Optimal Scheduling for Flying Taxi Operation

Panwadee Tangpattanakul

and Ilhem Quenel

Capgemini Engineering, 4 avenue Didier Daurat, 31700 Blagnac, France

Capgemini Sophia Antipolis, 950 avenue de Roumanille, E. Golf-Park, Bâtiment TECK, 06410 Biot, France

Keywords: Flying Taxi, Scheduling Problem, Metaheuristic, Genetic Algorithm, Simulated Annealing.

Abstract: According to the traffic congestion problem in big cities around the world, some sustainable transportation

technologies are researched and developed in these recent years. Flying taxi is a transportation mode, which

is being developed from several major brands. It can become an alternative transportation mode in the future.

In this work, a simplified optimization model of the flying taxi scheduling is proposed. Two algorithms, which

consists of genetic algorithm and simulated annealing, are used to solve the problem. The experiments are

conducted on 15 instances with different number of customer demands (between 10 and 200 demands) and

different number of available flying taxis in the system (from 2 to 10 taxis). The experimental results show

that both algorithms are efficient to solve the problem. The genetic algorithm obtains better quality solutions

for the small and medium size instances but it spends more computation time than the simulated annealing.

However, the simulated annealing can solve the large instances and obtain good solutions in reasonable time.

1 INTRODUCTION

Nowadays, there are not flying taxis in the daily life

yet, but the development of autonomous vehicles and

delivery drones is illustrated that the flying taxi

transportation mode will not be only the dream and

getting closer to reality. However, to reach the

success of the passenger transportation services,

several criteria, such as payload capacity, safety

constraint, noise pollution, and operation cost, must

be well-designed in the research process before it can

be launched in the real life.

This work will present a simplified scheduling

problem model for the future flying taxis by

mimicking the characteristics of parcel delivery

drones. It concerns two main subjects of literature

review, consisting of the delivery drone scheduling

and the taxi demand responsive scheduling. For the

drone delivery scheduling part, different

characteristics and constraints of parcel delivery

drones are studied. For the taxi demand responsive

scheduling part, the method to respond to the demand

is studied. We can see in details as follow:

1.1 Delivery Drone Scheduling

In this part, we want to understand the general

characteristics of drone and scheduling model of

delivery drones. We plan to mimic the delivery drone

model to obtain the simplified flying taxi model and

then combine this part with the taxi demand

scheduling to design the flying taxis scheduling

formula. The drone delivery concept was proposed by

Amazon Prime Air in 2013 (Amazon, 2013) and other

companies, such as Wing, UPS flight forward,

Flytrex, Wingcopter, Zipline also develop their own

commercial delivery drones later (Ueland, 2021).

In the research of parcel delivery, there are three

ways of delivery problem from depot center to

customers (Kuang, 2019). The first one is a traditional

method which delivers the packages by trucks only.

The second one is a drone delivery where the drones

bring the packages directly from the depot to

customers. The last one is a truck and drones

combination which brings the large number of

package by trucks from the depot to customer zone

and drones deliver package from the truck to the

customers. (Marray & Chu, 2015) proposed two

models of parcel delivery with drone, which are the

Flying Sidekick Traveling Salesman Problem

(FSTSP) and the Parallel Drone Scheduling Traveling

Salesman Problem (PDSTSP). In (Saleu et al., 2018),

they want to minimize the completion time for

PDSTSP by using an iterative two-step heuristic. In

(Ponza, 2016), the FSTSP is considered and the

simulated annealing is used to solve the problem.

There are several important drone constraints that

should be satisfied in the real operations such as the

Tangpattanakul, P. and Quenel, I.

Optimal Scheduling for Flying Taxi Operation.

DOI: 10.5220/0010677200003063

In Proceedings of the 13th International Joint Conference on Computational Intelligence (IJCCI 2021), pages 141-148

ISBN: 978-989-758-534-0; ISSN: 2184-3236

141

reliability of drone delivery network, battery

consumption, and delivery time windows. As in

(Torabbeigi et al., 2018), the reliability concerning

drone failure is considered. They assume the drone

failure follow an exponential distribution. In (Cheng

et al., 2020), the robust scheduling is proposed by

considering the wind condition uncertainty. In (Kim

et al, 2020), the drone operation planning model is

destination and also the battery level of the drone.

