Fuzzy Multi-objective Optimization for Ride-sharing Autonomous

Mobility-on-Demand Systems

Rihab Khemiri

and Ernesto Exposito

Univ. Pau & Pays Adour, E2S UPPA, LIUPPA, EA3000, Anglet, 64600, France

Keywords: Ride-sharing Autonomous Mobility-on-Demand Systems, Multi-objective Possibilistic Linear Programming,

Fuzzy Logic, Goal Programming, Dispatching, Rebalancing.

Abstract: In this paper, we propose a novel three-phase fuzzy approach to optimize dispatching and rebalancing for

Ride-sharing Autonomous Mobility-on-Demand (RAMoD) systems, consisting of self-driving vehicles,

which provide on-demand transportation service, and allowing several customers to share the same vehicle at

the same time. We first introduce a new multi-objective possibilistic linear programming (MOPLP) model for

the problem of dispatching and rebalancing in RAMoD systems considering the imprecise nature of the

customer requests as well as two conflicting objectives simultaneously, namely, improving customer

satisfaction and minimizing transportation costs. Then, after transforming this possibilistic programming

model into an equivalent crisp multi-objective linear programming (MOLP) model, the Goal Programming

(GP) approach is used to provide an efficient compromise solution. Finally, computational results show the

practicality and tractability of the proposed model as well as the solution methodology.

1 INTRODUCTION

Nowadays, urban systems are characterised by the

expansion of cities and by the growth of their

population. This affects the current mobility trends

marked by the continued growth of demand for

personal mobility as well as the increasing of

privately owned automobile.

This trend leads to many social and environmental

severe problems including traffic congestion,

increased travel times, air pollution as well as the

growth of the greenhouse gas emissions, especially in

the densely populated areas with limited space for

parking and road infrastructure.

To deal with these problems, an efficient

transportation system that responds to the mobility

demands of people and that is more sustainable,

reliable and efficient becomes essential.

In this context, Autonomous Mobility-on-

Demand (AMoD) systems represent a very promising

solution in meeting these needs. This emerging

system is a fleet of self-driving electric vehicles

designed to provide personal on-demand

transportation service for passengers. AMoD systems

offer many potential benefits such as minimizing

pollution, avoiding the need for further routes and

parking spaces. Moreover, autonomous vehicles may

be safer than traditional vehicles as they can avoid

accidents due to human errors, well known to be the

main reason of traffic accidents.

These several advantages have led recently a

number of works to investigate the potential of

AMoD systems. A key challenge in this context is the

design of dispatching strategies that entail to

optimally assign the customers to vehicles, thus

satisfying the customer's request at each given station

and at each time period. To do this, the number of

vehicles available at each station and each period

must satisfy customer requests.

Nevertheless, when some stations are more in

demand than others, at the end of the trip, vehicles

will tend to be accumulated at these stations and

become exhausted at others. This can lead to a spatial-

temporal distribution of vehicles, which will probably

not be in line with the distribution of the customer

requests in the following periods.

Therefore, it becomes inevitable to devise

efficient policies to deal with this problem of

imbalance. Such rebalancing policies entail

redistributing empty vehicles from overload stations

to underloaded stations. However, AMoD systems

might aggravate the congestion problem given the

presence of these empty vehicles (Tsao et al., 2019).

284

Khemiri, R. and Exposito, E.

Fuzzy Multi-objective Optimization for Ride-sharing Autonomous Mobility-on-Demand Systems.

DOI: 10.5220/0009779602840294

In Proceedings of the 15th International Conference on Software Technologies (ICSOFT 2020), pages 284-294

ISBN: 978-989-758-443-5

This has prompted some AMoD systems to integrate

the emerging transportation paradigm of ride-sharing

to improve traffic flow.

Another important challenge is to deal well with

rapidly varying customer requests, given several real-

world constraints. Accordingly, it becomes

mandatory to forecast customer requests to compute

efficient strategies, having the robustness to

inaccuracies and uncertainties due to several external

factors such as traffic and weathers.

These various challenges have recently led to a

considerable amount of studies to address the

potential of AMoD systems. However, the majority

of these researches do not allow easy to account for

real-world phenomena such as the uncertain futures

of customer demand, which limits their practical

applications.

Although a few recent studies have been

developed to cope with demand uncertainty, the latter

are usually based on probability distributions, which

requires knowledge of historical data. When such

information was lacking, the Fuzzy Set Theory

(Zadeh, 1978) and the Possibility Theory (Dubois and

Prade, 1988; Zadeh, 1965) can help to handle

epistemic uncertainty.

