Heuristic Methods for the Antenna-Constrained Beam Layout

Optimization on Multibeam Broadcasting Mission

Camille Lescuyer

, Christian Artigues

, Jean-Thomas Camino

and C

edric Pralet

Airbus Defence and Space, 31 Rue des Cosmonautes, 31402, Toulouse, France

CNRS, LAAS, 7 Avenue du Colonel Roche, F-31400, Toulouse, France

ONERA/DTIS, University of Toulouse, 2 Avenue Edouard Belin, 31400 Toulouse, France

Keywords:

Telecommunication Satellite, Television Broadcasting, Linguistic Beam, Heuristic, Graph Coloring.

Abstract:

In this paper, we tackle a payload design problem for a broadcasting mission where a telecommunication

satellite must provide television services to distinct regions deﬁned as polygons. To cover these polygons,

several telecommunication beams are emitted by the satellite, with the risk that they mutually degrade their

performance while also being hard to accommodate mechanically on the spacecraft. The problem is to de-

termine a set of non-conﬂicting beams that cover all the regions and optimize a performance metric related

to the sizes of the beams used. The ﬁrst method is a matheuristic exploiting iterative solv of an ILP model.

The second method, called the merge-and-split heuristic, is inspired by Iterated Local Search and reuses a fast

graph coloring algorithm to analyze conﬂicts among selected beams. These two methods are evaluated on

realistic instances, the largest one involving more than one hundred regions to cover.

1 INTRODUCTION

Telecommunication satellites are major assets used

by operators to provide coverage in remote areas at

a lower cost than terrestrial networks. As operators

require an increasing quantity and quality of the ser-

vices they provide, satellites must optimize their on-

board resources. Therefore, satellite manufacturers

need the support of optimization techniques to help

them ﬁnd payload design solutions that reach the mis-

sion requirements while complying with the full set of

operational and design constraints. One of these so-

lutions consists in using multiple beams to increase

the capacity without expanding the frequency spec-

trum, as the same frequency band can be used by sev-

eral beams. Moreover, using multiple beams requires

using dedicated antenna technologies such as Single-

Feed-Per-Beam (SFPB), which is cost-effective and

can reach high-performance radio frequencies. How-

ever, SFPB introduces beam layout constraints. Each

beam is created by a so-called feed horn (or feed, see

Fig. 1), whose size and shape can be computed be-

forehand given the target region to be served, and each

beam is allocated to a certain reﬂector (parabolic an-

tenna available onboard). As shown in Fig. 2, one

issue is that it can turn out that the positions required

for two feeds on the same reﬂector are not compatible

with each other for geometrical reasons if the target

regions are too close.

To optimize the design of telecommunication

satellites while considering beam layout constraints,

several approaches have been studied. (Kyrgia-

zos et al., 2013) explores non-uniform beam sizes

and bandwidth allocation among beams to maximize

inter-beam distance. (Camino, 2017) incorporates

antenna constraints into a non-uniform beam lay-

out, leveraging graph coloring techniques for beam-

to-reﬂector allocation. A patent by (Hammill and

Dishaw, 2004) presents a method for generating non-

uniform beams based on population density within

speciﬁc polygons. (Contardo and Hertz, 2021) pro-

poses an exact algorithm to cover polygons with a set

of disks, but do not address the beam-to-reﬂector al-

location problem.

In this paper, we introduce two heuristic meth-

Figure 1: Beams of differ-

ent sizes and their associ-

ated feeds.

Figure 2: A conﬂict be-

tween two feeds.

294

Lescuyer, C., Artigues, C., Camino, J. and Pralet, C.

Heuristic Methods for the Antenna-Constrained Beam Layout Optimization on Multibeam Broadcasting Mission.

DOI: 10.5220/0012380200003639

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

In Proceedings of the 13th International Conference on Operations Research and Enterprise Systems (ICORES 2024), pages 294-301

ISBN: 978-989-758-681-1; ISSN: 2184-4372

ods for beam layout design in SFPB antennas, aim-

ing to fulﬁll the requirements of telecommunication

missions. Our primary objective is to provide ser-

vice to as many requested regions as possible while

ensuring a satisfactory quality of service. In this de-

composed optimization scheme where the beam lay-

out is the ﬁrst step, the quality of service cannot be

accurately evaluated at this stage where important de-

sign steps subsequent to the beam layout optimiza-

tion are missing (frequency allocation, deﬁnition of

all payload routes, TV channels assignment, ampli-

ﬁers sizing, power distribution, etc.). However, we

can still create beams having a small size to maxi-

mize the power spectral densities and enhance signal

quality for ground receivers. In the end, our goal is

to select small beams that cover as many regions as

possible, while preventing feed conﬂict assignments

among beams allocated to the same reﬂector.