Each drone is assigned to either deliver the parcels or

recharge the battery. In the operation planning of

(Torabbeigi et al., 2020), the battery consumption rate

of drones is modeled depending on the carrying

payload. The drones pick the parcels from the depot,

deliver to customers and then return to the depot.

(Dukkanci et al., 2020) studies the energy

consumption of drone according to its speed function.

For the delivery time window constraint, (Han et al.,

2020) considers a vehicle routing problem by

satisfying the time window. In (Huang et al., 2020),

the parcel delivery system which considers the

cooperation of public transport and drones is

considered. They model the system characteristic

including the delivery time, energy consumption and

battery recharging.

For the simplified model of flying taxi, we have

to select some constraints from drone operations to be

considered. We will concentrate to the battery

consumption function, battery recharging, and time

window of demands, but ignore other constraints in

this step.

1.2 Taxi Demand Scheduling

We study how to manage taxi demands efficiently in

this part. The passengers require the taxis to pick up

in the specific location and specific time window.

Then, the taxis bring them to the destination in the

limitation of tardiness.

A classical problem in the domain of operations

research, which concerns the taxi demand scheduling,

is the vehicle routing and scheduling problem with

time window constraints or VRSPTW. This problem

needs to find the minimum-cost vehicle routes to

serve a set of customers by satisfying the vehicle

capacity constraints. Moreover, each customer must

have a service in the specific available time window.

The VRSPTW is an NP-hard problem for which

heuristic algorithms are widely used. (Solomon,

1987).

We can also model the taxi demand scheduling

problem like the fleet management problems as in

(Bielli et al., 2011). It presents different mathematical

models for variant transportation modes and

characteristics. In (Glaschenko et al., 2009), it

considers a multi-agent approach to real-time

scheduling to be able to re-schedule and update

schedule in real time. The orders have to be matched

to drivers, vehicles, resources and work practices.

The schedule must ensure fair and proportional jobs

distribution for drivers. In (Shen et al. 2017), the

dispatching system to design a demand-responsive

schedule is considered. It consists of a planning of

travel path (routing) and customer pick-up and drop-

off times (scheduling) by considering certain

constraints such as vehicle capacity limitation and

available time windows.

The profit of taxi service is an important objective

for the business. The fare of general taxi services is

calculated from a base rate (for some first kilometers

as initial charge), a distance rate (multiply to the next

kilometer count), and a minute rate (multiply to the

minute count of the waiting time). However, the fare

of flying taxi will be calculated from only the base

rate and the distance rate. We ignore the congestion

on the low altitude air traffic in this step.

According to the related work, we design a

simplified model of flying taxi scheduling problem. It

is presented in Section 2. Some delivery drone

specifications such as the vertical take-off and

landing, operating hours, recharging time, battery

capacity, power consumption rate will be borrowed

from the literature for the experiments. We need to

maximize the working time that the flying taxis serve

the clients on the selected demands and it is used to

be the objective function in our work. In Section 3 and

4, two metaheuristics, which are a genetic algorithm

and a simulated annealing, are presented. We apply

these two algorithms to optimize the scheduling

problem. The computational results on fifteen

generated instances are illustrated in Section 5.

Lastly, the conclusion and the future work are

presented in Section 6 and 7, respectively.

2 FLYING TAXI SCHEDULING

OPTIMIZATION MODEL

In this section, an optimization model for the flying

taxi scheduling problem will be presented. The

simplified flying taxi characteristics are designed

from the commercial delivery drones in several

research works. We assume that the flying taxis

operate to serve the customers as same as the drone

service for transporting the parcel from an origin to a

destination. The main operating time of a task is to

ECTA 2021 - 13th International Conference on Evolutionary Computation Theory and Applications

142

move in horizontal axis between two geometric

points, which represent the origin and the destination

in x-y coordinate. The operating time is calculated by

using the Euclidean distance and the average speed of

the flying taxi. Moreover, we consider the fixed

additional time in the beginning of the task for

picking up the passenger and vertical takeoff, and in

the end of the task for vertical landing and dropping

the passenger as in Figure 1. Some unprofitable trips

(for changing the taxi location to pick up the next

passenger or to go to the center for battery recharging)

are also taken into account for the schedule

management.