To the best of our knowledge, fuzzy logic has

never been used to model the uncertainty in the

context of AMoD systems.

The aim of this paper is to present a novel fuzzy

approach for dispatching and rebalancing RAMoD

systems. Specifically, the paper has three important

contributions. First, it introduces a MOPLP model

which contemplates the uncertainty affecting future

demand. Two primary goals are considered

simultaneously in the MOPLP model, namely,

improving customer satisfaction and minimizing

transportation costs. Second, in order to find an

efficient compromise solution to the proposed

MOPLP model, we suggest the exploitation of the

well-known goal programming approach (Charnes

and Cooper, 1961), which integrates the desire of the

decision-maker with the logic of optimization to

satisfy various goals (Pati et al., 2008). Third, we

demonstrate the applicability of our proposed

approach through numerical computations.

The remainder of this article is organized as

follows. In the next section, we briefly review

existing works and present their limitations. Section

3 illustrates some fundamental concepts used in this

work. In section 4, we present the considered

dispatching and rebalancing problem in RAMoD

systems. In section 5, the proposed multi-objective

possibilistic linear programming model for RAMoD

systems is developed. In section 6, we exploit

appropriate strategies for converting the proposed

fuzzy model into an equivalent crisp one. Section 7

aims at finding an efficient compromise solution for

the problem, thus exploiting the goal programming

approach. We validate the proposed three-phase

approach through numerical tests being exploited in

Section 8. Finally, section 9 concludes the paper and

provide future directions.

2 RELATED WORK

The problem of dispatching and rebalancing has

received great attention over the last few years. The

proposed studies can be classified into three main

approaches, namely, simulation-based models;

queuing-theoretical models and model predictive

control (MPC) algorithms.

Simulation-based models (Hörl et al., 2018; Levin

et al., 2017; Maciejewski et al., 2017; Javanshour et

al., 2019) can accurately describe AMoD systems, but

being unable to provide optimal solutions.

Queuing-theoretical models (Zhang and Pavone,

2015; Zhang and Pavone, 2016; Iglesias et al., 2019;

Belakaria et al., 2019) have the advantage of

capturing the uncertainty of the customer requests.

These models are based on the Jackson network

concept (Serfozo, 2012), in which all arrivals at each

queuing station should follow a Poisson process

(Moran, 1952). This concept assumes constant rates

of occurrence of each random variable. That is, if the

random variable is customer’s arrival times, it

assumes customers arrive at stations at a constant rate

(Javanshour et al., 2019). However, in reality, the

customer arrival process for the various origin–

destination pairs is time-variant nature. Therefore, we

can deduce that the Queuing-theoretical models

prevent the AMoD system modelers from capturing a

realistic vision into these systems.

In contrast, Model predictive control (MPC)

algorithms (Zhang et al., 2016; Alonso-Mora et al.,

2017 ; Iglesias et al., 2018 ; Tsao et al., 2018 ; Tsao

et al., 2019) can efficiently accommodate time-

varying future demand. However, the majority of

existing MPC algorithms assume that future customer

demand is deterministic and the rare studies that

accommodate uncertainty mainly suggest the use of

stochastic programming. The probabilistic reasoning

approaches are usually based on evidence/data

recorded in the past. However, in many practical

situations, this evidence/data is unavailable or

subjectively specified, and the standard probabilistic

approach would not be appropriate to deal with them.

Thus, Fuzzy set theory and possibility theory provide

Fuzzy Multi-objective Optimization for Ride-sharing Autonomous Mobility-on-Demand Systems

285

an appropriate framework to handle uncertainty in

such situations. Accordingly, it has been successfully

used to model and treat uncertainties in many fields

such as supply chain planning (Khemiri et al., 2017;

Nemati and Alavidoost, 2019; Lima-Junior and

Carpinetti, 2020), Business Process modelling

(Yahya et al., 2017; Sarno et al., 2020), web services

(Rhimi et al., 2016; Bagga et al., 2019), image

processing (Ali and Lun, 2019; Nagi and Tripathy,

2020), etc.

Despite all this progress, the fuzzy logic and the

possibility theory have never been exploited to handle

uncertainties in AMoD systems.

To the best of our knowledge, this paper is the first

one to leverage the strengths of such techniques and

introduce a novel strategy for solving the dispatching

and rebalancing decisions problem with imprecise

travel demand in RAMoD systems.

In the next section, the basic concepts of the fuzzy

logic are provided.

3 THEORETICAL

BACKGROUND

This section briefly outlines the fuzzy set theory, the

triangular fuzzy numbers and the goal programming

method used in this paper.