The rest of the paper is organized as follows: Sec-

tion 2 presents the formal problem deﬁnition using In-

teger Linear Programming (ILP). Section 3 introduces

a matheuristic approach that iteratively enlarges a set

of candidate beams. Section 4 describes a heuristic

algorithm called merge-and-split. Section 5 provides

experimental results on various instances. Finally,

Section 6 concludes the paper and outlines potential

future developments.

2 PROBLEM DEFINITION

2.1 Telecommunication Mission

The mission is deﬁned by a set of regions on the

Earth’s surface. The geometrical deﬁnition of a re-

gion is a polygon deﬁned by the set of successive ver-

tices placed on its boundary, and each region has its

own demand expressed in terms of number of televi-

sion channels. The segmentation of the market in ge-

ographical areas can be made for linguistic reasons or

because of the economical context. In the following,

the set of polygons to cover is referred to as P.

2.2 Beams

General Beams. We can deﬁne a beam as a disk

that covers a speciﬁc area on the Earth’s surface

served by the satellite communication system. More

formally, a beam is a pair b = (c

, r

) where c

the center of the beam (deﬁned by a longitude and

a latitude) and r

is the radius of the disk associated

with the beam, expressed in degree angle. For signal

quality reasons, the radius of all the beams considered

must not be greater than a maximum value referred to

as MaxRadius.

Smallest Covering Beams. In the following, we

consider speciﬁc kinds of beams that are directly de-

ﬁned by the set of polygons they cover. Using the

Welzl algorithm (Welzl, 2005), we can ﬁnd the small-

est beam b = (c

, r

) covering a subset P

′

⊆P in O(n)

time. Here, n is the number of polygons in P

′

. The

Smallest Covering Beam for a set P

′

⊆ P is denoted

as SCB(P

′

), and the covered polygons are denoted as

One combinatorial challenge arises: if a beam can

be deﬁned from any set P

′

⊆ P, then there are 2

|P|

candidate beams.

Conﬂicts Between Beams. As mentioned in the in-

troduction, the SFPB antenna is characterized by a

limited set of reﬂectors, usually three or four. Each re-

ﬂector is associated with a cluster of feeds, as shown

in Figure 2. Each feed is the hardware equipment as-

sociated with a single beam, and the reﬂector focuses

the signal and reﬂects the beam on the Earth’s surface.

Two beams whose disks are close on the Earth’s

surface and that are allocated to the same reﬂector

can create a conﬂict on their associated feeds. To pre-

vent conﬂicts, a minimum separation distance is im-

posed on the Earth’s surface. Formally, for two beams

= (c

, r

) and b

= (c

, r

) assigned to the same

reﬂector, the CONFLICT(., .) predicate is deﬁned as

follows:

CONFLICT(b

, b

) : dist(c

, c

) < κ(r

+ r

) (1)

Here, dist(c, c

′

) is the Euclidean distance between

points c and c

′

in the longitude-latitude plane, and

κ > 1 is a ﬁxed parameter determined by the satellite

manufacturer. In this context, κ =

√

3 is considered,

reﬂecting recent observations in related satellite man-

ufacturing activities.

From an operations research perspective, the beam

separation constraints can be represented using a

graph coloring problem. Indeed, let B

denote the set

of beams selected. We can build a graph G(B

) called

the feed conﬂict graph, containing one node per beam

in B

and one edge between two nodes associated with

beams b

, b

such that CONFLICT(b

, b

) takes value

true. Then, a set of beams is mechanically imple-

mentable if the chromatic number of G(B

), referred

to as γ(G(B

)), is less than or equal to the number of

reﬂectors, so that the beams can be distributed among

the different reﬂectors.

