Enhanced Optimal Beacon Placement for Indoor Positioning: A Set

Variable Based Constraint Programming Approach

Sven L

ofﬂer, Ilja Becker, Carlo B

uckert and Petra Hofstedt

Programming Languages and Compiler Construction, Brandenburg University of Technology Cottbus - Senftenberg,

Konrad-Wachsmann-Allee 5, 03046 Cottbus, Germany

{sven.loefﬂer, ilja.becker, carlo.bueckert, hofstedt}@b-tu.de

Keywords:

Beacon Placement, Constraint Programming, Indoor Positioning, Trilateration, Optimization Algorithms,

Decision Support Systems.

Abstract:

Indoor localization is of increasing importance in various environments, including hospitals, retirement homes,

and emergency situations. To achieve efﬁcient and accurate positioning of mobile individuals indoors, the opti-

mized distribution of sensors is crucial. The task of manually placing beacons (sensors) for indoor positioning

in a building can be challenging and time-consuming. Several researchers have tackled this issue using differ-

ent algorithms and considering various use cases. In our previous work (L

ofﬂer et al., 2022) at the ACS/IEEE

International Conference on Computer Systems and Applications (AICCSA 2022), we introduced a novel

approach that leverages constraint programming with exclusively Boolean variables to efﬁciently place Blue-

tooth Low Energy (BLE) beacons in indoor scenarios. We evaluated the quality of our results by comparing

them against manually optimized beacon placement and assessing their performance in a real-world building.

This paper extends the ﬁndings of (L

ofﬂer et al., 2022) by introducing a new constraint-based approach that

incorporates only set variables and Boolean variables, a more elaborate and balanced evaluation, i.e. on further

buildings, and certain reﬁnements of our overall method.

1 INTRODUCTION

Indoor localization is an increasingly important topic,

particularly in hospitals, retirement homes, and highly

industrialized work areas. The ability to quickly lo-

cate patients, disoriented individuals, or injured per-

sons can be crucial in emergency situations, poten-

tially saving lives. For example, in many medical fa-

cilities individuals are currently brought to the treat-

ment area well in advance of their scheduled appoint-

ments to ensure seamless execution of treatments

without time gaps. However, from the patient’s per-

spective, the resulting waiting time is often extremely

frustrating and can even have negative health effects.

In nursing homes and retirement facilities, where in-

dividuals have freedom of movement, staff members

often face challenges in locating patients in a timely

manner for treatment. By employing indoor tracking

systems, individuals can be located and promptly di-

rected to their destinations, reducing treatment wait-

ing times and improving overall circumstances. An-

other application of indoor location services is as-

sisting customers in navigating large public or pri-

vate buildings, such as libraries, parking lots, or of-

ﬁce complexes as well as workers in large industrial

plants, e.g. in rescue situations.

The availability of smaller Bluetooth Low Energy

(BLE) beacons with extended battery lifetimes, along

with the integration of Bluetooth and GPS technolo-

gies in even the most affordable smartphones, enables

rapid and reliable determination of positions in vari-

ous scenarios. Given that GPS signals are typically

unavailable or unreliable inside buildings, indoor po-

sitioning plays a crucial role in facilitating seamless

transition and navigation within such structures. In-

door positioning applications encompass indoor nav-

igation, asset tracking, people or personnel tracking,

and more. Among the prevalent techniques used in

indoor positioning algorithms, BLE beacons and sig-

nal strength measurements are commonly employed

for triangulation, trilateration, or simple proximity-

based approaches. This paper focuses on trilatera-

tion, a measurement method for determining the po-

sition of a point based on its distances to three refer-

ence points. The placement of beacons for trilatera-

tion should strive for optimality, aiming for minimal

beacon usage while maintaining efﬁcacy. The place-

ment and overall quantity of the beacons should be as

Löfﬂer, S., Becker, I., ückert, C. and Hofstedt, P.

Enhanced Optimal Beacon Placement for Indoor Positioning: A Set Variable Based Constraint Programming Approach.

DOI: 10.5220/0012203400003543

In Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2023) - Volume 1, pages 70-79

ISBN: 978-989-758-670-5; ISSN: 2184-2809

close to optimal as possible for these methods. More-

over, beacon placement should be cost-efﬁcient and

unobtrusive.