3.1 Fuzzy Set Theory

Fuzzy set theory was originally introduced by Zadeh

(Zadeh, 1965) to deal with the imprecision,

uncertainty, and vagueness of subjective information.

From a mathematical point of view, a fuzzy set is

characterized by a membership function. Such a

function attributes to each object in the fuzzy set a

specific grade of membership ranging from zero to

one.

In this study, triangular fuzzy numbers are used to

represent the imprecise data. As shown in Figure 1, a

triangular fuzzy number 𝑍

can be represented by the

triplet (a, b, c) where a, b, c are the most pessimistic,

the most possible and the most optimistic value of 𝑍

The triangular fuzzy number 𝑍

can be represented

by the following membership function:











⎩

⎪

⎨

⎪

⎧

0, xa





, axb





, bxc

0 , xc

(1)

Figure 1: The triangular possibility distribution of 𝑍

3.2 Goal Programming

There are several methods in the scientific literature

for dealing with multi-objective models. Among

them, the goal programming (GP) method which is

originally developed by Charnes et al. (Charnes and

Cooper, 1961) and successfully used in several

problems (Lee and Kim, 2000; Amin et al., 2019;

Colapinto et al., 2020).

The popularity of this method is based on its

mathematical flexibility, its robustness, and its

accuracy.

The goal programming method consists in

introducing for each criterion a goal to be achieved

and to identify the solution that minimizes the sum of

the deviations from these goals.

Several variants of the GP have been proposed in

the literature. Here we use the Weighted Goal

Programming (WGP) method. The WGP can be

represented as follows:

Min

∈

 w









 w















(2)

Subject to:



(x) 0 , l1,2,..,L







- δ





 δ





= g



, i1,2,..,n





,δ





0

Where:

 C



(x) is the set of constraints.

 δ





and δ





are respectively the positive and

negative deviation from the target value g



 w





and w





are respectively the weight

attached to the positive and negative deviation.

 F







is the evaluation of the solution x against

the criterion i.

 𝐠

𝐢

is the aspiration level of the objective

function i.

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286

4 PROBLEM FORMULATION

Despite the major progress that has occurred in recent

years, these various initiatives do not take into

account the specificities of the low-density areas. In

the Tornado Mobility research project (Tornado,

2020), that we are working on, the objective is to

study the interaction between autonomous vehicles

and connected intelligent infrastructures for serving

mobility in low population density areas.

For this purpose, we consider an urban area

discretized into multiple stations and served by

several on-demand vehicles. Each vehicle can serve

one or more passengers without exceeding their

capacity. The considered fleet of vehicles is

characterized by a high level of heterogeneity:

transportation costs, speeds, and capacities of each

vehicle can be different.

In the context of Tornado project, customers first

request transportation from a pickup to a drop off

location in the predefined urban area via a mobile

application.

If there are available vehicles, one of them will be

dispatched to drive this passenger towards its

destination. Instead, if there are no available vehicles,

the user instantly leaves the system (i.e. without any

waiting time). Therefore, as in (Zhang and Pavone,

2016; Iglesias et al., 2019), our RAMoD system

operate according to the passenger loss model. Such

a model is well suited for systems where a high degree

of service is desired (Iglesias et al., 2019).

At the end of the trip, the vehicle could be

dispatched to accomplish other mobility demands. It

could also rebalance itself or even park in the drop-

off station for a certain period of time.

For simplicity, it is assumed that each station has

sufficient space so that vehicles can immediately be

parked and recharged at all times.

Unlike traditional approaches, the proposed

model does not assume complete knowledge about

future customer demand; instead, it assumes that such

critical parameters are estimated by the decision-

maker using Triangular fuzzy numbers.

Finally, it is assumed that the time is discretized

into an ordered set of time periods.

To deal with this challenging problem, we devise

a three-phase approach, where the main steps are

presented in Figure 2 and detailed in the following

sections.

Figure 2: Framework of the proposed approach.

5 PHASE I: PROPOSED MULTI

OBJECTIVE POSSIBILISTIC

LINEAR PROGRAMMING

MODEL

5.1 Notation

 The set of indices

₋ S: Number of stations (s = 1, 2, …, S).

₋ V: Number of vehicles (v = 1, 2… V).

₋ T: Number of time periods (t = 1, 2…, T).

 Decision variables

₋ Miss

v,t

: Binary variable indicating if vehicle

v is on mission during period t.

₋ Park

v,t,s

: Binary variable indicating if

vehicle v is parked in station s during period

₋ Miss_T

v,s1,s2,t1,t2

: Binary variable indicating

if vehicle v is on customer transport mission

traveling from station s1 to station s2

beginning at period t1 and arriving at period

t2.

