Weber, 2012), multiple-drivers

and single-rider (Drews and Luxen, 2013), and ﬁnally

multiple-drivers and multiple-riders (Herbawi and

Weber, 2012). In each variant, the matching between

riders and drivers depends on one or more objective

functions, such as the maximization of the number of

matchings, the maximization of the cost saving, the

maximization of distance saving, etc.

The ridesharing problem with intermediate lo-

cations can be seen as an extension of the slugging

problem and the classical ridesharing problem.

Firstly, the driver’s route can change to accommodate

riders compared to slugging problem. Secondly, in

contrast to the classical ridesharing problem, the

pick-up and drop-off locations are not necessarily the

origins and the destinations of rider, respectively.

The problem of ridesharing with intermediate lo-

cations is addressed in (Aissat and Oulamara, 2014b),

(Bit-Monnot et al., 2013) and (Aissat and Oulamara,

2014a). In (Aissat and Oulamara, 2014b), the authors

consider the round trip ridesharing problem with an

intermediate meeting location. The rider drives to the

intermediate meeting location using his private car

and parks it there, so in the return trip, he must be

dropped off at that location to get his car back. Thus,

for a given demand, the optimization system deter-

mines the best meeting location, the best driver in out-

going trip and the best driver in return trip passing via

the intermediate meeting location where the rider’s

car was left. Authors develop an efﬁcient approach

where the objective is to minimize the total cost in

round trip. Their approach was validated by exper-

iments based on real data of ridesharing. The main

advantage of their approach is increasing the oppor-

tunity of matching between riders-drivers and then a

signiﬁcant reduction of the total travel cost compared

to the classical approach of round trip ridesharing.

In (Bit-Monnot et al., 2013), the authors consider

the problem of ridesharing with intermediate loca-

tions, where the rider can use the public transportation

either in order to reach the meeting location from his

starting location, or to reach his ending location com-

ing from the separate location. The authors propose

an optimal method to ﬁnd the pick-up and drop-off

locations in O(m·n

), where n is the number of nodes

and m is the number of edges in the graph, in which

the objective is to minimize the cumulated travel time

for both the driver and the rider. However, the time

complexity of this method prevents its use in real-

time ridesharing, and their model does not take into

account the detour time constraint (i.e., the total time

of the detour should be less than a given value ﬁxed by

the driver (rider)) and detour cost constraint (i.e., the

incurred cost of the driver (rider) using ridesharing

mode is more attractive than the incurred cost when

they travel alone).

In this paper, we consider the problem of rideshar-

ing with intermediate locations. The objective func-

tion is to minimize the total travel cost in scenario in-

volving transportation modes with time-independent

arc costs, while ensuring that their detour costs and

times remain reasonable. Thus, for a given rider, we

determine the best driver, the best meeting and sepa-

rate locations that minimize the total travel cost under

constraints of time and cost detour of rider and driver.

We suggest some heuristic methods that reduce the

number of shortest paths computations, based on the

APosterioriApproachofReal-timeRidesharingProblemwithIntermediateLocations

exact pruning algorithm.

The remainder of the paper is structured as fol-

lows. Section 3 details the model of ridesharing with

two intermediate locations. Section 4 proposes two

solving approaches. Section 5 introduces an exact

method and an offer selection heuristic. Section 6

generalizes our approach by introducing some param-

eters. Section 7 analyses the performance of our al-

gorithms. Finally, concluding remarks and future re-

search are included in Section 8.

3 PROBLEM DESCRIPTION

A road network is represented by a weighted graph

G = (V, E), where V is the set of nodes, E the set

of edges. Nodes model intersections and edges de-

pict street segments. With each edge (i, j) ∈ E two

weights are associated. c

(i, j) depicts the traveling

cost and τ

(i, j) depicts the traveling time. A path in

a graph G is represented by a vector µ = (u, . . . , v) of

nodes in which two successive nodes are connected

by an edge of E. The cost c(µ) of path µ is given by

the sum of costs of all edges in µ. A shortest-path be-

tween a source node u and a target node v is a path

with minimal cost among all paths from u to v. In the

following, a shortest-path between node u and node v

will be represented by u → v.

We consider an offer and a demand of rideshar-

ing represented by o = (s,t, [t

min

max

], ∆

) and d =

, [t

min

max

], ∆

), respectively, where s and s

and t

) are the starting (ending) locations of the driver

and the rider, respectively, [t

min

max

] ([t

min

max

]) is

the departure time window of the driver (rider), ∆

(∆

) is the maximal detour time that can be accepted

by the driver (rider), i.e. the extra-time that the driver

(rider) can accept additionally to the travel time cor-

responding to his shortest path.

An edge (i, j) of G has a nonnegative traveling

cost c

(i, j) depending on the fact that the edge is used

by a driver (k = o) or by a rider (k = d).

A basic way to implement the ridesharing (also

called classical ridesharing) is the case where the

driver picks up the rider at his starting location, drops

him off at his ending location and the driver continues

to his target location. Thus, an offer o is matched with

a demand d, if and only if there exists a shortest-path

= s →s

→t

→t in graph G, such that the driver’s

detour cost (c

(γ

)−c

(s →t)), remains reasonable,

i.e.,

(γ

) −c

(s →t) ≤ε ·c

→t

) (1)

where ε in (1) is a detour factor (Geisberger et al.,

2010). The term ε ·c

→ t

) is the reward that

rider provides to driver. Speciﬁcally, it represents the

maximal detour cost that still guarantee an incentive

to the driver to pick-up the rider at his starting

location s

and to drop him off at his ending location

. Clearly, if c

(γ

) −c

(s → t) > ε ·c

→ t

the driver does not accept the demand d. Authors in

(Geisberger et al., 2010) show that the reasonable

choice of ε is at most 0.5.