Figure 1: Flying taxi passenger service in the considered

area.

For the flying taxi scheduling problem, we have

to find the optimal task schedule for all available

flying taxis in the system. The customers submit the

requests with the origin points, destination points, and

pick up time to the taxi center, and then the taxi center

will manage the demands to the flying taxis in order

to obtain the most advantage of the existing resources.

The assigned tasks can be a demand response for

passenger service or a battery recharging. The

location changes before starting tasks either for

moving to the next customer for preparing the

demand response or to the center for the battery

recharging are computed. The tasks are assigned to

the available flying taxis depending on the required

demands from the customers and the remaining

battery level of the flying taxis.

We propose a flying taxi scheduling model by

considering the above assumptions and borrowing

some drone characteristics from the literatures. The

objective of our work is to maximize the operational

time of the flying taxis to serve the customers and

gain the fare. The task assignment must satisfy the

flying taxi constraints that the taxi must be available

in required period and also has enough battery level

for finishing its task and return back to the center for

recharging, if it is necessary. The required parameters

and variables are shown as follow:

Indices

𝑖 demand index 𝑖1,2,…,𝑛

𝑗 flying taxi index 𝑗1,2,…,𝑚

𝑘 flying taxi operation index

𝑘1,2,…,𝐾

Parameters

𝑑



demand 𝑖

𝑢



pick-up time of demand 𝑖

𝑝_𝑜



𝑥_𝑜



,𝑦_𝑜



 origin point of demand 𝑖

𝑝_𝑒



𝑥_𝑒



,𝑦_𝑒



 destination point of

demand 𝑖

𝑙



distance between origin and destination

points of demand 𝑖 (unit:metre)

𝑙







𝑥_𝑜



𝑥_𝑒







𝑦_𝑜



𝑦_𝑒







𝑝_𝑐



𝑥_𝑐,𝑦_𝑐



center point for battery

recharging

𝐵 battery consumption rate (unit:

%/minute)

𝑣 average speed in horizontal axis (unit:

m/minute)

𝑠 required time for vertical takeoff or

landing, including time to pick up and

drop passenger, if need (unit: minute)

𝑅 battery recharging time to full (unit:

minute)

𝑏



minimum remaining battery level to

prevent full discharge (unit: %)

𝑇 available time for flying taxi operations

in 24 hours (1440 minute)

Decision Variables

𝑥



1 if flying taxi 𝑗 serves the customer of

demand 𝑖 as the 𝑘th operation, and 0

otherwise

𝑦



1 if flying taxi 𝑗 recharges its battery at

the center as the 𝑘th operation, and 0

otherwise

𝑏



battery level of flying taxi 𝑗 after flying

taxi 𝑗 finishes its 𝑘th operation (unit:

The initial battery charge is set to

100%.

𝑡



time of flying taxi 𝑗 after flying taxi 𝑗

finishes its 𝑘th operation (unit: minute)

The initial time is set to 0 and it

increases as the operations are

performed.

𝑝



𝑥_𝑝



,𝑦_𝑝



 position in x-y

coordinate after flying taxi 𝑗 finishes

its 𝑘th operation

The initial position is set to center

position 𝑝_𝑐

𝑔



distance to change the location of

flying taxi 𝑗 to origin point of demand

𝑖 or to the recharging center as the 𝑘th

operation (unit: meter)

Optimal Scheduling for Flying Taxi Operation

143

The flying taxi scheduling problem is formulated as below:

max  

𝑙



𝑣

2𝑠𝑥















(1)

Subject to:

𝑥















,∀𝑖

(2)

𝑥



𝑦



1,∀





𝑗,𝑘

(3)

𝑦



𝑥







,∀𝑗,𝑘, 𝑤ℎ𝑒𝑟𝑒𝑘2

(4)

𝑔



 𝑥





𝑥_𝑝









𝑥

𝑜







𝑦

𝑝









𝑦

𝑜







𝑦





𝑥_𝑝









𝑥

𝑐



𝑦

𝑝









𝑦

𝑐



,∀𝑗,𝑘

(5)

𝑏









𝐵

𝑔



𝑣

2𝑠

𝑙



𝑣

2𝑠





𝑥



 100%𝑦



𝑏



,∀𝑗,𝑘

(6)