Heuristic Methods for the Antenna-Constrained Beam Layout Optimization on Multibeam Broadcasting Mission

295

2.3 Input Data

For a particular beam layout problem, we consider the

following input data:

• P = {p

, p

, . . . , p

}: set of polygons to cover;

• R = {1, 2, . . . , N

}: set of reﬂector indices;

• B = {b

, b

, . . . , b

}: set of candidate beams; the

radius of all these candidate beams is assumed to

be consistent with parameter MaxRadius;

• I ⊂ B ×B: pairs of beams which cannot be as-

signed to the same reﬂector (incompatible beams).

Moreover, for every p ∈ P, we denote as B

⊆ B the

set of beams in B that cover polygon p, that is the set

of beams b ∈ B such that p ∈ P

2.4 ILP Deﬁnition

Deﬁning a solution of the beam layout problem means

(1) selecting a subset of the candidate beams to cover

polygons, and (2) associating a reﬂector with each se-

lected beam so that the reﬂector-to-beam allocation

is feasible. Such a problem can be formalized as an

Integer Linear Program using the following variables:

• z

∈ {0, 1}, p ∈ P : binary variable taking value 1

if and only if polygon p is covered by a selected

beam;

• x

b,r

∈ {0, 1}, b ∈ B, r ∈ R : binary variable taking

value 1 if and only if beam b is allocated to reﬂec-

tor r.

The ILP model proposed is given in Equations 2 to 5.

maximize M ·

∑

p∈P

−

|B|

∑

b∈B,r∈R

b,r

(2)

subject to ∀p ∈P,

∑

(b,r)∈B

×R

b,r

≥ z

(3)

∀p ∈P, ∀b ∈ B

∑

r∈R

b,r

≤ z

(4)

∀r ∈ R, ∀(b

, b

) ∈ I, x

+ x

≤ 1 (5)

The objective function given in Equation 2 tries to

both maximize the number of polygons covered (term

∑

p∈P

) and minimize the size of the beams used

(term −

|B|

∑

b∈B,r∈R

b,r

). To express that the main

goal is to maximize the coverage of the polygons,

a large constant M = MaxRadius

is used to weight

the ﬁrst term, which leads to a lexicographic objec-

tive function. Constraint 3 ensures that if a polygon

is covered, then at least one of its covering beams is

selected. Constraint 4 ensures that if a polygon p

is not covered, then none of the beams covering p

is selected. As z

is a binary variable, it also states

that a beam is associated with at most one reﬂec-

tor. Last, Constraint 5 expresses that two conﬂicting

beams cannot use the same reﬂector.

2.5 Complexity

While we plan to prove the NP-hardness of the beam

layout problem considered, we can refer to closely

related problems from the literature that are already

known to be NP-hard. One notable similarity can

be observed with various NP-complete covering and

packing problems (Fowler et al., 1981). For example,

the 3-colorable unit disk covering problem involves

ﬁnding a set of unit disks and assigning a distinct

color to each disk such that the union of the disks

covers all given points, and overlapping disks have

different colors. This problem is proven NP-hard in

(Biedl et al., 2021). In another direction, the beam

layout problem addressed in (Camino, 2017) consid-

ers a set of points to be covered by beams, with each

point having a trafﬁc demand. The objective is to ﬁnd

a beam layout, including the beam-to-reﬂector assign-

ment, that maximizes the total trafﬁc. This problem is

proven NP-hard through a polynomial reduction from

the Circle-Covering problem. It is important to note

that our problem differs in that we aim to maximize

the number of polygons covered by beams of small

sizes, rather than maximizing the total trafﬁc associ-

ated with a set of covered points.

3 MATHEURISTIC METHOD

As mentioned before, enumerating all candidate

beams is exponential in the number of polygons and

is not practicable. For example, for an instance in-

volving only 20 polygons, there can be up to 1048576

possible subsets of polygons and as many potential

beams. This is why we propose a matheuristic method

that solves the ILP several times, on a restricted pool

of candidate beams B that evolves during the itera-

tions. Iterations are performed until a solution cover-

ing all the polygons is found or until a maximum CPU

time is reached. Also, one of the modelling issues

in the ILP presented in Section 2.4 is the fractional

part of the criteria in Equation 2 (

|B|

∑

b∈B,r∈R

b,r

referred as MSRS for Mean Squared Radius Sum in

this paper. The latter is fractional and difﬁcult to im-

plement, we will then use the following criterion in

this ﬁrst method:

maximize M ·

∑

p∈P

−

∑

b∈B,r∈R

b,r

with M =

∑

b∈B

+ 1

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296

3.1 Detailed Description

The matheuristic is provided in Algorithm 1. Ini-

tially, the set of candidate beams B only contains all

the smallest beams covering a single polygon in P

and all the smallest beams covering two polygons in

P. Based on these candidate beams, it is possible to

compute the set I containing the pairs of incompat-

ible beams and to solve the ILP model given before

for input data P, R, B, I. In Algorithm 1, this is

achieved through a call to a function solveILP, that

returns the set of beams B

selected in the solution

of the ILP presented in Section 2.4. More formally,

if y denotes the value of a variable y after optimiza-

tion, then B

← solveILP(P, R, B, I) is equivalent to

← {b ∈ B |

∑

r∈R

b,r

= 1}. Altogether, the beams

in B

cover a set of polygons P

= ∪

b∈B

. The algo-

rithm iteratively improves the current solution by gen-

erating a set of new beams to expand the set B while

there are uncovered polygons and available compu-

tation time. It updates beam incompatibilities and

solves the enlarged ILP model, potentially reﬁning the

beam selection strategy. The algorithm returns the last

beam selection found and, if there’s a maximum CPU

time for each solveILP call, it can also return the best

solution across iterations based on the objective func-

tion provided in Equation 2. Notably, in Algorithm 1,

cpuTime() denotes the current computation time, and

TimeLim represents the global CPU time limit.

Input: P: set of polygons to cover; R: set of

reﬂector indices

B ← initBeams(P);

I ← initIncomp(B);

← solveILP(P, R, B, I) ;

← ∪

b∈B

;

while P

̸= P and cpuTime() ≤ TimeLim do

← generateNewBeams(B

, P);

← generateNewIncomp(B

, B);

(B, I) ← (B ∪B

, I ∪I

);

← solveILP(P, R, B, I) ;

← ∪

b∈B

end

return B

Algorithm 1: Pseudocode of the matheuristic.

3.2 Beam Generation Heuristic

In the proposed matheuristic method, the main chal-

lenge lies in devising an effective strategy for gen-

erating beams that can be added to the current pool.

Several approaches have been tested, but in this para-

graph we only discuss the one that yielded the best

results thus far. The method involves enlarging the

pool of beams with new beams that cover subsets of

polygons with increasing cardinality during each it-

eration. To achieve this, we introduce a parameter

called NbPolyLimit. Initially, we set NbPolyLimit to

2, enabling the generation of beams that individually

cover one or two polygons. In each iteration, solving

the Integer Linear Programming (ILP) problem pro-

duces a set of selected beams denoted as B

. If B

does

not cover all the polygons, we increment NbPolyLimit

by one unit and generate beams b by merging two

beams from the solution. The resulting beams b cover

at most NbPolyLimit polygons and are added to the

pool. Formally, we compute

= {b = SCB(P

∪P

)| (6)

, b

∈ B

) ∧(r

≤ MaxRadius)

∧(|P

| ≤ NbPolyLimit)}

4 MERGE-AND-SPLIT

The ILP manipulated in the previous sections has lim-

itations on instances containing more than 100 poly-

gons. To overcome this limitation, we deﬁne a spe-

ciﬁc heuristic that is independent of the ILP formal-

ization, called the merge-and-split heuristic.

4.1 Global Description

Globally, the merge-and-split heuristic is inspired

from Iterated Local Search (Lourenc¸o et al., 2003),

which alternates between optimization phases where

local moves are performed to try and improve the cur-

rent solution (beam merging moves in our case), and

a perturbation phase where the features of the cur-

rent solution are randomly updated (split operations

in our case). The merging phase consists in replac-

ing two beams b

and b

by the smallest beam cov-

ering all the polygons covered by b

and b

. The

selection of the beams involves randomness to di-

versify the exploration of the search space. Beam

merging operations aim at providing a new conﬂict

graph that is easier to color than the current graph

and are performed until reaching a colorable graph

or until a maximum number of merging operations

is reached. At that point starts the splitting process

of some beams, where splitting a beam b covering a

set of polygons P

means replacing b with the set of

individual beams {SCB({p}) |p ∈ P

} covering each

polygon served by b. Each time a feasible solution has

been found, it is evaluated according to the MSRS cri-

teria. We ﬁx a total time limit, and we return the best

solution found at the end of the process. The merge-

and-split heuristic proposed is described in Figure 3.