However, in order to get more opportunities of

ridesharing, a rider can accept two different inter-

mediate locations, one for pick-up and another for

the drop-off. Especially, the rider travels by his

own to the ﬁrst intermediate location r

with a cost

→ r

), where he will be picked up by the driver

and dropped off at the second intermediate location

, then continue his journey from r

to t

on his own.

Thus, the driver and the rider will share the common

travel cost c

→ r

). If the rider rewards the driver

with the amount ε ·c

→ r

), then the cost of rider

is c

→ r

) + ε ·c

→ r

) + c

→t

Depending on how the value of ε is set, we dis-

tinguish two cases, (i) A priori approach in which

the user (driver and/or rider) ﬁxe the detour factor ε,

for instance, the rider wants to share equitably the

common path cost with the driver, i.e. ε = 0.5, (ii) A

posteriori approach in which the user (driver/rider)

does not set the detour factor ε in advance, but the

system calculates the most attractive value of ε that

guarantees the best matching between the rider and

the driver.

The a priori approach is adressed in (Aissat and

Oulamara, 2014a). Given a ﬁxed value of ε, the

authors developed efﬁcient heuristics that determine

the best matching between the driver and the rider,

while ensuring that the detour cost of the rider and the

driver remains reasonable. Their objective function is

to minimize the total traveling cost.

In this paper, we address the a posteriori ap-

proach of a ridesharing with intermediate locations,

while considering additional constraints, namely, the

time windows, detour time and the desired minimum

cost-saving of users. More formally, we consider the

general problem of ridesharing with two intermediate

locations r

and r

in which driver picks up the rider

at the intermediate pick-up location r

and drops him

off at the intermediate drop-off location r

. The rider

travels on his own from s

to r

and from r

to t

(see

Figure 2).

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

Figure 2: The solid lines symbolize the path of the driver

and the rider in order to form a joint trip, whereas, the

dashed ones stand for the shortest paths of the driver and

the rider.

The objective is to minimize the total travel cost,

while ensuring the cost and time detour constraint for

the driver and the rider.

3.1 Matching Constraints

In our problem, a matching between a driver and a

rider can be established only under timing constraint

of the ride (Deﬁnition 1), and the travel cost constraint

(Deﬁnition 2).

Deﬁnition. 1. (Time Synchronization)

We say that an offer o and a demand d form a time

synchronization at location v if and only if β ≥ 0,

where

β=min

(

max

+ τ

(s → v) −(t

min

+ τ

→ v)) (2a)

max

+ τ

→ v) −(t

min

+ τ

(s → v)) (2b)

Equation (2a) guarantees that if the rider leaves

his origin location s at t

min

to reach the intermediate

pick-up location v, he must arrive no later than

the latest arrival time of the driver at v. The same

argument is applied to the driver in equation (2b).

Thus, if β ≥ 0, the propagation of the departure

time windows of the driver and the rider at the

intermediate pick-up location v will coincide. So,

they can meet each other at the earliest at this location

at time max



min

+ τ

→ v),t

min

+ τ

(s → v)



Deﬁnition. 2. (Reasonable Fit)

We say that an offer o = (s, t, [t

min

max

], ∆

) and a

demand d = (s

, [t

min

max

], ∆

) form a reasonable

ﬁt if and only if there exist two intermediate locations

and r

6= r

) such that, o and d form a time

synchronization at location r

and

(s →t) + c

→t

) −(c

(s → r

) +

→ r

) + c

→ r

) + c

→t) +

→t

)) ≥ 0 (3)

(s → r

) + τ

→ r

→t) ≤ τ

(s →t) + ∆

(4)

→ r

) + τ

→ r

→t

) ≤ τ

→t

) + ∆

(5)

The inequality (3) represents the cost-saving of

matching the offer o with the demand d. The cost-

saving is deﬁned as the difference between the travel

cost of the driver and the rider when each of them

travels alone without ridesharing and the total travel

cost of the driver and the rider with a shared path be-

tween r

and r

The term (τ

(s → t) + ∆

) in (4) (resp. τ

→

) + ∆

in (5)) allows to limit the amount of time that

the driver (resp. rider) passes in traveling. We ignore

any extra-time during pick-up or drop-off of riders.

In the following lemma, we show that if the gener-

ated cost-saving is positive, then it is always possible

to allocate the cost-saving among the driver and the

rider so that each of them has an individual beneﬁt.

Lemma. 1. If an offer o = (s, t, [t

min

max

], ∆

) and a

demand d = (s

, [t

min

max

], ∆

) form a reasonable

ﬁt, then there exists a detour factor ε ∈[0, 1] such that

the gains of the rider and the driver are positive.