𝑏



𝐵

⎝

⎛



𝑥_𝑝



𝑥

𝑐



𝑦

𝑝



𝑦

𝑐



𝑣

2𝑠

⎠

⎞

𝑥







𝑏



,∀𝑗,𝑘

(7)

𝑡













𝑔



𝑣

2𝑠







𝑥



𝑢



,∀𝑗,𝑘

(8)

𝑢





𝑙



𝑣

2𝑠





𝑥







𝑡











𝑔



𝑣

2𝑠𝑅



𝑦



𝑡



,∀𝑗,𝑘

(9)

𝑡





𝑇,∀𝑗,𝑘

(10)

𝑥







∈



0,1



,∀𝑖,𝑗,𝑘

(11)

𝑦





∈



0,1



,∀𝑗,𝑘

(12)

𝑏





,𝑡





0,∀𝑗,𝑘

(13)

𝑝





∈ℝ



,∀𝑗,𝑘

(14)

The mathematic formula of the flying taxi scheduling

problem presents an objective function in Equation

(1) and the constraints in Equation (2) to (14) that

must be satisfied. In Equation (1), the objective

function of this problem is to maximize the

operational time that the flying taxi fleet can serve the

customers according to the selected and scheduled

demands in the solution sequences. This objective

value can represent the obtained profit from their

proportional relation. Equation (2) ensures that the

demand cannot be served more than once. Equation

(3) guarantees that the flying taxi can be assigned

only one task, demand response or battery recharging,

but it is not allowed to operate two tasks

simultaneously. Equation (4) ensures that only the

demand response task will be assigned to the flying

taxi if the previous task is the battery recharging.

Equation (5) calculates the unprofitable distance

when the flying taxi changes the position from the last

position of the previous task, to the origin position of

the next selected demand if the flying taxi is assigned

to serve the customer or to the recharging center if the

flying taxi is assigned to recharge the battery. This

unprofitable distance value will be used in the next

equations. Equation (6) calculates the remaining

battery level after the flying taxi finishes the assigned

task. If the assigned task is a demand response, the

battery consumption is calculated from the location

move to pick up the passenger and bring them to the

destination, and also including the vertical takeoff

and landing. If the assigned task is a battery

recharging, the battery level is always 100% after the

flying taxi finishes the tasks. Equation (7) guarantees

that the flying taxi has enough battery to finish the

passenger service task and also go back to the center

if the battery must be recharged. Equation (8) ensures

ECTA 2021 - 13th International Conference on Evolutionary Computation Theory and Applications

144

that the flying taxi has enough time to change the

location to the pick-up point of the next assigned

demand. Equation (9) calculates the finishing time

after the flying taxi operates the assigned demand

response task or the assigned battery recharging task

as in Figure 2. Equation (10) guarantees that the

finishing time of the last assigned task is in the

available operation time. Equations (11) and (12)

illustrate the binary decision variables of the demand

response and battery recharging tasks. Equation (13)

guarantees that the decision variables of battery level

and operating time of the flying taxis are nonnegative.

Finally, Equation (14) shows that the current position

of the flying taxi corresponds to real values in two

dimensions (x-y coordinate).

Figure 2: Operation time of possible assigned tasks:

demand response or battery recharging.

We do not have the real flying taxi characteristics

since the flying taxi service does not exist in real-life

yet. Therefore, we borrow some drone specifications

from Kim et al, 2020 and modify some specification

values for our experiments. We ignore the drone

failure reliability and weather uncertainty in this step.

The specifications for the experiments is shown as in

Table 1.

Table 1: Specifications for scheduling experiments.

Item Value

Battery consumption rate



𝐵



0.67%/minute (≈ 100% for

2.5 hours)

Average speed in

horizontal axis



𝑣



833 m/minute (≈ 50 km/h)

Required time to takeoff

or landing



𝑠



5 minutes

Battery recharging time to

full



𝑅



60 minutes

Minimum remaining

attery level



𝑏





Available operating time

𝑇

1440 minutes/day (from

midnight to midnight of

the next day)

The algorithms, which are used to solve this

scheduling problem, will be explained in the next two

sections.