Presently, beacon placement is often performed

manually. Manual beacon placement is not only time-

consuming but also prone to errors. Even after ex-

tensive testing, there is no guarantee of complete

coverage throughout the entire building, nor certain

knowledge of a minimum (or acceptably small) num-

ber of beacons. With our new constraint-based ap-

proaches, assuming appropriately pessimistic param-

eter choices, we can ensure complete coverage of all

positions with a minimum number of beacons at the

same time. Our model can also be used to verify man-

ually created beacon placements for their coverage

adequacy.

In this paper, we present a new method to stream-

line and enhance the beacon placement process, lever-

aging constraint optimization and programming tech-

niques to compute optimal beacon positions suitable

for common indoor positioning algorithms. We have

modeled a constraint problem based on set variables

and boolean variables, which employs highly effec-

tive channeling constraints.

The remainder of this article is organized as fol-

lows. Section 2 provides an overview of constraint

programming fundamentals essential for our pro-

posed method. Section 3 reviews prior research on

BLE beacon placement and indoor localization. Sec-

tion 4 describes the beacons, hardware, and soft-

ware used in developing our method. The subsequent

section (Section 5) outlines the developed constraint

problem for beacon placement. Section 6 presents

experimental results obtained from real-world sce-

nario evaluations. Finally, we conclude the paper

and discuss potential future improvements to our new

method (Section 7).

2 BASICS OF CONSTRAINT

PROGRAMMING

Constraint programming (CP) is a powerful method

for declaratively modeling and solving NP prob-

lems. Common research areas in CP include roster-

ing, graph coloring, optimization, and satisﬁability

(SAT) problems (Marriott, 1998). In this section, we

provide the necessary deﬁnitions of constraint pro-

gramming for our approach, which can be summa-

rized in two steps:

1. Declarative representation of a problem as a

constraint model.

2. Independent solving of the constraint model by a

solver.

The CP user’s responsibility is to model the ap-

plication problem with constraints (1) and initiate the

solver, which runs independently like a black box (2).

We begin by introducing constraints based on ﬁ-

nite domain variables before expanding this concept

to include set variables.

Finite Domain Variables and Constraints

A constraint (X, R) consists of a relation R and an

ordered set of variables X on which the relation is de-

ﬁned (Dechter, 2003a). Examples include ({x, y}, x >

y), ({x, y, z}, x ∗y = z), or ({A, B}, A → B). In this pa-

per, we only specify the relation of a constraint since

the variables used in a constraint are clearly identiﬁ-

able.

A constraint satisfaction problem (CSP) is a 3-

tuple P = (X, D,C), where X = {x

, x

, . . . , x

} is a

set of variables, D = {D

, D

, . . ., D

} is a set of

ﬁnite domains where D

is the domain of x

, and

C = {c

, c

, . . . , c

} is a set of constraints involving

one or more variables from X (Apt, 2003a).

A solution to a CSP is an assignment of values d

from their corresponding domains D

to the variables

, satisfying all constraints. A constraint optimiza-

tion problem (COP) is an extension of a CSP, where

an optimization variable x

opt

identiﬁed as such will be

minimized or maximized.

The following COP 1 is an example for a COP,

which describes the problem of ﬁnding a rectangle

a × b with only integer values {1, 2, 3, 4, 5} for a and

b, which has a maximum ﬂoor area while at the same

time the perimeter must not be greater than 15.

COP 1 = (X, D,C) with X = {a, b, A, P}, D =

= D

= {1, 2, 3, 4, 5}, D

= {1, 2, ..., 25}, D

{1, 2, ..., 15}} and the constraints C = {(a ∗ b =

A), (2 ∗ a + 2 ∗ b ≤ P)}. The optimization variable is

A and should be maximized. An optimal solution of

this COP is a = 4, b = 3, A = 12, and P = 14.

A speciﬁc constraint relevant to the upcom-

ing work is the count constraint. The constraint

count(X, occ, v) limits the variables of the set X such

that the value v only occurs occ times (GCC, 2022;

van Hoeve and Katriel, 2006). For example, the

constraint count({x

, x

}, 3, 1) guarantees, that

the value 1 occurs exactly 3 times in the variables

, x

and x

. Instead of a constant, we also al-

low a set of constants as input for the occurrence

in the count constraint. In this case the occurrence

must be equal to a value in the set. For example

count({x

, x

}, {2, 3}, 1) is only satisﬁed, if

two or three of the variables are assigned the value 1.