Proof. Assume that an offer o and a demand d form

a reasonable ﬁt. From deﬁnition (2), there exist two

intermediate locations r

and r

, such that o and d

form a time synchronization at location r

and

(s →t) + c

→t

) −(c

(s → r

) +

→ r

) + c

→ r

) + c

→t) +

→t

)) ≥ 0

Assume that there exists a detour factor ε ∈ [0, 1] in

which the common trip cost of the driver and the rider

is shared according to ε, i.e., the rider rewards the

driver with amount ε ·c

→ r

On the one hand, the driver can accept to share a ride

with the rider, only if his total cost using ridesharing

modality is less than his cost if he travels alone, i.e.,

(s → r

) + (1 −ε) ·c

→ r

) + c

→t)

≤ c

(s →t) (6)

From (6) we obtain, ε ≥ ε

, where

(s → r

) + c

→ r

) + c

→t) −c

(s →t)

→ r

)

On the other hand, the rider accepts to be picked up

and dropped off by the driver in intermediate loca-

tions, only if his total cost using ridesharing modality

is less than his cost if he travels alone, i.e.,

→ r

) + ε ·c

→ r

) + c

→t

)

≤ c

→t

) (7)

From (7) we obtain, ε ≤ ε

, where

→t

) −c

→ r

) −c

→t

)

→ r

)

APosterioriApproachofReal-timeRidesharingProblemwithIntermediateLocations

Furthermore, it is easy to see that the existence of the

solution is constrained by ε

≤ ε ≤ε

In order to satisfy the constraint (3), the ε must be

in the interval [ε

, ε

]. Thus, any value of ε in the

interval [ε

, ε

] satisﬁes the constraints of matching

between the rider and the driver. A reasonable value

of ε might be the average value of the interval [ε

, ε

i.e. ε =

+ε

3.2 Objective of Ridesharing System

As in (Aissat and Oulamara, 2014a), we use the term

global-path (s, s

, r

,t, t

) to describe the concate-

nation of paths s → r

, s

→ r

, r

→ r

, r

→ t

and r

→ t

. i.e. (s, s

, r

,t, t

) = (s → r

⊕s

→

⊕r

→ r

⊕r

→ t ⊕r

→ t

). A shortest global-

path between source nodes s, s

and target nodes t, t

is a global-path with minimal cost c(s, s

, r

,t, t

)

among any global-path from s, s

to t, t

, where

c(s, s

, r

,t, t

) = c

(s → r

) + c

→ r

)

→ r

) + c

→t)

→t

) (8)

Thus, the objective of the ridesharing system is to de-

termine the best intermediate locations r

and r

that

minimize the shortest global-path such that the offer

o and a demand d form a reasonable ﬁt.

4 PROPOSED APPROACH

The proposed methods are based on computing sev-

eral shortest paths. Shortest path algorithms have

been extensively studied, since they are used in almost

all real cases of transportation problems. Efﬁcient

algorithms are especially needed when geographical

information systems (GIS) are involved. The well-

known algorithm to ﬁnd the shortest path is the Di-

jkstra algorithm (Dijkstra, 1959). In (Sanders and

Schultes, 2007), the authors note that the Dijkstra al-

gorithm can in theory be used to ﬁnd the shortest path

on a road network, but for large networks it would be

far too slow. Different speed-up techniques, such as

bidirectional search, goal direction, etc., can be used

to improve performance. An overview of such tech-

niques is available in (Sanders and Schultes, 2007).

4.1 Search Space of Potential

Intermediate Locations

The matching constraints deﬁned in section 3.1 allows

us to limit the search space of intermediate locations.

In the following, we provide characteristics of poten-

tial intermediate locations.

Deﬁnition. 3. (Potential Intermediate Nodes).

Let N

↑

(s), N

↑

), N

↓

(t) and N

↓

) be deﬁned as

↑

(s) = {v ∈V | c

(s → v) ≤ c

(s →t)

∧ τ

(s → v) + τ

(v →t) ≤τ

(s →t) + ∆

}

↓

(t) = {v ∈V | c

(v →t) ≤c

(s →t)

∧ τ

(s → v) + τ

(v →t) ≤τ

(s →t) + ∆

}

↑

) = {v ∈V |c

→ v) ≤ c

→t

)

∧ τ

→ v) +

τ(v →t

) ≤ τ

→t

) + ∆

}

↓

) = {v ∈V | c

(v →t

) ≤ c

→t

)

∧

τ(s

→ v) + τ

(v →t

) ≤ τ

→t

) + ∆

}

↑

(s) represents the set of potential pick-up loca-

tions of the driver. Indeed, if v is the pick-up location,

the best situation for the driver is to share the cost

(v →t) with the rider as long as v satisﬁes the trav-

eling time constraint τ

(s → v) + τ

(v →t) ≤ τ

(s →

t) + ∆

. Thus, if a node v does not respect the cost

and the traveling time constraints, then v cannot be an

pick-up location. Set N

↓

(t) represents the potential

drop-off locations of the driver. The same reasoning

as for the set N

↑

(s) is applied to N

↓

(t).

On the other hand, the sets N

↑

) and N

↓

) rep-

resent the potential pick-up and drop-off locations of

the rider, respectively.

Remark that in the traveling time constraint of set

↑

) (resp. N

↓

)), we use

τ(v →t

) (resp.

τ(s

→

v)), i.e., the traveling time corresponding to the esti-

mated distance between two locations v and t

(resp. s

and v) using Haversine formula, instead of the short-

est traveling time τ

(v →t

) (resp. τ

→ v)).

In fact, for a given node v, if τ

→v)+τ

(v →t

) >

→ t

) + ∆

, we can’t deduce that v cannot be a

pick-up location. This is due to the fact that there may

be another node v

, where v

is a potential drop-off lo-

cation and t

(v → v

) +t

→t

) < τ

(v →t

The Haversine formula estimates the distance be-

tween two locations based on their latitude/longitude.

This formula uses a spherical model to estimate the

distance between two points on the earth surface.