₋ Miss_R

v,s1,s2,t1,t2

: Binary variable indicating

if vehicle v is on a rebalancing mission

traveling from station s1 to station s2

beginning at period t1 and arriving at period

t2.

₋ S_Cr

t,s1,s2

: The number of satisfied customer

requests traveling from station s1 to station

s2 departing at time period t.

 Certain parameters:

₋ Dist

s1, s2

: distance between stations s1 and s2

(considering the shortest way).

Fuzzy Multi-objective Optimization for Ride-sharing Autonomous Mobility-on-Demand Systems

287

₋ Cap

: Transport capacity of the vehicle v.

₋ SP

: speed of the vehicle v.

₋ Tr_cost

: transportation cost of the vehicle v.

₋ Local_init

v,s

: represents the initial

availability of vehicle v at station s. If

vehicle v is available at station s in the first

periode, Local_init

v,s

=1 and 0 otherwise.

 Fuzzy parameters:

₋ 𝑪𝒓

t,s1,s2

: number of customer requests who

wish to travel from station s1 to station s2

departing at time period t.

5.2 Objective Functions

 Objective 1: Improving customer satisfaction,

which is to minimize the number of lost

customer requests.

Minimize LCr

∑∑

𝐶



,





t,s1,s2

(3)

 Objective 2: Minimizing the overall

transportation cost.

Minimize TC=

∑

∑∑



















_cost

(4)

*(Miss_T

v,s1,s2,t1,t2

+ Miss_R

v,s1,s2,t1,t2

)* Dist

, s2

5.3 Model Constraints

S_Cr

, s1, s2

≥ 0 and integer ∀t , ∀s1, s2 ϵ [1, S] (5)

Miss

v,t

, Park

v,t,s

, Miss_T

v,s1,s2,t1,t2

Miss_R

v,s1,s2, t1, t2

ϵ [0,1] ∀t, v, s, s1, s2, t1, t2

(6)

Equations (5) and (6) guarantees the non-negativity

of the various decision variables: S_Cr

t,s1,s2

is an

integer, while other variables are binary.

∑





ark

v,t, s

+ Miss

v, t

= 1 ∀v, t

(7)

Equation (7) models the two possible states each

autonomous vehicle can take namely parked at a

station and be on a mission from one station to

another. On the other hand, this constraint ensures

that a vehicle can have only one state at any one time.

Miss

v,t

∑∑

,





,

iss_T

v,s1,s2,t1,t2

+ Miss_R

v,s1,s2,t1,t2

∀v, t

(8)

When a vehicle is on a mission, two possible

actions can be achieved i) transport one or more

customers from one station to another, and ii) travel

without customers for rebalancing the system. These

actions are modeled using equation (8), which also

guarantees that the vehicle can only perform one

action at a time.

Miss_R

v,s1,s2,t1,t2

≤ Park

v,t1-1,s1

∑



iss_R

v,s3,s1,t3,t1-1

∑



iss_T

v,s4,s1,t4,t1-1

∀ v, s1, s2, t1>1, t2 = t1 + (Dist

s1,s2

/ SP

t3=

-(Dist

/SP

) -1, t4 = t1-(Dist

/SP

)-1

(9)

Miss_T

v,s1,s2,t1,t2

≤ Park

v,t1-1,s1

∑



iss_R

v,s3,s1,t3,t1-1

∑



iss_T

v,s4,s1,t4,t1-1

∀ v, s1, s2, t1>1, t2 = t1 + (Dist

s1,s2

/ SP

t3=t1-(Dis

/SP

)-1, t4 = t1-(Dis

/SP

)-1

(10)

When vehicle v is on a mission traveling from

station s1 to station s2 beginning at period t1, it is

necessary that v is physically located in station s1 at

the beginning of period t1. In other words, either the

vehicle v i) arrived at a station during the last period

(i.e. Miss_R

v,s3,s1,t3,t1-1

=1 Or Miss_T

v,s4,s1,t4,t1-1

=1 ) , or

ii) parked at a station during the last period (i.e.

Park

v,t-1,s1

=1 ). The equations (9) and (10) ensure that

this constraint is respected respectively for

rebalancing missions and customer transport

missions.

Park

v,t,s

≤ Park

v, t-1,s

∑



iss_R

v,s1,s,t1,t-1

∑



iss_T

v,s2,s,t2,t-1

∀v, s, t >1, t1=t-(Dist

s1,s

/ SP

)-1,

t2=t+

(

Dist

s2,s

/SP

)

-1

(11)

Equation (11) guarantees that if a vehicle v is

parked at a station s during a time period t (i.e. Park

t,s

=1), it is necessary that it be physically located in s at

the beginning of t (i.e. Park

v,t-1,s

+ Miss_R

v,s1,s,t1,t-1

Miss_T

v,s2,s,t2,t1-1

=1).