3 GENETIC ALGORITHM

The genetic algorithm is a popular metaheuristic

method, which is very efficient for solving complex

combinatorial optimization problems (Goldberg,

1989). It mimics the natural survival in life evolution.

The genetic algorithm will start with the first

generation, which generates an initial population. The

initial population is often generated randomly. The

population contains several chromosomes (solutions)

and each chromosome contains several genes. In this

work, we used a biased random key genetic

algorithm, which are proposed in (Gonçalves et al.,

2011), for solving the flying taxi scheduling problem.

The gene values are real values in the interval 0 and

1. The number of genes in each chromosome is equal

to number of demands multiply to number of flying

taxis. Each gene represents a demand, which is

responded by a flying taxi. In each iteration of the

genetic algorithm, a new population will be generated

from three sets of chromosomes: elite set from

selection process, crossover set from crossover

process and mutant set from mutation process. The

elite set copies the best chromosomes from the

previous population. The crossover set contains the

offspring chromosomes in which the gene values

come from two different parent chromosomes. The

first and the second parents are selected randomly

from the elite set and the non-elite set, respectively.

The mutant set is randomly generated as in the initial

population. This mutant set helps to escape local

optima. We use the number of iterations since the last

archive improvement as a stopping criterion. The

parameter values are tuned by the experiments and

they are set as in Table 2.

Table 2: Parameter values of GA.

Parameters Values

Population size 2 times of chromosome size

Elite set size 10% of population size

Crossover set size 70% of

ulation size

Crossover

robabilit

0.7

Sto

criterion 30

A decoding method with ideal priority

combination are designed for obtaining the solution

from each chromosome.

Each gene of the GA chromosome represents a

demand, which is assigned to a flying taxi. We

borrow the decoding method from (Mendres et al.,

2009). The expression of the decoding with ideal

priority combination of the demand 𝑖 is presented as:

𝑃𝑟𝑖𝑜𝑟𝑖𝑡𝑦





𝑇𝑢



𝑇



1𝑔𝑒𝑛𝑒





Optimal Scheduling for Flying Taxi Operation

145

where 𝑇 is the available operation time of the flying

taxi and 𝑢



is the pick-up time of the demand 𝑖. The

concept of the ideal priority is to give higher priority

to the demand, which has the earlier pick-up time.

In the decoding step, the demand and the specific

flying taxi, which has the highest priority, will be

considered firstly. If all necessary constraints are

satisfied, the demand will be assigned to the sequence

of the specific flying taxi. The gene concerning the

same demand of the assigned one will be removed

from the waiting list. Then, the next lower priority

demand will be considered until the all demands are

examined. Finally, we will obtain the sequence of the

selected demanded to be responded by the flying taxis

and their battery recharging tasks.

4 SIMULATED ANNEALING

The simulated annealing is a metaheuristic algorithm,

which is also efficient for solving combinatorial

optimization problems. It was proposed to solve the

traveling salesman problem in 1983 (Kirkpatrick,

1983). It mimics the heating and controlled cooling

down process of the material by decreasing the

probability to accept the worse solutions according

the temperature. The temperature is slowly

decreased from the initial value to zero. The

algorithm uses the perturbation strategies to

generate a new candidate close to the current

solution, measure its quality, and decide to move to

it according to the probability from the current

temperature.

In this work, we use the fast simulated annealing

from Brownlee, 2021. The temperature is calculated:

𝑡𝑒𝑚𝑝𝑡𝑒𝑚𝑝



/𝑖𝑡𝑒𝑟



1

where 𝑡𝑒𝑚𝑝 is the current temperature, 𝑡𝑒𝑚𝑝



the initial temperature, and 𝑖𝑡𝑒𝑟



is the

iteration number. Since we consider the maximization

problem in this work, the acceptance probability of

the worse candidate is calculated from the difference

of the objective values between the candidate and

current solutions as follow:

𝑝𝑟𝑜𝑏



exp

𝑜𝑏𝑗



𝑜𝑏𝑗



𝑡𝑒𝑚𝑝



It shows that the algorithm can escape from basins of

attraction, since it allows accepting the candidate

solution even its quality is worse than the current

solution when the temperature parameter is still high.

