Research on Reloading Airdrop Strategy Based on Multi-Objective

Programming

Yue Yang, Xiaolong Wu and Zengyang Wang

*

Dalian Naval Academy, Dalian, China

Keywords: Airdrop, Multi-Objective Optimization, Genetic Algorithm.

Abstract: In order to realize the rapid assembly of equipment, it is necessary to use formation for airdrop. When the

formation aircraft are too close to each other, the wake of the front aircraft may affect the safety of the rear

aircraft and the cargo. In order to make the landing point of the rear aircraft and the target point as small as

possible and the landing time as close as possible. In this paper, a multi-objective optimization problem with

target distance and landing time as objective functions is established. After analyzing the specific flight

problems and establishing a series of constraints, a suitable intelligent optimization algorithm-genetic

algorithm is selected to solve the multi-objective problems.

1 INTRODUCTION

In modern war, the key to determine victory or

defeat in a war is the rapid transfer and delivery of

military equipment and materials. The heavy load

airdrop with medium and large transport aircraft as

the transport platform has become an important

strategic action. The so-called heavy airdrop refers

to the use of large transport aircraft and parachute

landing equipment to quickly drop heavy weapons

and equipment from a certain height to the

designated ground. As the main way for airborne

troops to carry out airborne combat weapons,

ammunition, vehicles and other follow-up supplies,

it plays a key role in the deployment of airborne

troops in depth and breaking through the enemy

defense line.

The main factors that affect the safety and

accuracy of airdrop include the local meteorological

conditions (temperature, pressure, air density), the

position of the aircraft, the driving distance in the

cargo deck, the loss height of the cargo deck, the

steady descent rate of the cargo deck, and the time

from the cargo to the landing. With the new changes

in the battlefield environment, to meet the needs of

future airdrop operations, heavy air drop is being

given new requirements and new connotations,

among which the precision of heavy air drop point

will become a trend.

2 PROBLEM ANALYSIS

2.1 Analysis of Maximum Wake

Intensity

Assuming that the aircraft speed v, aircraft gravity

G, and air density ρ are all known. So, the lift

coefficient of the aircraft is

2

.

144

y

G

G

v

(1)

If the radius

w

r

and spacing of the vortex core

w

L

are known, the maximum intensity of the trailing

vortex

0

and the maximum tangential velocity of

the trailing vortex circumference

max

V

can be

obtained.

0

w

G

vL

(2)

0

2

max

w

V

r

(3)

2.2 Analysis of Wake Fully Formed

Position

The complete formation distance of the forward tail

vortex is

552

Yang, Y., Wu, X. and Wang, Z.

Research on Reloading Airdrop Strategy Based on Multi-Objective Programming.

DOI: 10.5220/0012287400003807

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

In Proceedings of the 2nd International Seminar on Artiﬁcial Intelligence, Networking and Information Technology (ANIT 2023), pages 552-555

ISBN: 978-989-758-677-4

Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.

13.85384

.

y

S

C

(4)

The initial sinking velocity of the wake vortex

is

0

.

2

d

w

V

L

(5)

Then the sinking height of the wake vortex can

be calculated as follow.

0

1

0.025

2

y

w

C

HS

L

(6)

Wake vortex formation time is

S

T

v

(7)

2.3 The Relation Between the Rear

Wing Tip and the Forward Tail

Vortex

The transverse distance between the section where

the aircraft is located and the boundary of the tail

vortex is

3

47 .

8

w

k lr fb

L

L L L

S

(8)

The sinking height of the forward tail vortex

at the rear wing tip is

1

1

0.025 .

y fb

fb

CL

H L V

v

下

(9)

The linear distance between the aircraft and

the tail vortex boundary is

2

2

1

.

k

r L H h

(10)

The intensity of the tail vortex attenuation to

the position of the rear wing tip is

0.05

2

0

2

2

,.

2

0.5

t

w

e

r

v r t

r

r r t

(11)

3 MODELING

3.1 Constructing Objective Function

In order to ensure that the front and rear cargo are

as close as possible, we can make the two aircraft

in the formation of heavy aircraft as close as

possible, that is,

2 2 2

min

lr fb

L L h

(12)

If the relative position vector of two aircraft is

T

lr fb

X L L h

， ，

, then the above formula can be

expressed as the L-2 norm of X.

2

min X

(13)

The difference between the delivery time of the

aircraft formation is

t

. In order to ensure that the

landing time of the cargo before and after the aircraft

is as close as possible, there is

min t

(14)

3.2 Constructing Constraints

Considering the safety of air delivery, the following

constraints are formulated.

1) The steady decline rate of the cargo platform

shall not exceed

8ms

, that is

8.v

(15)

2) The tangential wind speed of the cargo

platform subject to the wake shall not exceed

2ms

,

that is

0.05

2

0

2

2

, 2.

2

0.5

t

w

e

r

v r t

r

r r t

(16)

3) The relative distance between aircraft is

greater than 0, that is

2

0.X

(17)

3.3 Constructing Constraints

So, a multi-objective optimization model of

formation is obtained

2

min X

min t

s.t.