Enhanced Optimal Beacon Placement for Indoor Positioning: A Set Variable Based Constraint Programming Approach

To solve constraint problems (CSPs and COPs),

so-called solvers usually use backtracking search

nested with propagation. Popular ﬁnite domain

solvers are among others Google OR tools (Per-

ron and Furnon, 2023), Gecode (Christian Schulte,

2019), JaCoP (Kuchcinski and Szymanek, 2013) or

the Choco solver (Prud’homme et al., 2017), with

the latter beeing used here. More information about

solvers and their mode of operation can be found in

(Rossi et al., 2006; Dechter, 2003b; Apt, 2003b).

Set Variables and Constraints

So far, we have only considered integer variables that

can take on a single value. However, going forward,

we will also allow set variables that can be associated

with a collection of values. A set variable x is a vari-

able that has a discrete domain D(x) = [lb(x), ub(x)],

where lb(x) is a set of values representing the lower

bound and ub(x) is a set of values representing an

upper bound for x. Thus, the domain of a set vari-

able consists mandatory elements (exactly lb(x)) and

possible elements beyond that, i.e. the elements of

ub(x) \ lb(x). The value assigned to x should be a

set s(x) such that lb(x) ⊆ s(x) ⊆ ub(x) (van Hoeve

and Katriel, 2006). An example of a set variable is x

with a domain D(x

) = {{}, {0, 1, 2}}. This indicates

that the set variable can take the following valid as-

signments: {}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2},

{0, 1, 2}. Thus, the variable x

is not required to con-

tain any value, but it can include any number of values

from the set {0, 1, 2}. If the lower bound were set to

{1}, it would mean that all assignments must include

the value 1: {1}, {0, 1}, {1, 2}, {0, 1, 2}.

Set variables are somtimes already considered in

ﬁnite domain constraint solvers like Choco. It is also

possible to handle set variables as set of Boolean vari-

ables. In this case for each value d

of each variable

a Boolean variable x

i, j

is created, which represent

whether the original set variable x

contains value d

i, j

= True) or not (x

i, j

= False).

Regardless of how set variables are handled by

the solver, they enable us to utilize additional useful

methods and constraints. For example, the method

setCard(IntVar c) sets the cardinality of a set var

equal to an integer variable c. On the other hand,

the boolsChanneling(B = [x

, x

, ..., x

], x

) con-

straint binds the set variable x

to the Boolean vari-

ables x

, x

, ..., x

, as described before (i ∈ x

⇔

B[i] = x

= True).

3 RELATED WORK

In this section, we delve into current research in the

areas of sensor technology, automatic beacon place-

ment, and indoor positioning.

3.1 Different Sensor Technology

There are two primary approaches to indoor local-

ization: infrastructure-based and infrastructure-less

methods (Taskan and Alemdar, 2021). Infrastructure-

less methods rely on environmental features (sound,

light, magnetic ﬁelds, or smartphone sensors) to ob-

tain location ﬁngerprints. Infrastructure-based meth-

ods leverage pre-installed visual sensors or vari-

ous wireless technologies like ZigBee (Fang et al.,

2012), WiFi, Ultra-Wideband (UWB), RFID (Zhang

et al., 2016), and BLE (Subedi and Pyun, 2020).

Infrastructure-based indoor positioning systems can

be costly due to required methodologies or expensive

hardware.

Among infrastructure-based technologies, BLE

has gained widespread adoption in ubiquitous com-

puting and IoT applications due to its low power con-

sumption and cost-effectiveness (Kuxdorf-Alkirata

et al., 2019). BLE beacons are portable, easy to de-

ploy, and offer high positioning accuracy. While Zig-

Bee consumes less energy, its support on smartphones

is limited. WiFi has broad support but higher energy

requirements compared to BLE (Subedi and Pyun,

2020). UWB, an emerging technology, is not yet sup-

ported on smartphones.

3.2 Automatic Beacon Placement

A signiﬁcant amount of research has been conducted

on the placement of beacons for indoor positioning

algorithms. In the following, we outline the most no-

table approaches.

The ﬁrst step in beacon placement is determin-

ing a metric to assess the quality of the placement.

One commonly used metric is the Geometric Dilu-

tion of Precision (GDOP). Originally used for navi-

gation satellites, GDOP quantiﬁes the spread of mea-

sured values and depends on the relative positions of

the satellites and the observer. In the work by (Ra-

jagopal et al., 2016), they introduce a method to adapt

GDOP for indoor spaces. This augmented GDOP is

then utilized in a toolchain to evaluate various bea-

con placement algorithms. The study reveals a sig-

niﬁcant reduction in the number of required beacons

compared to standard trilateration techniques.

