Grobelny J., Michalski R. (2022) Linguistic patterns as a framework for an expert knowledge representation in agent movement simulation, KnowledgeBased Systems, 243, 108497. Cited (JCR): 0, Other Cites: 0 IF:8.8 5yIF: Pt:200
Abstract
The article shows how the concept of linguistic patterns (LP) combined with expert knowledge can be used to model the dynamics of social groups. Linguistic expressions similar to natural language are the basis for generating virtual forces (VF) that govern the movements of the individual agent. Using logical sentences and providing the methodology to determine the degree of their truth led to the development of rules of agent behavior. Such an approach allows for constructing flexible simulation models. This paper describes this new idea in detail and illustrates its capabilities and properties by simple examples. Furthermore, a series of simulation experiments involving problems of known solutions with stable final configurations is performed and qualitatively analyzed. For additional validation, a suboptimal scattered plot generation method proposed by Drezner is used. A set of simulations for classic problems showed the convergence of agent dynamic behavior in the proposed framework with the solutions provided by the Drezner approach.
1. Introduction
Agent movement simulations are used to model complex systems and
observe their overall behavior based on local relations and interactions
of their components. Such a way of modelling has a rich tradition and
has been widely applied to explore the dynamics of social as well as
abstract systems. Studying segregation problems has greatly contributed
to the development of this trend.
This current article presents a novel conceptual framework for a
study of agent movement within relatively small social groups. In
general, the fundamental objective of this concept is to find a stable
spatial configuration of groups of agents. The mutual attitudes of the
agent pairs are known and represented in the form of the so–called
Moreno matrix. This matrix may be defined by an expert or researcher in
a flexible manner by means of linguistic expressions that describe the
degree of acceptance or antipathy. The agents can constantly move across
the plane starting from a randomly generated layout. This movement is
driven by success. The rules controlling the movement and each
step payoff are also defined by natural languagelike phrases formulated
according to an expert knowledge. Such expressions are called linguistic
patterns (LPs). The virtual social forces generated by these patterns
exert an influence on agents and affect their behavior.
In specific application problems, one has to create LPs that reflect
logical relationships and describe the desired state of the examined
system in reality. Since a formal description by classic mathematical
formulae may be difficult due to the information uncertainty, the fuzzy
sets and LPs appear to be well fitted to this job. The determination of
the appropriate patterns can be obtained, for instance, by finding a
consensus between knowledge of different experts within the given field
or in concrete situations.
This paper was inspired by works on modeling migration behavior
[1]–[3] and the initial concept of linguistic patterns (LPs) proposed in
[4], [5]. Although the presented idea shares some assumptions and
properties similar to those of previous works, there are several
profound differences. The novelty and specific distinctive features of
our proposal that constitute a contribution of this research are
highlighted below.
To the best of our knowledge, the LPs and linguistic variables,
which are at the core of our study, have not been previously used in
this research area. This novel solution, which allows in a flexible way
to encode and apply expert knowledge, made it possible to resign from
rigid mathematical relations that describe the agents’
behavior.
In other studies, regular grids are usually used to model
migration behavior. Their particular cells specify possible agents’
locations, which is a substantial restriction. In our approach, it is
possible to place agents at any point in an unconstrained plane. It is
feasible thanks to exploiting an analogy to physical attractive and
repulsive forces.
The resulting flexibility and the freedom of agents’ movement
allow for a significant extension of the proposed approach. Our method
goes far beyond the analysis of agent behavior in the classical sense.
It is, for instance, possible to obtain the socalled scattered plots
which is unfeasible in classical migration models due to the mentioned
limitations. Our approach enables the construction and analysis of such
scattered plots for a variety of criteria and tasks. Thus, it can be
used much more widely used in economic and social practice than
traditional techniques. In this paper, the proposed idea is presented
along with simple exemplary models that show some of its possible
applications and highlight its potential flexibility.
In the context of searching the space of possible solutions to
find the best one, the assessment criteria defined in the form of LPs
provide an additional advantage over other methods. According to the
axioms of fuzzy set theory and multivalued logic, the proposed criterion
of the mean truth of LP fulfillment cannot exceed the value of one for a
specific solution. This feature is not available in classic approaches.
Therefore, when the maximum value is obtained, one can be sure that
there exists no better solution. There can be, however, some other
equivalently good configurations, given specific LPs
expressions.