Speciﬁcally, given two locations x = (θ

, λ

) and

y = (θ

, λ

) where θ

, i = 1, 2, is the latitude and λ

i = 1, 2, is the longitude. The distance between x

and y is given by

d(x, y) = 2 ×R ×arcsin(

√

a) where

R ≈6371(km) is the earth’s radius in kilometers, and

a = sin

(

−θ

) + cos(θ

) × cos(θ

) × sin

(

−λ

Thus, the estimated smallest duration from x to y is

noted by

τ(x → y) =

d(x,y)

max

, such that v

max

is the max-

imal speed among all used modality of the driver and

the rider.

The following lemma simultaneously character-

izes the set of potential intermediate locations for both

driver and rider.

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

Lemma. 2. A node v ∈V is a potential intermediate

node if and only if v ∈C = C

∪C

where C

= N

↑

(s)∩

↑

) and C

= N

↓

(t) ∩N

↓

4.2 Solving Methods

In this section, we propose two solving methods for

our model. The ﬁrst method is based on bidirectional

search, and the second method is based on one-to-all

shortest path. Both methods are based on the modiﬁ-

cation of the graph G.

Once the potential intermediate locations class

C has been determined, we add to the graph G two

dummy nodes S

and T

. The node S

is connected

to each node v, v ∈C

with an arc (S

, v) and the cost

c(S

, v) = c

(s → v) + c

→ v). The node T

connected to each node v, v ∈ C

with an arc (v, T

)

and the cost c(v, T

) = c

(v →t) + c

(v → t

). The

arc (S

, v) represents the driver and the rider moves

out from their starting locations to the pick-up node

v. The arc (v, T

) represents the driver and the rider

moves out from the drop-off node v to their ending

locations. In the following, we describe our methods.

Bidirectional Search Algorithm - BSA. The

BSA Algorithm works with two simultaneous

searches of the shortest path, one from the source

node S

and the second from the target node T

the same time, until ”the two search frontiers meet”.

More precisely, BSA Algorithm maintains two

priority queues, one for the search from the source

denoted by Q

, as in simple Dijkstra Algorithm,

and one for the backward search from the target

denoted by Q

, which is a forward search in the

reversed graph G

−1

, in which each arc (u, v) of G is

replaced by (v, u) in G

−1

In each iteration, we settle the node with the smallest

overall tentative cost, either from the source S

or to

the target T

. In order to do this, a simple comparison

of the minima of the two priority queues Q

and Q

sufﬁces. Once we settle a node v in one queue that

is already settled in the other queue, we get the ﬁrst

tentative cost of a shortest path from S

to T

. Its cost

is the cost of the path found by the forward search

from S

to v, plus the cost of the path found by the

backward search from v to T

Even when the two parts of the tentative solution of

the forward and the backward meet each other, the

concatenated solution path is not necessarily optimal.

To guarantee optimality, we must continue until the

tentative cost of the current minima of the queues is

above the current tentative shortest path cost (which

then corresponds to the cost of the shortest path).

Thus, at each iteration when the tentative cost of a

shortest path from S

to T

is updated, we check

the feasibility of constraints (3), (4) and (5), and

ﬁnally we keep the admissible path that minimizes

the global-path.

In the BSA Algorithm, when a node v is settled,

then for each outgoing arc (v, u) from the node v, we

check whether via this arc the node u can be reached

with a cost less than or equal to the current cost of

u. If yes, then the cost of u is updated to the new

lower cost. This procedure is called relaxing an arc.

The difference with the traditional relaxing an arc

procedure of the literature is simply on updating the

cost of the node even if the new cost is equal to the

current cost of the node. Indeed, this modiﬁcation

allows us to ensure that if the shortest path from

to T

contains only one intermediate node i.e.,

arc

, then the driver and rider will never

be able to form a reasonable ﬁt. Speciﬁcally, the

constraint (3) will never be satisﬁed.

Finally, in the found shortest path S

→ T

(i.e.

arc

→ r

arc

), we recover the best pick-up

location r

and the best drop-off location r

that

correspond to the successor of S

and the predecessor

of T

, respectively.

Shortest Path One-to-All - SPOA. In this method,

instead of computing one shortest path from S

to T

we compute for each node v in C

the shortest path

→ v. This approach allows us to ﬁnd all paths that

satisfy constraint (3). Once shortest paths satisfying

the constraint of cost-saving have been enumerated,

we select the global-path with minimal cost that

satisﬁes constraints (4) and (5).

The main advantage of the ﬁrst method (BSA) lies

in the fact that its running-time is less important than

the second method (SPOA), which requires to com-

pute all shortest paths to each node in class C

. On

the other hand, the second method has a greater con-

trol over the constraints of time detour for the driver

and the rider. We note that the relaxing arc procedure

applied in this method is the same as that described

above.

5 THE BEST MATCHING

SELECTION

The approach developed above and in (Aissat and

Oulamara, 2014a) correspond to the case where an

offer and a demand are already ﬁxed. However, in

ridesharing system, several demands and offers arrive

in the system. Thus, in this section, we consider the

APosterioriApproachofReal-timeRidesharingProblemwithIntermediateLocations

problem of the best driver selection. Especially, given

a set of offers already in the system, when a demand

request d arrives in the system, the objective is to se-

lect the offer candidate which is able to fulﬁll the ride

request.

Given a demand request d, we start by determin-

ing the two sets N

↑

) and N

↓

) of potential inter-

mediate pick-up and drop-off locations, respectively.

(see Deﬁnition 3).

The selection rule of the best offer requires the

storage of some informations for each offer: when

an offer o

is added to the system, we store all

possible intermediate locations for this offer. Thus,

we determine the costs c

→ v) and c

(v → t

The procedure of storage is detailed below.