One category of algorithms for position calcu-

lation relies on the Time-of-Flight (ToF) of signals

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

transmitted between the beacons and the target de-

vice. In the research conducted by (Wang et al.,

2019), beacon position optimization for such algo-

rithms is explored. They propose a greedy algorithm

that initially places O(OPT ln(m)) beacons, where m

represents the number of discrete location points in

the region and OPT denotes the size of the optimal

solution. Furthermore, a random sampling algorithm

is introduced, which reduces the number of required

beacons to O(OPT ln(OPT )). These algorithms, on

average, place between 6% to 23% fewer beacons

compared to prior works.

The study conducted by (McGuire et al., 2021) fo-

cuses on the self-localization of autonomous vehicles

in scenarios where traditional methods such as GPS

positioning are not available. In contrast to the pre-

viously mentioned approaches, they utilize the Angle-

of-Arrival (AoA) for position calculation. The inclu-

sion of the heading angle sets this method apart from

others and adds complexity to the resulting optimiza-

tion problem. The paper ﬁrst presents the determinant

of the Fisher information matrix for an arbitrary num-

ber of beacons. Subsequently, the optimal angular

separation for three beacons is analytically derived.

The optimality of the solution is then demonstrated

through numerical simulations.

In a different approach, (Sharma and Badarla,

2018) adopts a unique strategy by considering the

placement domain of beacons as a grid of candidate

locations on the surfaces of ceilings and walls in the

target indoor environment. They formulate the er-

ror propagation resulting from the geometric arrange-

ment between anchor beacons and target devices as

an optimization objective. To minimize the total bea-

con count while satisfying the GDOP constraint men-

tioned earlier, they employ Mixed Integer Linear Pro-

gramming (MILP). The proposed method was com-

pared against the conventional linear placement of

beacons, which assumes a planar geometry between

device and beacon locations. The results demon-

strated an improvement in the minimum GDOP while

maintaining the same number of placed beacons.

However, to the best of our knowledge, no other

work has employed a comparable speciﬁc constraint

programming approach for beacon placement using

solely Boolean variables or set variables.

3.3 Indoor Positioning

One of the main use cases for an automatic placement

of beacons in a building is indoor positioning. Hence

this section examines some of the current work re-

garding this use case.

Many of these systems also utilize information ob-

tained from nearby BLE beacons for indoor position-

ing. One such method is developed by (Toma

c and

Skrjanc, 2021). They introduce a novel visual-inertial

localization algorithm that automates the collection of

Bluetooth data required for positioning. This data is

subsequently utilized in a constrained nonlinear opti-

mization algorithm. The developed algorithm is im-

plemented on a smartphone, enabling real-time deter-

mination of beacon locations and the creation of path

loss models. The paper highlights the advancement

of their innovation by allowing users to assess if suf-

ﬁcient data has been computed for achieving the de-

sired positioning accuracy.

A distinct approach was explored by (Grottke and

Blankenbach, 2021). Their research employs an evo-

lutionary optimization strategy to implement an in-

door positioning algorithm, utilizing the Wireless Lo-

cal Area Network (WLAN), Bluetooth (BLE), and the

smartphone’s Inertial Measurement Unit (IMU). Each

new source of information modiﬁes the probability

values of particles through a probability-map-based

approach. Their results showed a reduction of the

maximum position error 31%, while the RMSE and

the 95-percentile positioning errors could be reduced

by 34% and 52% respectively.

The authors of (Amsters et al., 2019) conducted a

comprehensive evaluation of efﬁcient indoor position-

ing systems, considering factors such as cost and ac-

curacy. Their ﬁndings indicated signiﬁcant advance-

ments in the ﬁeld of indoor positioning. Simple in-

door positioning systems, based on cameras, can be

obtained for a few hundred dollars, while higher ac-

curacies can be achieved with technologies like HTC

Vive at a slightly higher cost. However, these systems

may not be scalable for larger environments. To ad-

dress this, the authors recommended systems such as

Marvelmind or Pozyx, which utilize ultrasound rang-

ing and Ultra Wide Band beacons, respectively. No-

tably, none of the recommended efﬁcient systems re-

lied on BLE beacons, underscoring the need for fur-

ther improvement in positioning using this technol-

ogy.

4 EXPERIMENTAL SETUP

In this section we brieﬂy examine the hardware that

was used to test the beacon placement in buildings

and introduce the software we used to implement our

algorithm. Finally, in this section we explain how our

input building plans look like. In order to be compara-

ble with (L

ofﬂer et al., 2022), we made no changes to

the setting, but simply included additional ﬂoor plans

in the evaluation.