Park

v,t,s

+ Miss_T

v,s,s1,t,t1

+ Miss_R

v,s,s2,t,t2

≤ Local init

v,s

∀v, s, t=1, s1, s2, , t1=t+(Dist

s,s1

/ SP

t2=t+(Dis

)

(12)

Equation (12) indicates that a vehicle may only be

parked in a station s during the first period (i.e.

Park

v,1,s

=1) if it is initially available at this station (i.e.

Local_init

v,s

=1). Besides, a vehicle v may only travel

on a rebalancing mission (i.e. Miss_R

v,s,s1,1,t1

=1) or a

customer(s) transport mission (i.e. Miss_T

v,s,s1,1,t1

=1)

if it is initially available at this station

(Local_init

v,s

=1).

S_Cr

t1,s1,s2

≤

∑





iss_T

v, s1, s2, t1, t2

* Cap

∀s1, s2, t1, t2= t1 + (Dist

s1,s2

/SP

)

(13)

Equation (13) ensures that the number of satisfied

customer requests traveling from station s1 to station

s2 departing at time period t1 can not exceed the total

capacity of the vehicles transporting customers from

station s1 to station s2 beginning at period t1.

S_Cr

t, s1, s2

≤ Cr

t, s1, s2

∀t, s1, s2

(14)

Finally, equation (14) guarantees that vehicles

transporting customer(s) from station s1 to station s2

ICSOFT 2020 - 15th International Conference on Software Technologies

288

beginning at time period t cannot transport more

customers than it has been requested.

In this study, it is assumed that the imprecise

customer demand in the first objective function and

constraint (14) is modeled using a triangular-shaped

possibility distribution. As explained in section 3,

triangular possibility distribution 𝐶𝑟



can be

represented by the triplet (Cr

, Cr

) where Cr

and Cr

are the most pessimistic, the most

possible and the most optimistic value of 𝐶𝑟



6 PHASE II: STRATEGY FOR

PROCESSING THE FUZZINESS

CUSTOMER REQUESTS

6.1 Treating the Imprecise Objective

Function

Given the imprecise customer’s request coefficients

in the first objective function, it is generally not

possible to determine an ideal solution to the problem

constrained by (3)-(14).

In the scientific literature, several approaches for

identifying compromise solutions are proposed

(Luhandjula, 1989; Sakawa and Yano, 1989; Tanaka

and Asai, 1984; Tanaka et al., 1984; Lai and Hwang,

1992). As mentioned by Hsu and Wang in (Hsu and

Wang, 2001), the first four approaches (Luhandjula,

1989; Sakawa and Yano, 1989; Tanaka and Asai,

1984; Tanaka et al., 1984) are based on restrictive

assumptions and are generally difficult to implement

in practice, we then use Lai and Hwang's approach

(Lai and Hwang, 1992; Liang, 2006).

Since the imprecise customer demand has

triangular possibility distributions, the

𝐿𝐶𝑟

objective

function would also have a triangular possibility

distribution. This imprecise objective is represented

by the three important points (LCr

, 0), (LCr

, 1) and

(LCr

, 0), geometrically. Therefore, minimizing the

fuzzy objective can be achieved by pushing these

critical points in the direction of the left-hand side.

According to Lai and Hwang’s approach solving

this problem becomes the process of minimizing

LCr

, maximizing (LCr

- LCr

) and minimizing

(LCr

- LCr

). In this way, our first objective function

can be transformed into a multiple crisp objective as

follows:

Minimize Z

=LCr

LCr

∑∑



,





,,



- S_Cr

t,s1,s2

(15)

Maximize Z

= LCr

- LCr

LCr

- LCr

∑∑



,







Cr



,,



- Cr



,,





- S_Cr

t,s1,s2

(16)

Maximize Z

= LCr

- LCr

LCr

- LCr

∑∑



,







Cr



,,



- Cr



,,





- S_Cr

t,s1,s2

(17)

6.2 Treating the Fuzzy Constraint

Recalling that equation (14) considers the situation in

which the crisp left-hand side is compared to the

fuzzy right-hand side. In this study, we implement the

well-known weighted average method for dealing

with this situation and approximating the 𝐶𝑟

parameter by crisp number. This method is originally

introduced by (Lai and Hwang, 1992) and has been

successfully used in several research studies (Wang

and Liang, 2005; Liang, 2006; Torabi and Hassini,

2009; Khemiri et al., 2017a) due to its simplicity and

efficiency in defuzzification.