Informations Storage: We denote by M

↑

) and

↓

) the f orward search space from a source s

and the backward search space from target t

, re-

spectively. A forward search space M

↑

) is a set

of triplets (v, c

↑

, τ

↑

), where (v, c

↑

, τ

↑

) means that the

shortest path from s

to v has a cost c

↑

and a travel

time τ

↑

. A backward search space M

↓

) is a set

of triplets (v, c

↓

, τ

↓

) where (v, c

↓

, τ

↓

) means that the

shortest path from v to t

has a cost c

↓

and a travel

time τ

↓

Using detour time constraint (4) of the driver, we

limit the search space. Then the triplets (node, cost,

time) are stored in M

↑

) and M

↓

) without consid-

ering any demand.

In order to determine the sets M

↓

) and M

↑

we use a modiﬁed A

algorithm (Hart et al., 1968).

Firstly, we determine the set M

↓

) using reverse A

algorithm from the destination t

toward the origin lo-

cation s

. Then, we determine the set M

↑

) using

algorithm from the origin s

toward the destina-

tion t

. Recall that in A

algorithm, the nodes are re-

trieved from the priority queue by the sum of their ten-

tative cost from the origin to the nodes and the value

of the heuristic function from the nodes to the des-

tination. Thus, for a given node v, the cost function

used in the ﬁrst A

Algorithm to determine M

↓

) is

τ(s

→v)+τ

(v →t

), whereas in the second A

Algo-

rithm that determines M

↑

), we use the cost function

→ v) + τ

(v →t

Remark that in the second A

algorithm, we use

(v →t

) instead of the estimation duration

τ(v →t

since it is already calculated in the ﬁrst A

algorithm.

We note that the traditional A

algorithm stops after

the destination has been reached. However, our A

algorithm used to determine M

↓

) (resp. M

↑

))

continues even when the destination s

(resp. t

) is

reached. The algorithm stops once all nodes with cost

function less than τ

→t

) + ∆

are labeled.

Furthermore, we deﬁne the bucket B

(v) as the set

which stores all drivers’ trips which can pass via this

location without violating the constraint of time de-

tour. Then, in the second algorithm A

, when a node

v is labeled, we add the entries in

(v) := B

(v) ∪{(i, c

↑

, τ

↑

, c

↓

, τ

↓

)}. (9)

5.1 Exact Offer Selection

Let σ

be the cost of the optimal global-path, ini-

tially σ

= +∞. After constructing the sets N

↑

)

and N

↓

), we scan the nodes of N

↑

). For each

potential intermediate location v

in the set N

↑

we calculate c

→ v

) using the forward one-to-

all Dijkstra algorithm for all v

∈ N

↓

). For each

offer o

such that (i, d

↑

, τ

↑

, d

↓

, τ

↓

) ∈ B

), we scan

each node v

in the set N

↓

). Then, if i is in the set

) such that o

and d form a reasonable ﬁt with

) as an intermediate pick-up (drop-off) location,

and the cost of the global path c(s, s

, v

,t, t

) is less

than σ

, then we update the optimal global-path σ

c(s, s

, v

,t, t

). The detail of the procedure is given

in Algorithm 1.

Algorithm 1: Exact offer selection.

Require: Demand d, sets N

↑

), N

↓

), and B

(v), ∀v ∈G.

Ensure: The best driver i

, global-path σ

1: Initialization, i

← −1, σ

← +∞.

2: for all v

in N

↑

) do

3: Using the forward one-to-all Dijkstra Algorithm,

compute the cost c

→ v

), ∀ v

∈ N

↓

4: for all i in B

) do

5: if i and d form a time synchronization at location

then

6: for all v

in N

↓

) do

7: if i ∈B

) and driver i and rider d form a

reasonable ﬁt with v

as intermediate pick-

up location and v

as intermediate drop-off

location then

8: if σ

> c(s

, s

, v

) then

9: i

← i

10: σ

← c(s

, s

, v

)

11: end if

12: end if

13: end for

14: end if

15: end for

16: end for

Note that the runtime of Dijkstra’s Algorithm us-

ing Fibonacci Heaps is bounded by O(|V |log |V |+

|E|). So, the two sets N

↑

), N

↓

) can be computed

in O(2 ·(|V |log |V |+ |E|)). Furthermore, Algorithm

1 runs in O(|N

↑

)|(|V |log|V | + |E|) + |N

↓

)| ·

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

∑

∈N

↑

)

)|)-time. Hence, the worst-case com-

plexity of the Exact offer selection is O((|N

↑

)|+

2)(|V |log|V | + |E|) + |N

↓

)| ·

∑

∈N

↑

)

)|)-

time.

5.2 Offer Selection Heuristic

Even if the exact offer selection method provides a so-

lution in polynomial time, the computation time can

be too long for large instances with real road net-

work. In particular, when the proposed method is

used in real time ridesharing system, in which a so-

lution should be provided in a few seconds. In this

section, we propose an heuristic method with a lower

time complexity, that allows us to select the best offer

that will be matched with the demand d.

Algorithm 2: Scan the set N

↑

Require:Demand d, set N

↑

), set N

↓

) and B

(v),∀v ∈G.

Ensure: The best driver i

, global-path σ

1: Initialization, i

← −1, σ

← ∞.