Enhanced Optimal Beacon Placement for Indoor Positioning: A Set Variable Based Constraint Programming Approach

4.1 Hardware and Software

The system uses BLE beacons from the company

blukii called Smart Beacon Go Mini. The beacons are

based on Bluetooth 4.2 (Bluetooth Low Energy), use

a Texas Instruments CC2640 controller and are pow-

ered by a CR2032 battery (providing up to a half year

of runtime when using a 1 second advertising inter-

val). The beacons support the iBeacon and Eddystone

protocols and have a conﬁgurable advertising interval

of 0.1 to 1 second while providing a range of up to 50

meters.

Different smartphones were used to check the sig-

nal strengths in the building from (L

ofﬂer et al., 2022)

at different locations and to make test and calibration

measurements. The main smartphone used for tak-

ing measurements was a Google Pixel 3a running An-

droid 12 with Bluetooth 5.0 capabilities.

The software for calculating the optimal beacon

placement was written in Java and with the help of

the Choco solver (Prud’homme et al., 2017) library.

No further software was necessary. It is possible

to replace Java and the Choco solver by any other

programming language and constraint solver. The

method presented in the following Section 5 is inde-

pendent of the used software.

4.2 Building Plans

The building plans for the ﬂoors of the considered

building were available in the PDF ﬁle format. As

an extension to (L

ofﬂer et al., 2022), we considered

3 more ﬂoor plans (see Figures 1 to 3) from typical

school buildings in East Germany as tests for the sub-

sequent beacon positioning. We converted the build-

ing from (L

ofﬂer et al., 2022) and the three new build-

ings to PNG ﬁles with 1,000,000 to 16,000,000 pix-

els, such that every pixel represents an area of 4cm

by 4cm. We differentiate between different wall types

and areas within the ﬂoor plans: massive walls, dry-

walls, triple glazed windows, free spaces which must

be covered by beacons and free spaces which do not

need beacon coverage. The different wall types have

an inﬂuence on the range of the individual beacons

and are considered when later calculating the posi-

tions of the beacons.

The selected ﬂoor plans have deliberately been

chosen to be highly diverse. Thus, the ﬂoor plan of

the school in Dresden (Figure 2) consists of only one

building. In contrast, the schools in Dahlewitz (Figure

1) and Potsdam (Figure 3) consist of two buildings,

which are connected by a thick corridor (Dahlewitz)

or three small corridors (Potsdam). We have chosen

the same resolution of 4 cm by 4 cm for the three new

Figure 1: The ﬂoor plan of a school in Dahlewitz (Sch,

1999).

Figure 2: The ﬂoor plan of a school in Dresden (Sch, 1999).

ﬂoor plans as the ﬂoor plan of the building in (L

ofﬂer

et al., 2022). Similarly, all other parameters are se-

lected in a consistent manner.

5 METHOD

In this section, we brieﬂy explain the approach from

ofﬂer et al., 2022) and elaborate on our new con-

straint modeling.

5.1 The General Approach

Figure 4 gives an overview over the developed pro-

cess which is explained in the following subsections.

Figure 3: The ﬂoor plan of a school in Potsdam (Sch, 1999).

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

Map of the building

2-D env. resistance A

and reachability arrays A

Preprocessing

Scaled env. resistance A

and reachability arrays A

Different scalings s = 4, 5, 10, 15, ...,54, 60

Beacon coverage set S

RSSI based range calculation

Scaled COP

COP construction

P = (X , D,C, min(numOfBeacons))

Optimized COP

COP reﬁnements

opt

= (X , D,C, min(numOfBeacons))

Beacon positioning

COP solving

Figure 4: Overview of the beacon positioning process from

ofﬂer et al., 2022).

First, the input building plan is preprocessed to

capture all relevant wall properties, including wall

thicknesses and wall types. This process generates

two arrays, one (A

) reﬂecting whether a region

should be covered by beacons (areas within the build-

ing) or not (areas outside the building, walls, etc.),

and a second one (A

) indicating the environmental

resistance for beacon signals in each region (environ-

ment factor E). These arrays are then scaled to dif-

ferent sizes. Different scalings ensure whether the in-

put image is accurately reconstructed or abstracted.