To do so, we first need to determine a minimal

acceptable possibility degree of occurrence for the

fuzzy/imprecise parameter, α. Then the original fuzzy

constraint (14) can be represented by a novel crisp

constraint as follows:

S_Cr

t,s1,s2

≤ w

,,,



,,,



,,,



∀t, s1, s2

(18)

Where w

+ w

= 1, and w

, w

and

denote

respectively the weights of the most optimistic, the

weights of the most possible and the weights of the

most pessimistic of the fuzzy demand. In practice, the

values of these weights, as well as the minimal

acceptable possibility degree α, can be defined

subjectively based on the knowledge and experience

of the decision-maker.

In our work, we adopt the concept of most likely

values, which is widely used in the literature (Lai and

Hwang, 1992). According to this concept, the most

pessimistic and optimistic values required a lower

weight than the one assigned to the most possible

value. Thus, as in (Lai and Hwang, 1992) we set these

parameters to: w

= w

= 1/6 ; w

= 4/6 and α = 0.5.

7 PHASE III: GOAL

PROGRAMMING-BASED

SOLUTION APPROACH

In the previous section, the original fuzzy MOLP

model was converted into an equivalent auxiliary

Fuzzy Multi-objective Optimization for Ride-sharing Autonomous Mobility-on-Demand Systems

289

crisp multi-objective linear programming model. To

deal with this multi-objective model, we use the

Weighted Goal Programming (WGP) method,

introducing specific weights for each criterion.

Accordingly, we can reformulate our problem as

follows:

Minimize F

* δ





* δ





*δ





*δ





(19)

Subject to:

(5) - (13), (18)

𝑍



- 𝛿





= 𝑍



∗

(20)

𝑍



+ 𝛿





= 𝑍



∗

(21)

𝑍



+ 𝛿





= 𝑍



∗

(22)

TC - 𝛿





=𝑇𝐶

∗

(23)

Where:

 𝑍



∗

is the goal calculated using the

mathematical model with objective function

(15) subject to constraints (5) - (13), (18) and

𝛿





is the positive deviation from this goal.

 𝑍



∗

is the goal calculated using the

mathematical model with objective function

(16) subject to constraints (5) - (13), (18) and

𝛿





is the negative deviation from this goal.

 𝑍



∗

is the goal calculated using the

mathematical model with objective function

(17) subject to constraints (5) - (13), (18) and

𝛿





is the negative deviation from this goal.

 𝑇𝐶

∗

is the goal calculated using the

mathematical model with objective function

(4) subject to constraints (5) - (13), (18) and

𝛿





is the positive deviation from this goal.

 W

Z2,

and W

are the importance

weights of the various goals, usually

determined by the decision makers such that

+ W

=1.

8 SIMULATION RESULTS

In this section, we display two sets of simulation

results to illustrate the validity and applicability of the

proposed approach. First, we demonstrate that the

dispatching and rebalancing problem in RAMoD

systems can indeed be resolved using the proposed

three-phase approach, especially in the presence of

imprecise customer requests. Then, we compare the

performance of our methodology with other dispatch

strategies by varying customer demand over time.

For all experiments, we consider a fleet size of 15

autonomous vehicles and 5 stations. The planning

horizon is decomposed into 10 periods. These periods

correspond to 10 different predicted request demands

with triangular distributions, synthesized in Table 1.

Initially, the vehicles were distributed equally among

the various stations, i.e. 3 vehicles for each station.

For reason of simplification, we consider that the

travel time between two stations is one time step. The

capacity of the vehicles is characterized by a high

degree of heterogeneity which varies from a

maximum capacity of a single passenger to a

maximum capacity of 8 passengers. Additionally, we

consider for simplicity that the weights of the various

criteria are the same (i.e. W

= W

=1/4).

For all simulations, the proposed approach has

been implemented using the LINGO optimization

package.

8.1 Detailed Results for the Proposed

Approach

Figure 4 summarizes the results provided by the

proposed approach by detailing vehicle statuses

according to the planning horizon. We remind that the

vehicle can be parked at one station, be on a

customer(s) transport mission and be on a rebalancing

mission. For the last two states, the departure and

arrival stations were also mentioned. These decisions

are guided by the criteria of the customer satisfaction

maximization and the transportation cost

minimization at each period of the planning horizon.