2: Using backward one-to-all Dijkstra Algorithm, com-

pute the cost c

→t

), ∀ v

∈ N

↑

3: for all v

in N

↑

) do

4: for all i in B

) do

5: if i and d form a time synchronization at location

then

6: if i ∈ B

) and τ

→ v

) + τ

→ t

) ≤

(τ

→t

) + ∆

) and τ

→ v

) + τ

→

) + τ

→ t

) ≤ (τ

→ t

) + ∆

) and

→t

)+ c

→t

)−c(s

, s

, v

) ≥

0 then

7: if σ

> c(s

, s

, v

) then

8: i

← i

9: σ

← c(s

, s

, v

)

10: end if

11: end if

12: if τ

→ v

) + τ

→ t

) + τ

→ t

) ≤

(τ

→t

) + ∆

) and τ

→ v

) + τ

→

) ≤ (τ

→ t

) + ∆

) and c

→ t

) +

→t

) −c(s

, s

, v

) ≥ 0 then

13: if σ

> c(s

, s

, v

) then

14: i

← i

15: σ

← c(s

, s

, v

)

16: end if

17: end if

18: end if

19: end for

20: end for

The offer selection heuristic is provided by Algo-

rithm 4 composed of Algorithms 2 and 3.

Algorithm 2 allows us to determine an offer that min-

imizes the global-path, by considering the nodes of

the set N

↑

) as potential pick-up locations, while the

drop-off location is ﬁxed at the rider (and driver) des-

tination. Steps 6 and 12 of Algorithm 2 check the ad-

Algorithm 3: Scan the set N

↓

Require:Demand d, set N

↑

), set N

↓

) and B

(v),∀v ∈G.

Ensure: The best driver i

, global-path σ

1: Initialization, i

← −1, σ

← ∞.

2: Using forward one-to-all Dijkstra algorithm, compute

the cost c

→ v

), ∀ v

∈ N

↓

3: for all v

in N

↓

) do

4: for all i in B

) do

5: if i and d form a time synchronization at location

then

6: if i ∈ B

) and τ

→ v

) + τ

→t

) ≤

(τ

→ t

) + ∆

) and τ

→ s

) + τ

→

) + τ

→ t

) ≤ (τ

→ t

) + ∆

) and

→t

) + c

→t

) −c(s

, s

, v

,t) ≥

0 then

7: if σ

> c(s

, s

, v

) then

8: i

← i

9: σ

← c(s

, s

, v

)

10: end if

11: end if

12: end if

13: if i and d form a time synchronization at location

then

14: if τ

→ s

) + τ

→ v

) + (τ

→ t

) ≤

(τ

→t

) + ∆

) and τ

→ v

) + τ

→

) ≤ (τ

→ t

) + ∆

) and c

→ t

) +

→t

) −c(s

, s

, v

) ≥ 0 then

15: if σ

> c(s

, s

, v

) then

16: i

← i

17: σ

← c(s

, s

, v

)

18: end if

19: end if

20: end if

21: end for

22: end for

missibility of the found path based on the maximum

detour time allowed by the driver and the rider, as well

as the generated cost-saving.

On the other hand, Algorithm 3 determines the best

offer that minimizes the global-path, by considering

the nodes of the set N

↓

) as potential intermediate

drop-off locations, while the pick-up location of the

ride is ﬁxed to the origin location of the driver (and

rider). Finally, the best offer i

∗

is given by the best

result of Algorithms 2 and 3. In order to improve the

cost of the global-path of matching (i

∗

, d), we use the

method described in section 4.2 to ﬁnd the best in-

termediate locations. The detail of the procedure is

given in Algorithm 4.

Algorithm 2 and Algorithm 3 run in respectively,

O(|V |log|V | + |E| +

∑

∈N

↑

)

)|)-time and

O(|V |log|V | + |E| +

∑

∈N

↓

)

)|)-time, re-

spectively. Then, having the sets N

↑

), N

↓

) and

the best offer i

, the step 4 in the Algorithm 4 can be

APosterioriApproachofReal-timeRidesharingProblemwithIntermediateLocations

Algorithm 4: Offer selection heuristic.

Require:Demand d, set N

↑

), set N

↓

) and B

(v),∀v ∈G.

Ensure: The best driver i

, global-path σ

1: Find the best driver i

in N

↑

) using Algorithm 2.

2: Find the best driver i

in N

↓

) using Algorithm 3.

3: Select between the two drivers i

and i

the driver i

with shortest global-path σ

= min(σ

, σ

4: Find the best intermediate locations r

and r

between

driver i

and demand d using method described in sec-

tion 4.2.

done in O((|V |log|V| + |E|) + |N

↑

)| + |N

↓

)|)-

time. Finally, the offer selection heuristic

method allows us to reduce the complexity to

O(5 · (|V |log |V | + |E|) + |N

↑

)| + |N

↓

)| +

∑

∈N

↑

)

)|+

∑

∈N

↓

)

)|).

6 REWARD CONSIDERATION

In our a posteriori approach, the user (driver and/or

rider) does not set the detour factor ε in advance,

and the system calculates the most attractive value

of ε that guarantee the best matching between the

rider and the driver. However in some cases, ei-

ther the driver or the rider may ask for a minimum

rate of the saved cost in his trip. For instance, the

driver requires at least 10% of the saved cost rela-

tive to his initial travel cost. In that case, we ex-

tend model to take into account that requirement.

More precisely, an offer and a demand of ridesharing

will be represented by o = (s,t, [t

min

max

], ∆

, σ

) and

d = (s

, [t

min

max

], ∆

, σ

), respectively, where σ

(σ

) is the minimum percentage of cost-saving ﬁxed

by the driver (rider) relative to his shortest path. Con-

straint 3 of the deﬁnition 2 is modiﬁed as follow.

(s →t) + c

→t

) −(c

(s → r

) +

→ r

) + c

→ r

) + c

→t) +

→t

)) ≥ σ

·c

(s →t) + σ

·c

→t

) (10)

The values of ε

and ε

of lemma 2 are modiﬁed in

order to consider a constraint (10).