Without scaling, each 4cm x 4cm pixel would be di-

rectly taken into account. By scaling, multiple pixels

are merged together. Depending on the scaling, the

resulting COP arises with a varying number of vari-

ables and constraints, which needs different solution

times. The coverage of each scaled region by placing

a beacon in each area is determined (Beacon coverage

set S) by use of Equation 1 and serves as the basis for

the Constraint Optimization Problem (COP). Before

solving the COP, various reﬁnements are performed

on it to minimize the solution time. Finally, the COP

is solved with the objective of minimizing the total

number of beacons used.

distance = 10

−

RSSI

10∗E

(1)

The COP modelling starts with a very large scal-

ing factor, resulting in a low level of detail and a

small generated COP. This COP is solved consider-

ing a speciﬁc time limit, and the best solution found

is recorded. Subsequently, the scaling factor is re-

duced, and a new COP is created. The new COP, due

to its higher resolution, exhibits a higher level of de-

tail but also generates a larger COP. Since all calcu-

lations resulting from scaling are rounded pessimisti-

cally, COPs with a low grid density are the quickest

to solve due to its small size. However, it may lead to

inferior solutions compared to COPs with higher grid

density, which, due to a higher number of variables

and constraints, requires more time for solution com-

putation. This approach allows for quickly ﬁnding a

good solution and, over time, discovering even better

solutions.

In comparison to (L

ofﬂer et al., 2022), it turned

out that we were overly pessimistic by introducing

a dynamic component in the calculation of the en-

vironmental factor E for the RSSI value. Thus, we

only consider static values for E: 2 (freeSpaces), 2.5

(drywalls), 4.5 (massiveWalls), and 10 (glass) and re-

moved the dynamic part which depends on the wall

thickness.

The determination of the environmental factor re-

mains pessimistic, as we always consider the highest

occurring value along a given path. For instance, if

Enhanced Optimal Beacon Placement for Indoor Positioning: A Set Variable Based Constraint Programming Approach

there is a massive wall (E = 4.5) between the current

position and a beacon, while the rest of the space is

unobstructed (E = 2), the value of 4.5 will still be

assumed for the entire distance in the environmen-

tal factor calculation. The constant pessimistic esti-

mation ensures that every calculated solution corre-

sponds to a reliable real-world solution. For more

details about the general approach, please refer to

ofﬂer et al., 2022).

5.2 The Constraint Optimization

Problem

We use constraint programming to mathematically

model and optimize the beacon placement. Therefore,

we create a COP that describes the conditions for the

beacon placement (using trilateration), i.e. solutions

of the COP are all possible placements of beacons. An

optimization is then carried out over this very large

number of variants. This is realized by adding an ob-

jective function (minimize(x

count

)), such that we get a

COP.

In (L

ofﬂer et al., 2022) a COP based on boolean

variables and count constraints was created. This

paper presents a new approach which is based on

set variables, boolean variables, and channelling con-

straints between them. The aim of the COP is to

achieve a minimum number of beacons while com-

pletely covering every point in the building with (at

least three) radio signals. The new COP is described

below and given in Figure 5.

For our set variable based COP we work with an

additional grid G3 which represents the input map.

Each grid element in G3 has a size of 3m by 3m.

Therefore, the beacons we want to position can only

be placed at locations that are at least three me-

ters apart horizontally and vertically from each other.

Keep in mind that the original map can have a pixel

size of 4 cm by 4 cm. This means that a beacon can

only be placed at every 75th position in both hori-

zontal and vertical directions (0.04m ∗ 75 = 3m). So,

while the size of our scaled map is determined by the

scaling factor s and has n × m grid elements, the grid

G3 always has a size of n

× m

, where n

is the

length of the building (in meter) divided by 3 and m

is the width of the building (in meter) divided by 3.

The feasible positions for beacon placement, denoted

as S

x,y

, only need to contain positions that fall within

our grid G3.

For each position in the grid G3, we create a

boolean variable x

i, j

with a domain D

i, j

= {0, 1}.

Additionally, we have an optimization variable x

count

with a domain D

count

= {3, 4, ..., n

∗m

}, and a set

variable x

i, j

for every position on the scaled map. The

domain of each set variable x

i, j

consists of the posi-

tions in the set S

i, j

. When a boolean variable x

i, j

is set

to true, it indicates that there is a beacon at position

(i, j). If a set variable x

i, j

contains the value (a, b),

it means that the position (i, j) is covered by a signal

from a beacon at position (a, b).

A constraint (|D

i, j

| ≥ 3) is created for each set

variable to ensure that every set domain has at least

three values, guaranteeing that each position on the

map is covered by at least three beacons. Addition-

ally, we need to ensure that a set domain only con-

tains a value (i, j) if there is a beacon at position (i, j),

meaning the boolean variable x

i, j

is set to true. Fortu-

nately, the boolsChanneling constraint, described in

Section 2, precisely fulﬁlls this requirement. There-

fore, the constraint speciﬁcally requires the boolean

variables representing positions in S

i, j

and the set

variable x

i, j

as inputs. All of this collectively forms

the constraint optimization problem shown in Fig-

ure 5.