Heuristic Methods for the Antenna-Constrained Beam Layout Optimization on Multibeam Broadcasting Mission

297

It starts from a set of beams B containing the small-

est covering beam associated with each polygon in P,

i.e. B = {SCB({p})|p ∈P}. By deﬁnition, this set of

beams covers all the polygons in P. In the following,

we detail the three main components of the algorithm,

that is the coloring, merging, and split procedures.

Figure 3: Merge-and-split algorithm.

4.2 Coloring Method

Each time the current set of beams B is updated, a

function needs to color the corresponding feed con-

ﬂict graph. This function is called many times during

the algorithm and needs to be fast, even if determin-

ing whether a graph can be colored using a restricted

number of colors is NP-complete. Therefore, we de-

cided to reuse DSATUR (Br

elaz,1979). DSATUR is

a greedy algorithm that colors nodes with the highest

degree ﬁrst. Each node is assigned the lowest feasible

color, considering its neighboring nodes’ colors. If

DSATUR manages to color the graph using no more

than N

colors, a consistent beam-to-reﬂector alloca-

tion exists. We denote the number of colors used by

DSATUR as

γ(G(B)), which serves as an upper bound

on the actual chromatic number γ(G(B)).

4.3 Merging Mechanisms

The pseudocode of the merging process is sketched

in Algorithm 2. As expressed in the condition of the

while loop, merging operations are performed while

the chromatic number of current feed conﬂict graph

G(B) exceeds N

and there is some computation time

left and the number of merge operations does not ex-

ceed a limit referred to as NbMergeMax

At each step, the merging loop selects a candidate

pair of beams (b

, b

) in Cand, according to a merg-

ing method randomly chosen (more details later on

this point). The merged beam is accepted if and only

if it is feasible according to parameter MaxRadius,

but also if the estimated chromatic number of G(B

′

)

with B

′

←(B \{b

, b

}) ∪{b

} is lower than or equal

to the estimated chromatic number of the feed conﬂict

graph G(B) (i.e.,

γ(G(B

′

)) ≤

γ(G(B))). Indeed, merg-

ing two beams can also create new edges in the feed

conﬂict graph. As illustrated in Figure 4, beam 10

created by merging beams 3 and 4 has a conﬂict with

beam 9 that was not in conﬂict with the two beams

merged. Finally, the set of candidate beams Cand is

updated to avoid selecting several times the same pair

of beams, and the number of merging operations is

incremented.

Input: B: set of beams in the current solution

NbMerge ← 0

Cand ← {(b

, b

)|b

, b

∈ B, b

≺ b

}

while

γ(G(B)) > N

and

Cand ̸=

0 and

cpuTime() ≤ TimeLim and

NbMerge ≤ NbMergeMax do

m ← select merging

method(ProbMergMethod)

, b

) ← select a pair in Cand given m

Cand ← Cand \{(b

, b

)}

← SCB(P

∪P

)

if r

≤ MaxRadius then

′

← B ∪{b

}\{b

, b

}

γ(G(B

′

)) ≤

γ(G(B)) then

B ← B

′

Cand←Cand∪{(b

, b)|b ∈B

′

\{b

}}

NbMerge ← NbMerge + 1

end

Algorithm 2: Merging loop.

In the following, we deﬁne three heuristics to se-

lect the two beams b

, b

to merge at each step.

Merging Method 1 (M1). The ﬁrst merging heuris-

tic favors the creation of small beams. To diver-

sify the search process, we select a pair (b

, b

)

leading to a merged beam b

that is among the

⌈|B|

× RatioSelectedBeams⌉ smallest ones. Here,

As we only merge beams two by two, the highest num-

ber of merging operations is always |P|−1. We consider a

smaller value for NbMergeMax in order to favor the explo-

ration of solutions containing a larger number of beams

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298

(a) Before merging.

(b) After merging.

Figure 4: Impact of beam merging on the feed conﬂict

graph.

RatioSelectedBeams ∈]0, 1] is a parameter that allows

us to control the number of candidate pairs considered

at each step, that is the degree of diversiﬁcation.

Merging Method 2 (M2). The second merging

heuristic favors the selection of two beams b

, b

that

have the highest number of common neighbors in

graph G(B). As illustrated in Figure 4b, the under-

lying idea is to make the coloring of these common

neighbors easier. In Figure 4, beams 3 and 4 have

four common neighbors, the merging of this pair of

beams deletes 10 edges and creates 7 ones, reducing

the total number of edges in the graph. Similarly to

M1, in order to diversify the search process, merging

method M2 randomly selects a pair of beams among

the ⌈|B|

×RatioSelectedBeams⌉ highest quality ones.