7 COMPUTATIONAL RESULTS

In this section, we provide experimental results us-

ing the proposed methods, i.e, the two heuristics BSA

and SPOA, the exact method (EM) and the classical

ridesharing (CR).

Environment. The proposed methods were imple-

mented in C# Visual Studio 2010. The experiments

were done on an Intel Xeon E5620 2.4 Ghz processor,

with 8 GB RAM memory.

Offers-demands Data. In our experiments, we

use real data provided by Covivo company

. These

data concern employees of Lorraine region trav-

eling between their homes and their work places.

The real data instance is composed of 1513 partici-

pants. Among them, 756 are willing to participate in

ridesharing service only as drivers. The rest of par-

ticipants (i.e, 757) are willing to use other modes in

order to reach the intermediate pick-up location from

their origins, as well as from the intermediate drop-off

location to their destinations. However, given the lack

of available data of others transportation modes in the

French region Lorraine, we assume that the driver and

the rider have the same cost (i.e, each rider owns a

private car that he might use during a part of his trip,

either in order to reach the intermediate pick-up loca-

tion coming from his origin, or/and reaching his des-

tination coming from the intermediate drop-off loca-

tion). Thus, the data instance is composed of 756 of-

fers and 757 demands. The smallest and the greatest

trip’s distances are 2 km and 130 km, respectively.

The set of offers and demands are ﬁltered such

that we can never ﬁnd a driver’s offer and a rider’s

demand which have either the same starting or ending

locations. This way will allow us to test the ﬂexibility

of our approach compared to the classical ridesharing

approach. The time window for each trip is ﬁxed as

follows. The early departure time and the latest de-

parture time are ﬁxed at 7:30 a.m. and at 8:00 a.m,

respectively. The time detour of the driver (rider) is

ﬁxed to at most 20% of his initial trip duration.

Road Network. Several geographical maps are

available for geolocalised applications. The most fa-

mous ones are Google Maps, Bing Maps, Nokia Maps

and OpenStreetMap

(OSM). Our road network of the

French region Lorraine was derived from the publicly

available data of OpenStreetMap (OSM) and was pro-

vided by GeoFabrik

. It consists of 797830 nodes

and 2 394 002 directed edges. Each node in the road

network can be an intermediate pick-up (resp. drop-

off) location in which the driver and the rider can meet

(resp. separate). OSM is used in several projects and

it facilitates the map integration or exploitation. For

instance, OsmSharp is an open-source mapping tool

designed to work with OpenStreetMap-data. In our

experiments, we use OsmSharp’s routing and OSM

data processing library to test our shortest path com-

putation on real data set.

http://www.covivo.fr

http://www.openstreetmap.org

http://www.geofabrik.de/

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

Figure 3: Map visualization of offers of ridesharing. Red

and blue points are the geocoded home and work locations,

respectively.

Computational Results. Two computational ex-

periments were conducted. In the ﬁrst experiment,

a demand and an offer of ridesharing were ﬁxed, and

we evaluate the performance of four methods (BSA,

SPOA, EM and CR) in terms of cost-saving, number

of matchings and running-time (see Table 1). In this

ﬁrst experiment, two scenarios were tested. In the ﬁrst

scenario, the driver’s detour time was limited to 20%,

and in the second scenario, the time detour is limited

to 10%. In each scenario, several instances were gen-

erated as follow: we scan all demands and for each

demand we randomly select 10 offers, then 7570 in-

stances (757 (demands) × 10 (offers)) were tested.

For each method, the following results are reported:

Gap: the deviation of algorithms with respect to the

optimal solution, Time: the required CPU time in sec-

onds, Match(M ): the percentage of number matching

found. In Columns Gap and Time, the average values

on the 7570 instances are reported. In the row (|C |),

the average number of nodes contained in potential in-

termediate locations (class C ) over the 7570 instances

is reported.

As we can see, the proposed heuristics (BSA and

SPOA) detect all matchings and provide exact re-

sults for all instances when the detour time is less

than 20%. Whereas, when the detour time is less

than 10%, the Match(M ) of BSA method decreases

to 94% and SPOA method decreases to 97%. On the

other hand, the CR approach detects at most 55.7%

when the detour time is less than 20% and decreases

Table 1: Performance of our algorithm with ﬁxed offer.

Detour time ≤ 0.2 Detour time ≤ 0.1

Gap M Time Gap M Time

(%) (%) (s) (%) (%) (s)

EM - - 844 - - 620

BSA 0 100 1.78 0.4 94 1.22

SPOA 0 100 2.10 0.2 97 1.58

CR 27 55.7 0.92 34 15 0.83

|C | 450 742

more sharply to 15% with detour time less than 10%.

The Gap of CR can be considered as the additional

cost-saving generated by using methods with interme-

diate locations. The running time in BSA method is

less than in SPOA method, this is due to the one-to-all

Dijkstra algorithm used in SPOA compared to BSA.

In the second experiment, we evaluate the heuris-

tic selection of offers. This allows us to know in prac-

tice, if the selected offer found using one intermediate

location (either pick-up or drop-off location) is gen-

erally the same selected offer found using two inter-

mediate locations (see Table 2). The column “Time”

represents the running-time of the whole algorithm,

whereas the row “Same offer (SA)” represents the

success rate in which the selected offer in the step 3

(Algorithm 4) is the same offer selected by the exact

method (EM).

Table 2: Performance of the whole process.