We used the same optimizations to improve the

solution speed of the COP as in (L

ofﬂer et al., 2022).

This includes different scalings, the use of paralleliza-

tion through a portfolio approach, and the addition of

additional constraints.

6 RESULTS

In this section, we evaluate our new constraint-

based method by comparing the maximum numbers

of beacons calculated using the constraint optimiza-

tion method and its solution times against the COP

in (L

ofﬂer et al., 2022), and two greedy-based ap-

proaches.

The best solution for the building in (L

ofﬂer et al.,

2022) we could ﬁnd using our boolean variable-based

constraint modeling approach, is illustrated in Fig-

ure 6. Similarly, the best solution we could ﬁnd for

the set variable-based constraint modeling approach

is shown in Figure 7. The purple squares indicate the

areas where the placement of beacons is required to

ensure that at least three beacon signals cover each

position on the entire building level. Due to our pes-

simistic approach, a purple square divided by a wall

signals that the beacon can be placed on either side

and still ensure triple coverage.

The solution of the boolean-based COP uses 15

beacons and the solution of the new set variable-based

COP uses 18 beacons, which is ﬁve and two beacons

fewer than what we needed for our best manually

crafted solution. We thoroughly examined for both

solutions the entire ﬂoor and conﬁrmed that every

area consistently received signals from at least three

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

P = (X, D,C, f ) with:

X = {x

i, j

| ∀i ∈ {1, ..., n

}, j ∈ {1, ..., m

}} ∪ (one Boolean variable for each position (i, j) in the grid G3)

i, j

| ∀i ∈ {1, ..., n}, j ∈ {1, ..., m}} ∪ (one set variable for each position (i, j) in the n × m map)

count

} (a beacon counting variable)

D = {D

i, j

= {0, 1} | ∀i ∈ {1, ..., n}, j ∈ {1, ..., m}} ∪ (at position (i, j) is a beacon (1) or not (0))

count

= {3, ..., n

∗ m

}} ∪ (maximal number of beacons is between 3 and n ∗ m)

i, j

= S

i, j

| ∀i ∈ {1, ..., n}, j ∈ {1, ..., m}}

(every position (i, j) can be covered by all beacons with position in S

i, j

)

C = {|D

i, j

| ≥ 3 | ∀i ∈ {1, ..., n}, j ∈ {1, ..., m}} ∪ (every position (i, j) is covered by at least 3 beacons)

{boolsChanneling(boolvars(S

i, j

), x

i, j

) | ∀i ∈ {1, ..., n}, j ∈ {1, ..., m}} ∪

(if there is a beacon at position (i, j) then all set variables

corresponding to a position in S

i, j

contain the position (i, j))

{count({x

i, j

| ∀i ∈ {1, ..., n}, j ∈ {1, ..., m}}, x

count

, 1)} (sum up the used beacons)

minimize(x

count

)! (minimize the number of used beacons)

Figure 5: The set variable COP which represents the beacon positioning problem.

Figure 6: The best solution found with the boolean variable-

based approach for the building from (L

ofﬂer et al., 2022):

15 beacons.

different beacons. The improvement, which is three

beacons better compared to (L

ofﬂer et al., 2022), is

attributed to the removal of the dynamic component in

the RSSI calculation. This assumption, as mentioned

before, was overly pessimistic.

To enhance the comprehension and evaluation of

the effectiveness of our method, we conducted a com-

parative analysis with the two greedy approaches pre-

sented in (L

ofﬂer et al., 2022). The ﬁrst greedy ap-

proach prioritizes covering large areas in the mid-

dle of the buildings initially and subsequently focuses

on the boundaries of the buildings. Consequently,

the ﬁrst set of beacons covers signiﬁcant portions of

the building, but unfortunately, the borders are ini-

tially overlooked. This drawback results in the algo-

Figure 7: The best solution found with the set variable-

based approach for the building from (L

ofﬂer et al., 2022):

18 beacons.

rithm requiring a greater number of beacons to cover

smaller areas at the borders of the building towards

the end. The second algorithm assigns higher weights

to hard-to-reach places such as corners, ensuring they

are covered earlier. This approach results in fewer

critical areas remaining where a larger number of bea-

cons would be required at the end.