Merging Method 3 (M3). This third method ﬁrst

selects a beam b for which the index of the color as-

signed by DSATUR is strictly greater than the num-

ber of reﬂectors available. This beam is then merged

with a beam b

′

that is selected following either a rule

inspired from method M1, or a rule inspired from

method M2, or a rule that merges b with its closest

neighbor. The method for selecting the second beam

is chosen randomly among these three methods.

4.4 Splitting Mechanisms

The splitting procedure selects ⌈β ×|B|⌉ beams in B,

with a selection according to a probability distribution

that is proportional to the beam size. The procedure

can be called in two different cases:

• DSATUR manages to color the current set of

beams using no more than N

colors, then β = β

and should be low enough to try and improve the

current solution and not restart all over again, but

not too low to favor the exploration of other parts

of the search space

• NbMergeMax has been reached, then, β = β

and

should be higher in order to try and remove incon-

sistencies in the current solution.

4.5 Parameters

In the end, the merge-and-split heuristic uses sev-

eral parameters. TimeLim is the maximum duration

of the process; NbMergeMax is the maximum num-

ber of merging operations performed before splitting

some beams; ProbMergMethod = [p

, p

] is

the selection probability for each merging method;

RatioSelectedBeams is the proportion of beam pairs

among which the beam merging method selects a

good alternative; β

is the ratio of beams to split when

the current solution is colorable using no more than

colors; β

is the ratio of beams to split when the

current solution is not feasible.

5 EXPERIMENTS

Experimental Setup. The matheuristic and merge-

and-split methods have been implemented in Python.

The ILP problems have been solved using CPLEX

12.10. The runs were made on a server with 96 cores

of an Intel(R) Xeon(R) Gold 5318Y CPU @2.10GHz

processor and 62GB of RAM. CPLEX exploits all

96 cores, while the merge-and-split heuristic runs on

a unique core. This difference must be considered

in the analysis of the experimental results, but for

the heuristic our goal is to build a fast method any-

way. We consider a telecommunication satellite hav-

ing N

= 4 reﬂectors. The methods were tested on

six instances, corresponding to sets of polygons P

mapped on France, Italy, West Europe, and Central

Europe. The last two instances correspond to the Cen-

tral Europe instance split into two parts that are called

Central Europe Part1 and Central Europe Part2. If we

covered each of the polygons of the four instances us-

ing one beam per polygon, we would obtain a feed

conﬂict graph G

= (B

, E

) whose features are given

in Table 1.

Table 1: Features of the instances.

Instance Notation |P| |E

| γ(G

)

France F 22 144 10

Italy I 30 128 8

West Europe W 92 467 9

Central Europe C 103 893 12

C Part 1 C1 52 294 10

C Part 2 C2 51 402 11

To evaluate the efﬁciency of each algorithm, we

analyze the number of polygons covered by the solu-

tion found and the number of beams selected in this

Heuristic Methods for the Antenna-Constrained Beam Layout Optimization on Multibeam Broadcasting Mission

299

solution. We also analyze the Mean Squared Radius

Sum given by

MSRS =

∑

b∈B

where B

stands for the set of beams selected

in the ﬁnal solution. For the merge-and-split

heuristic, we use 0.2 for β

; 0.8 for β

;

[0.2, 0.7, 0.1] for ProbMergMethod and ﬁnally 0.2 for

RatioSelectedBeams.

Figure 5: Matheuristic result

on France.

Figure 6: Merge-and-split

heuristic on France.

Figure 7: Matheuristic result

on Italy.

Figure 8: Merge-and-split

heuristic on Italy.

Figure 9: Matheuristic result

on West Europe.

Figure 10: Merge-and-split

heuristic on West Europe.

Figure 11: Matheuristic re-

sult on Central Europe.

Figure 12: Merge-and-split

heuristic on Central Europe.

Results of the Matheuristic. The solutions found

by the matheuristic method are illustrated in Figures

5, 7, 9, 11, depicting the best sets of beams found,

each beam being colored according to the index of

the reﬂector to which it is assigned. From an opera-

tional point of view, the solutions appear to be of good

quality for the telecommunication satellite designers.