Detour time ≤ 0.2 Detour time ≤ 0.1

Gap SA Time Gap SA Time

(%) (%) (s) (%) (%) (s)

EM - - 9409 - - 7803

BSA 0.7 91 3.79 2.1 83 2.64

SPOA 0.3 91 5.22 1.4 83 3.87

∑

v∈C

|B(v)| 113836 84742

From Table 2, we can see that 83% (resp. 91%

when detour time ≤ 0.2) of selected offers found us-

ing one intermediate location are the same with se-

lected offers found using the two intermediate loca-

tions. Furthermore, when detour time ≤ 0.2, in BSA

(resp. SPOA) method, the solution is found in about

3 seconds (resp. 5 seconds) compared to 9409 sec-

onds for the exact method (EM). The Gap between

the two heuristics does not exceed 2.1% in both cases

of detour time values.

8 CONCLUSION

In this paper, we considered the problem of rideshar-

ing with intermediate locations. We proposed solu-

tion methods that allow us to determine the interme-

APosterioriApproachofReal-timeRidesharingProblemwithIntermediateLocations

diate pick-up and drop-off locations that minimize the

total travel cost of rider and driver. The particular in-

terest of our work is in making the service of rideshar-

ing more ﬂexible and efﬁcient. Our approach is sim-

ple to use and allows to reduce the driver’s detour by

using intermediate locations, and to increase the sav-

ings for both drivers and riders compared to the clas-

sical ridesharing.

REFERENCES

Agatz, N., Erera, A., Savelsbergh, M., and Wang, X. (2012).

Optimization for dynamic ride-sharing: A review. Eu-

ropean Journal of Operational Research, 223:295–

303.

Aissat, K. and Oulamara, A. (2014a). Dynamic ridesharing

with intermediate locations. In proceedings of SSCI

2014.

Aissat, K. and Oulamara, A. (2014b). Round trip rideshar-

ing with an intermediate meeting location. In proceed-

ings of MOSIM 2014.

Amey, A. (2011). Proposed methodology for estimating

rideshare viability within an organization: Applica-

tion to the mit community. In Transportation Research

Board Annual Meeting, pages pp. 11–25.

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

exact method for the car pooling problem based on

lagrangean column generation. Operations Research,

52:422–439.

Bit-Monnot, A., Artigues, C., Huguet, M.-J., and Killi-

jian, M.-O. (2013). Carpooling: the 2 synchronization

points shortest paths problem. In OASIcs-OpenAccess

Series in Informatics, volume 33. Schloss Dagstuhl-

Leibniz-Zentrum fuer Informatik.

Dijkstra, E. (1959). A note on two problems in connexion

with graphs. Numerische Mathematik, 1:269–271.

Drews, F. and Luxen, D. (2013). Multi-hop ride sharing. In

Proceedings of the Sixth Annual Symposium on Com-

binatorial Search, pages 71–79,.

Furuhata, M., Dessouky, M., Brunet, F. O. M., Wang, X.,

and Koenig, S. (2013). Ridesharing: The state-of-

the-art and future directions. Transportation Research

Part B: Methodological, 57:28–46.

Geisberger, R., Luxen, D., Neubauer, S., Sanders, P., and

Volker, L. (2010). Fast detour computation for ride

sharing. In 10th Workshop on Algorithmic Approaches

for Transportation Modelling, Optimization, and Sys-

tems - ATMOS’10, pages pp. 88–99. Th. Erlebach, M.

Lubbecke.

Hart, P., Nilsson, N., and Raphael, B. (1968). A formal

basis for the heuristic determination of minimum cost

paths. IEEE Trans. on Sys. Sci. and Cyb, 4:100–107.

Hartwig, S. and Buchmann, M. (2006). Empty seats trav-

eling: next-generation ridesharing and its potential to

mitigate trafﬁc-and emission problems in the 21st cen-

tury. NOKIA Research Center.

Herbawi, W. and Weber, M. (2012). The ridematching

problem with time windows in dynamic ridesharing:

a model and a genetic algorithm. In Proceedings

ACM Genetic and Evolutionary Computation Confer-

ence (GECCO), pages 1–8.

Ma, S. and Wolfson, O. (2013). Analysis and evaluation

of the slugging form of ridesharing. In Proceedings

of the 21st ACM SIGSPATIAL International Confer-

ence on Advances in Geographic Information Sys-

tems, pages 64–73. ACM.

Ma, S., Zheng, Y., and Wolfson, O. (2013). T-share: A

large-scale dynamic taxi ridesharing service. In Data

Engineering (ICDE), 2013 IEEE 29th International

Conference on, pages 410–421. IEEE.

Metz, B., Davidson, O., De Coninck, H., Loos, M., and

Meyer, L. (2005). Ipcc special report on carbon

dioxide capture and storage. prepared by working

group iii of the intergovernmental panel on climate

change. IPCC, Cambridge University Press: Cam-

bridge, United Kingdom and New York, USA, 4.

Mote, J. E. and Whitestone, Y. (2011). The social context

of informal commuting: Slugs, strangers and struc-

turation. Transportation Research Part A: Policy and

Practice, 45(4):258–268.

Sanders, P. and Schultes, D. (2007). Engineering fast route

planning algorithms. In Experimental Algorithms,

pages 23–36. Springer.

Sassi, O. and Oulamara, A. (2014). Joint scheduling and op-

timal charging of electric vehicles problem. In Com-

putational Science and Its Applications–ICCSA 2014,

pages 76–91. Springer.

Varone, S. and Janilionis, V. (2014). Insertion heuristic

for a dynamic dial-a-ride problem using geographical

maps. In proceedings of MOSIM 2014.

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