Indeed, we find that the increase in the cost of

transporting a vehicle leads to not using it (i.e. staying

parked in the station) if customer demand can be

satisfied by vehicles with a lower transport cost. For

example, for the first period, customer demands were

satisfied with the various stations. In particular for

station S3, this fuzzy demand has been satisfied by

using V7 and V8 with the use of ride-sharing, while

the V9 remains parked in S3 because it has much

higher transport cost. Also during the second period,

the vehicle V12 remains parked in the station S4 since

customer demand has been satisfied by vehicles with

a lower transport cost.

With the increase in customer demands during the

third and fourth periods and guided by the criterion of

maximizing customer demands satisfaction, all

vehicles in the fleet were launched on missions, even

the most costly ones.

However, beyond the fifth period,

the mobilization of all vehicles remains insufficient to

ICSOFT 2020 - 15th International Conference on Software Technologies

290

Table 1: Fuzzy demand for each period.

SiS

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

(

0,1,2

)

(

0,1,2

)

(

0,1,2

)

(

4,5,6

)

(

1,2,3

)

(

3,4,5

)

(

6,8,10

)

(

4,5,6

)

(

4,5,6

)

(

1,3,5

)

(

1,2,3

)

(

0,1,2

)

(

0,1,2

)

(

2,3,4

)

(

1,3,5

)

(

1,2,3

)

(

4,5,6

)

(

2,4,6

)

(

3,5,7

)

(

4,5,6

)

S4 (0,1,2) 0 (2,3,5) (2,4,6) (2,3,4) (4,5,6) (4,5,6) (2,4,6) (3,6,9) (2,3,4)

S5 0 0 (1,2,3) 0 (2,4,6) (2,4,6) (2,4,6) (10,12,14) (6,8,10) (2,3,4)

S1 (0,1,2) (0,1,2) (1,2,3) 0 (1,2,3) (6,7,8) (1,2,3) (1,2,3) (6,8,10) (2,5,8)

(

0,1,2

)

(

0,1,2

)

(

3,4,5

)

(

0,1,2

)

(

1,2,3

)

(

3,4,5

)

(

1,2,3

)

(

1,2,3

)

(

2,4,6

)

(

0,1,2

)

(

0,1,2

)

(

0,1,2

)

(

3,4,5

)

(

1,3,5

)

(

1,2,3

)

(

3,5,7

)

(

4,5,6

)

S5 0 0

(

0,1,2

)

(

3,4,5

)

0 0 0

(

1,2,3

)

(

3,5,7

)

(

4,5,6

)

S1 (1,2,3) (0,1,2) (2,4,6) (1,4,7) (1,2,3) (2,5,8) 0 (3,4,5) (3,5,7) (10,13,16)

S2 (0,1,2) (0,1,2) (1,2,3) (2,4,6) (1,2,3) (0,2,4) (1,2,3) (0,1,2) (0,1,2) (2,3,4)

S4 0

(

0,1,2

)

(

1,2,3

)

(

2,4,6

)

(

1,2,3

)

(

3,4,5

)

(

1,2,3

)

(

1,3,5

)

(

2,5,8

)

S5 0 0

(

1,2,3

)

0 0 0 0

(

2,3,4

)

0 0

(

0,1,2

)

(

0,1,2

)

(

0,1,2

)

0 0 0 0 0 0

S2 (0,1,2) (0,1,2) 0 0 0 0 0 0 0 0

S3 (0,1,2) (0,1,2) (0,1,2) (5,7,9) 0 0 0 0 0 0

S5 0 (0,1,2) (0,1,2) (1,2,3) 0 0 0 0 0 (2,5,8)

S1 0

(

0,1,2

)

(

1,3,5

)

(

0,1,2

)

(

1,2,3

)

0 0 0

(

1,2,3

)

(

1,2,3

)

S2 0

(

0,1,2

)

(

0,2,4

)

(

1,2,3

)

(

1,4,7

)

0 0 0 0

S3 0

(

0,1,2

)

(

3,4,5

)

0 0 0 0 0 0

S4 0 0 0 (1,3,5) 0 0 (3,5,7) 0 0 0

satisfy customer demand, especially when some

stations are more in demand than others, at the end of

the trip, vehicles are accumulating in these stations

and depleting in the others. This justifies the use of

rebalancing decisions from overloaded stations to

under loaded stations.

The rebalancing decisions are also subject to the

cost minimization criterion. Indeed, the least

expensive vehicles will be assigned first to

rebalancing missions

8.2 Performance of the Proposed

Approach

To evaluate the performance of the proposed

approach (D-R-RAMoD-Fuzzy), we conducted a

simulation study comparing it to other dispatch

strategies. These latter are concretely three versions

of our proposed approach:

 D-R-RAMoD-Perfect: The dispatching and

rebalancing approach proposed in previous

sections based on an exact customer request as

it appears in the data set as a "forecast" for the

next 10 time periods. This is an efficient

strategy to find the optimal dispatching and

rebalancing policies for the case when the

customer request is known in advance. Thus, it

can be used for providing performance upper

bounds of the system.