Both greedy algorithms perform signiﬁcantly

worse than the constraint-based approaches and yield

inferior solutions. For the building in (L

ofﬂer et al.,

2022), the ﬁrst greedy algorithm requires 27 beacons,

while the second greedy algorithm requires 28 bea-

cons. In comparison, the constraint-based approaches

outperform both greedy algorithms by achieving bet-

ter results.

Enhanced Optimal Beacon Placement for Indoor Positioning: A Set Variable Based Constraint Programming Approach

Table 1: A comparison of the results between the two constraint-based and greedy-based approaches.

Boolean COP Set COP Greedy 1 Greedy 2

Beacons Time Beacons Time Beacons Time Beacons Time

BTU Cottbus 15 207s 18 0.12s 27 0.08s 28 0.06s

Dahlewitz 9 185s 9 7.93s 11 0.01s 10 0,01s

Dresden 9 0.03s 10 0.02s 11 0.01s 12 0,01

Potsdam 43 168s 48 0.08s 50 0.48s 49 0.5s

Table 1 shows solutions with the fewest beacons

found by the various approaches for different build-

ings, along with their corresponding solution times.

It can be observed that the boolean variable-based ap-

proach from (L

ofﬂer et al., 2022) consistently ﬁnds

the best solution, requiring the fewest number of bea-

cons. The set variable-based approach, while only

achieving an equally good solution for the school in

Dahlewitz, quickly ﬁnds a good solution (in 3 out of

4 cases in under 1s and in one case in under 8s). The

runtime of each COP was limited to 10 minutes, and

the same solver settings were used. The two greedy

algorithms quickly ﬁnd a solution (always under 1s),

but in this comparison, the solutions they ﬁnd are con-

sistently the worst.

Of course, trying our approaches for only four

buildings against two simple greedy algorithms is no

proof that our constraint approach always ﬁnds an op-

timal solution, but it can be seen as conﬁrmation that

our approach poses a valid alternative. Furthermore, it

was demonstrated that our new constraint model (the

set variable-based COP) is capable of solving such

placement problems with good results in very short

time.

The greedy approaches compute only one solu-

tion each, while the COPs, on the other hand, com-

pute multiple solutions, from which they then select

the best one, resulting in a slightly longer computa-

tion time. However, the computation times are mini-

mal, and it can be expected or at least presumed that

this scalability will hold true for larger, more complex

buildings, as the beacon placements are ultimately

only inﬂuenced locally due to the layout of corridors

and rooms within the building.

For highly complex problem instances, two dif-

ferent scaling approaches can be employed. One is

already inherent in our developed process, involving

various scaling factors for environmental resistance

and reachability arrays A

. However, this scal-

ing approach is limited due to the restricted range of

the beacons. Since the range cannot be scaled accord-

ingly, in the more extensively scaled arrays, only a

few areas are covered, resulting in signiﬁcantly re-

duced accuracy of the overall computation and, due

to the pessimistic perspective, inferior outcomes.

The alternative approach to scaling large layouts

would be their division into sectors, with individ-

ual coverage calculations. At the interfaces between

two sectors, there may potentially arise areas of fre-

quent overlap, which could unnecessarily increase the

overall number of required beacons. Nevertheless, it

should still be possible to quickly achieve very good

solutions using this method.

7 CONCLUSION AND FUTURE

WORK

We introduced a new method for ﬁnding optimal so-

lutions to a sensor beacon placement problem for in-

door positioning. This method utilizes Boolean vari-

ables and set variables with Boolean channeling con-

straints. We conducted additional experiments to

determine environmental factors for different types

of walls, leading to the realization that the previ-

ously considered dynamic component was overly pes-

simistic and could be eliminated. We then modeled

and solved a new representative Constraint Optimiza-

tion Problem (COP) for the issue and outlined meth-

ods to improve the solution process. Our approach

was tested on ﬂoor plans from four different buildings

and compared against the approach of (L

ofﬂer et al.,

2022) and two greedy algorithms. The results demon-

strated the effectiveness of our method in achieving

good solutions fast.

Future work involves expanding our experiments

to include more buildings, conducting an experimen-

tal analysis of additional environmental factors for

different wall types, incorporating additional con-

straints to accelerate the solution process, enhancing

the signal strength estimation, and exploring the uti-

lization of a SAT solver. Additionally, we are inter-

ested in studying the interaction of beacons across dif-

ferent ﬂoors and in vertical spaces such as stairwells.

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Enhanced Optimal Beacon Placement for Indoor Positioning: A Set Variable Based Constraint Programming Approach