The acceptance probability will decrease slowly

when the temperature parameter is lower. To generate

the candidate solution, the perturbations are designed

by changing one of the taxi to respond to the demand

and changing the order to assign the task to the

sequence. The initial temperature of the simulated

annealing process is set to 100. The number of

iterations that the best solution is not improved is

defined as a stopping criterion and it is set to 1000.

5 COMPUTATIONAL RESULTS

The genetic algorithm and the simulated annealing

are used to solve the flying taxi scheduling problem.

The experiments are done on fifteen generated

instances. They concern different number of demands

and different number of flying taxis in the 24 hours of

time duration. The day time demands are generated

randomly with higher probability than the demands in

the night time. The format of instance name is

“Instancexx_yy.txt”, where xx shows the number of

demands and yy shows the number of flying taxis in

this instance. As the example of “Instance10_2.txt”,

this instance concerns ten demands and two flying

taxis. The origin-destination points and the passenger

pick-up time of each demand are defined in the

instances. The algorithms are implemented in Python

3.8.8 and Conda 4.9.2 on a personal computer with an

Intel Corei5 1.6 GHz CPU and 8 GB RAM.

Figure 3 shows an example of the approximate

optimal schedule for “Instance100_5”, which

examines 100 customer demands and 5 available

flying taxis. The green bars represent the taxi services

to respond the customer demands and the orange bars

illustrate the battery recharging tasks. The obtained

schedule also satisfies all operational constraints such

as the necessary time to change the location for on

time picking up the next customer and the flying taxis

have enough remaining battery level after finishing

the mission and they can return to the center for

battery recharging, if they need.

Figure 3: A result example of the instance100_5.

The computational results of the genetic

algorithm and the simulated annealing are compared.

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The boxplots of eleven runs of each instances are

shown in Figure 4. The results show that the genetic

algorithm with the ideal priority combination

decoding method obtains better solutions for the

small and medium instances. Since the genetic

algorithm is a population-based algorithm that

maintains and improves multiple solutions in each

iteration, it spends higher computation time than the

simulated annealing. On our parameter setting, the

genetic algorithm takes more than 24 hours to obtain

the approximate optimal solution for the large

instances and it is not reasonable for the dispatching

system of the flying taxi scheduling problem. So that

why, the genetic algorithm has not results for the

large instances. However, the simulated annealing,

which is a single solution approach, can solve the

large instances in acceptable time.

6 CONCLUSIONS

The simplified model of the flying scheduling

problem is presented in this work. Since there are not

the flying taxi in the daily life yet, the characteristics

of the flying taxi are mimicked from the parcel

delivery drone properties. The model concerns two

main subjects, consisting of the delivery drone

scheduling and the taxi demand responsive

scheduling. For the drone delivery scheduling part,

different characteristics and constraints of parcel

delivery drone are studied. The method to respond to

the demand is studied from the taxi demand

responsive scheduling part.

We assume that the flying taxis operate to serve

the customers as same as the drone service for

transporting the parcel from an origin to a destination.

The main operating time of a task is to move in

horizontal axis between two geometric points, which

represent the origin and the destination in x-y

coordinate. Moreover, we consider the fixed

additional time in the beginning of the task to pick up

the passenger and vertical takeoff and in the end of

the task to vertical landing and drop the passenger.

The objective of our work is to maximize the

operational time that the flying taxi fleet can serve the

customers according to the selected and scheduled

demands in the solution sequences. The task



Figure 4: Objective value and average computation time comparison between two algorithms: Genetic algorithm (GA) and

simulated annealing (SA).

Optimal Scheduling for Flying Taxi Operation

147

assignment must satisfy the flying taxi constraints

that the taxi must be available in specific period and

has enough battery level for finishing its task and

return back to the center for recharging, if it is

necessary. Two metaheurics are used to solve this

problem and the computational results are compared.

The results show that the genetic algorithm obtains

better solutions for the small and medium instances,

but it spends higher computation time than the

simulated annealing. However, the simulated

annealing can solve the large instances in reasonable

time.

7 FUTURE WORKS

For the future work, the characteristics of the flying

taxis and their scheduling model can be improved to

represent closer to the real operations in the future.

New technology of battery can be taken into account.

The air obstacles of the flying taxi trajectory can be

considered.

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