 D-R-AMoD-Fuzzy: This version uses the

same model described in section 5 for single

capacity vehicles (without the use of ride

sharing).

 D-RAMoD-Fuzzy: This version is exclusively

concerned with the “Dispatching” problem and

vehicles do not rebalance in any situation.

The summary results of this comparison are

presented in Figure 3, illustrating the number of lost

customer requests for each dispatch strategies as a

function of time.

Figure 3: The number of lost customer requests for each

dispatch strategies as a function of time.

As expected, the strategy with exact customer

requests has the best performance, with a minimum

number of lost requests and a reduced transport cost.

The "D-R-AMoD-Fuzzy" strategy has the worst

performance, with mean lost requests sixfold than

that of "D-R-RAMoD-Perfect" strategy and

multiplied by four compared to that of our proposed

Fuzzy Multi-objective Optimization for Ride-sharing Autonomous Mobility-on-Demand Systems

291

Figure 4: Vehicle scheduling as a function of time.

approach (i.e. "D-R-RAMoD-Fuzzy" strategy). This

is not surprising, given that the single capacity

strategy is here compared to the ride-sharing policies

where the maximum capacity of vehicles is extended

to eight.

We can also see the marked difference in

performance between the "D-RAMoD-Fuzzy"

strategy and the "D-R-RAMoD-Perfect" strategy

from Figure 3 showing the number of lost customer

requests at any given period. Notably, the "D-

RAMoD-Fuzzy" strategy has significantly more lost

customer requests at any given time period, with

mean lost requests multiplied by four compared to the

optimal strategy and multiplied by three compared to

that of our proposed approach. This is also not

unexpected, since we can gain much of performance

by incorporating rebalancing trips ensuring a balance

between the number of vehicles available in each

station and customer requests.

A significant performance gain is attributed by

incorporating rebalancing trips and the fact that

several customers can share the same vehicle. Indeed,

we can notice that out of 10 experiments, the

proposed approach generates an optimal solution for

six experiments. It also offers solutions that are very

close to the optimal solution for the other periods with

a deviation of 35%. This highlights the robustness of

the proposed approach for operating the fleet and

satisfying customers, even when forecasts of

customer requests are uncertain.

9 CONCLUSION

Despite the significant advances in AMoD and

RAMoD systems, the existing studies still display a

lack of approaches dealing with the uncertainty

affecting travel demand forecasts. The rare studies

dealing with this drawback mainly suggest the use of

stochastic programming that is usually based on the

statistical data. However, in practice, historical data

may not be reliable or even unavailable. Accordingly,

these traditional programming models may not be the

best tool to deal with uncertainty.

Thus, this work provides a new point of view on

the problem of dispatching and rebalancing in the

RAMoD systems by using a new alternative approach

for managing uncertainty. Specifically, we first

formulated the problem as a multi-objective

possibilistic linear programming model in which

customer requests are evaluated in an imprecise way

using triangular possibility distribution. The proposed

fuzzy formulation is then transformed to an

equivalent crisp multi-objective linear programming

model by combining appropriate strategies. In the

third phase, the well known goal programming

approach is being exploited to obtain a compromise

solution. Through experiments, we show that the

proposed approach has the capability to deal with

realistic situations in an uncertain environment and

provides an efficient decision tool for the dispatching

and rebalancing decisions in RAMoD systems.

This work leaves opens for considerable

extensions for future research.

First, the proposed approach can be extended in

situations when RAMoD systems are faced with

fluctuations of several parameters. This research area

will require introducing forecasting models that are

able to model not only the uncertain customer

requests but also other critical parameters such as

vehicle availability, costs, the states of charge of

vehicles, etc.

ICSOFT 2020 - 15th International Conference on Software Technologies

292

Second, we plan to explore the integration of

routing policies within a capacitated road network.

This, in turn, can be subject to important uncertainties

due to several external factors such as traffic

congestion. Thus, the goal of this research axis is to

devise a robust dispatching-rebalancing and routing

policy that leverages forecasting parameters while

considering the uncertainty that can arise in the road

network.

Finally, further research can study the couplings

that could occur between public transit and the

AMoD systems.

ACKNOWLEDGMENT

This work is financed by national funds FUI 23 under

the French TORNADO project focused on the

interactions between autonomous vehicles and

infrastructures for mobility services in low-density

areas. Further details of the project are available at

https://www.tornado-mobility.com/.

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