The detailed results of the matheuristic are given in

Table 2. In this table, we indicate the time limits of

the matheuristic method and the time limits speciﬁed

for the call to the ILP solver at each iteration (col-

umn ILP time limit). This parameter is set manually

following preliminary experiments. It is also worth

noting that the matheuristic method manages to ﬁnd

good solutions on all instances, but can have a long

computational time on the instance containing more

than 100 polygons, where the time limit is increased

to 1000 seconds.

Results of the Merge-and-Split Heuristic. Exam-

ples of solutions found by the merge-and-split heuris-

tic are given in Figures 6, 8, 10, 12. The detailed re-

sults are given in Table 2. Even if the solutions found

cover all polygons using a number of beams that is

rather low given the number of polygons to be cov-

ered, the merge-and-split heuristic ﬁnds several fea-

sible solutions during the process. Figure 13 displays

the evolution of the best solution’s quality for the West

Europe instance. Various TimeLimit values ranging

from 10s to 1000s (in increments of 10s) are consid-

ered. The number of selected beams (Figure 13a) and

the mean squared radius sum (Figure 13b) represent

the best values obtained for each run. The results in-

dicate that the improvement of both criteria becomes

less signiﬁcant as the total time limit is increased.

(a) Number of beams.

(b) Squared radius sum.

Figure 13: Impact of the time limit on the performance of

the merge-and-split heuristic (time limit on the x-axis).

Comparison Between the Two Methods. Given

the time limit to consider for each instance, the

matheuristic method outperforms the merge-and-split

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

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Table 2: Results of the matheuristic and the merge-and-split heuristic.

Instance

name

Time

limit

Matheuristic results Merge-and-split results

ILP time

limit

it.

Best solution

NbMergeMax Nb sol.

Best solution

| |B

| MSRS |P

| |B

| MSRS

F 100 10 7 22/22 7 0.062 5 8 22/22 7 0.069

I 10 5 3 30/30 12 0.017 5 88 30/30 15 0.027

W 200 50 3 92/92 37 0.022 20 14 92/92 21 0.49

C 1000 300 7 103/103 18 0.069 50 5 103/103 12 0.12

C1 50 10 4 52/52 19 0.037 5 15 52/52 18 0.38

C2 50 10 2 51/51 6 0.13 5 1 51/51 8 0.10

heuristic in terms of number of beams selected and

mean squared radius sum, except on instance C2.

However, the merge-and-split heuristic ﬁnds several

solutions covering all polygons, while the matheuris-

tic takes a few iterations to cover all polygons. The

heuristic method considers MSRS directly, while the

matheuristic focuses on SRS, as the formulation of

MSRS is not directly linear. We can show that the

matheuristic improves the SRS criterion from itera-

tion 2 while the MSRS is improved from iteration 2

to 3 on the Central Europe instance but then increases

until the end of the process.

6 CONCLUSION

In this paper, we considered a problem associated

with the design of a telecommunication satellite used

for television broadcasting on regions, for which a set

of beams of different sizes must be deﬁned to cover a

set of polygons, considering antenna mechanical con-

straints represented as a graph coloring problem. To

face the combinatorial issues and ﬁnd solutions that

are industrially feasible, we proposed two different

methods. The ﬁrst one is a matheuristic method that

is built upon an ILP formulation and uses an evolving

pool of candidate beams until ﬁnding a feasible solu-

tion. This matheuristic produces good-quality solu-

tions, but can require a long computational time. The

second method, called the merge-and-split heuristic,

iteratively constructs the beam layout by updating a

set of beams step-by-step through local merging op-

erations. This second method is faster and robust

to large scale instances. Several perspectives can be

listed for this work. For the matheuristic approach, we

could design other methods to ﬁll the pool with rele-

vant beams, and some unused beams could be deleted

to alleviate the ILP model. Moreover, to handle

the mean squared radius sum in the matheuristic in-

stead of the squared radius sum, we could use Linear-

fractional Programming. We could also try other cri-

teria, such as minimize the maximum radius or raise

the radius to higher exponents. For the merge-and-

split heuristic, we could parallelize the process to ben-

eﬁt from all cores and other merging methods should

be looked for, in particular to better identify the rea-

sons for the non-colorability at each merging step. A

last perspective is to deﬁne a hybrid method where the

matheuristic and the merge-and-split heuristic could

share solutions or sets of relevant beams.

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