0.05

2

0

2

2

,2

2

0.5

t

w

e

r

v r t

r

r r t

(18)

2

0X

In order to obtain the solution, the linear

weighting method is used to transform the multi-

objective problem into a single-objective problem,

that is,

12

2

min Xt

(19)

Research on Reloading Airdrop Strategy Based on Multi-Objective Programming

553

4 SIMULATED ANNEALING

ALGORITHM

Simulated annealing algorithm is a heuristic

algorithm designed to randomly search the global

optimal solution in the feasible solution space by

combining probabilistic jump characteristics.

If the new feasible solution

j

x

is found to be

better than the current feasible solution

i

x

, the new

feasible solution is accepted. Otherwise, the

Metropolis criterion determines whether to accept

the new feasible solution. In order not to reject

directly, define the acceptance probability

P

.

P

lies between [0,1], and measures the distance

between

j

fx

and

i

fx

. The closer is the

distance, the larger is

P

. Here we make

assumptions.

exp

ji

P f x f x

(20)

In order to improve the efficiency of the

algorithm, in the early stage of the algorithm search,

it is necessary to improve the scope of the algorithm

search to avoid falling into local optimal. In the later

stage of the search, it is necessary to reduce the

search scope of the algorithm as much as possible.

That is, it just searches locally, because at this time

it is close to the global optimal. We make a

deformation of the above formula (20).

exp

t j i

P C f x f x

(21)

t

C

in the formula (21) can be regarded as a time-

dependent coefficient. Then the probability P of the

algorithm accepting the new feasible solution

establishes a relationship with the time parameter.

If t is small in the early stage of search, and the

search scope is large enough, then the corresponding

P needs to be larger. And

t

C

is set to be negatively

correlated with

P

, so it should be small. If

P

is

smaller in the late search period,

t

C

should be

larger. Obviously, the longer time goes, the bigger

t

C

gets.

The flow of the search process is as follows.

1) Generate an initial solution A randomly, and

calculate the objective function

fA

corresponding to the initial solution.

2) A solution B is generated near the initial

solution according to the probability mechanism,

and the objective function

fB

corresponding to

the new solution B is calculated.

3) If

f B f A

, the new solution overwrites

the original solution and repeat the above steps.

If

f B f A

, it calculates the probability of

accepting the newer solution B, that is

exp .

tt

P f B f A C

Then it randomly

generates number

0,1r

. If

rP

, the initial

solution A is overwritten by the new solution B. And

the above steps are repeated. Otherwise, it returns to

the second step. A newer solution

1

B

is re-generate

near the initial solution, and it continues to iterate.

However, there is a problem in the above

process, that is, the setting of key coefficient

t

C

. So

we define the initial temperature

0

100T

.

According to thermodynamics, the formula for

temperature drop is

1tt

TT

(22)

In the formula (22),

is usually 0.95, then the

temperature at time t is

0

100 0.95

tt

t

TT

(23)

To ensure that

t

C

increases about t, we have

11

100 0.95

t

t

t

C

T

(24)

Then

exp exp

100 0.95

t

t

t

f B f A f B f A

P

T

(25)

Let

f f B f A

, when the temperature

is constant, the smaller

f

is, the greater the

probability

t

P

is. That is, the smaller the difference

from the existing solution is, the greater the

possibility of accepting the newer solution is. When

f

is constant, the higher the temperature is, the

greater the acceptance probability is. Therefore, it is

easier to accept the newer solution when the

temperature is high in the early stage of search.

5 SIMULATION CALCULATION

The theoretical basis of Monte Carlo method is the

law of large numbers. The law of large number

describes the results of a considerable number of

repeated experiments, and according to this law, the

larger the number of samples, the closer the average

will be to the true value.

ANIT 2023 - The International Seminar on Artiﬁcial Intelligence, Networking and Information Technology

554

One type of Monte Carlo method is that the

problem can be converted into some random

distribution of characteristic numbers, such as the

probability of a random event, or the expected value

of a random variable. By random sampling method,

the probability of random events is estimated by the

frequency of occurrence, or the numerical

characteristics of random variables are estimated by

the numerical characteristics of sampling, and it is

used as the solution of the problem. Here the initial

solution is selected based on Monte Carlo

simulation.

Figure 1: Initial value solution selection based on Monte

Carlo simulation.

Under the condition that

0.989

w

r

,

38.84

w

L

,

3

130Ge

,

88.8v

, temperature is 18

0

C

and air

pressure is 1014 hpa. The optimization model is

solved by MATLAB, shown in figure2.

Figure 2: Optimization procedure.

The optimal function value obtained is 25.6396,

and the optimal solution is

8.459 0.51310.5666

T

X ， ，

.

The time difference between the front and back

engines is 17.1458s, which is in good agreement

with the actual situation.

ACKNOWLEDGMENTS

This work was financially supported by Dalian

Naval Academy research and innovation team fund

DJYKYKT2021-018 ， and student research fund

DJYKYKT2022-003.

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