The remainder of the paper is organized as follows. First, we review
the relevant literature that outline the background of our proposal and
highlight differences with other studies. Next, in Section 3, we present
our approach and explain its relations with fuzzy set theory and
concepts of LPs. Their application to define our virtual forces and
evaluation criteria are illustrated with party participant
configurations inspired by [6]. The following section demonstrates the
application of our methodology to classic migration simulations
introduced by Sakoda [1] and Schelling [2], [3]. The final stable
configurations generated by the proposed LPbased method take the form
of scattered plots. Therefore, in Section 6, we show and
compare them with solutions provided by suboptimal eigenvectorbased
scattered plots, generated according to the classic Drezner [7]
idea. Then, we further discuss the simulation results obtained in
illustrative examples and provide a broader scope of potential
applications of our proposal’s potential applications. In addition, we
outline possible future extensions and research directions. The article
ends with concise conclusions.
2. Background and literature review
2.1. Agentbased modeling
Agentbased modeling (ABM) is still very popular, with new approaches
constantly being explored and developed in various fields. For example,
such techniques are currently being used to model the dynamics of online
social networks. An extensive discussion on the use of agentbased
methods in this area is presented by Lymperopoulos and Ioannou [8]. Some
of the recent advances in modeling social phenomena are discussed in
[9]. Castro et al. [10] reviewed more than 60 proposals of using
agentbased models to determine climate mitigation policies. Agents are
also employed in systemsofsystems analysis. Recent developments in
this area are described by Silva and Braga [11].
Other exemplary reviews on agentbased modeling concern product
lifecycle [12], collective intelligence [13], education [14], decision
making in manufacturing [15], sociotechnical energy transitions [16],
and even mosquito behavior and disease [17]. The ongoing pandemic led
some researchers to also focus on this type of modeling technique, for
instance, [18], and a review in this field [19]. North [20] gives some
theoretical aspects of agentbased models along with the analysis of
recent free software environments that support this approach.
2.2. Moving agentbased modeling with crisp knowledge representation
A special case of ABM is moving agentbased modeling (MABM). Since it
constitutes the direct background for our proposal, we provide a brief
review of the key advances and the most important research trends in
this area.
The macrolevel analysis of social group behavior based on
interactions and local displacements of agents living on a
checkboard was suggested by Sakoda [1]. In his concept, agents inhabit
cells of an 8×8 checkboard. The mutual attitude for each pair of agents
can be defined as neutral (0), positive (+1), or negative (–1). Because
not all cells are occupied, in subsequent steps, each agent can move to
a neighboring free place where it would feel better than at the location
he left. The movements of the individual agent in a given step are
restricted to a 3×3 square with the current agent position at the
center. The feeling in a particular place is defined by
valence, which is computed as the sum of attitude indicators
between the moving agent and all other agents divided by the sum of
distances between them. The main objective of Sakoda’s analysis was to
identify group formation processes by analyzing various parameters of
the model, particularly segregation and suspicion. The Sakoda concept
represents the trend in modeling the dynamics of social system dynamics
based on cellular automata (CA) theory. The proposed checkboard model
and the rules of agent interactions in the local environment (Moore’s
neighbourhood in CA theory) clearly refer to the idea of CA. The
history, potential, and advantages of implementation of this methodology
in the dynamics of analysis of the social systems dynamics have been
discussed, for instance, by Hegselmann and Flache [21].
Schelling [2], [3] tackled group behavior modeling in a similar way,
with a strong focus on segregation processes. Just as in Sakoda’s study,
the simulation of the group formation process, with agents deployed on a
regular grid, was based on the local interactions and movements of
agents within Moore’s neighborhood (3×3). In basic experiments, the
agents were divided into two classes. The objective of the agents’
movements was to increase the proportion of individuals in the agent’s
own class to foreign individuals in the new habitable location
(Moore’s neighborhood cell). The Sakoda and Schelling modeling concepts
were developed prior to CA computer implementations, so this study
involves only small populations and follows simple rules.
The development of computer modelling has boosted interest in this
research field. It resulted in a number of models and computer programs
that allowed free modification and analysis of their parameters.
Hegselmann and Flache [21] presented various examples of the
applicability of the Sakoda and Schelling analysis. They also involved
the possible implementation of different parameter definitions to model
social group formation processes.
In the spirit of CA, Klüver and Stoica [22] presented a variety of
social group behavior simulations based on the definitions of agents’
attitudes, specified by sociomatrices, also called Moreno matrices [23].
The arrays allow for taking into account attitude dynamics of different
intensities, also those asymmetric. This was a significant extension in
comparison to earlier approaches where only binary classification was
possible. Klüver and Stoica [22] demonstrated the correlation between
the dynamics of the agents' behavior on the CA grid and the matrix
structure with the Moore neighborhood size. They also proved a
farreaching compatibility of this approach with the models based on
neural networks and genetic algorithms.
Hegselmann and Flache [21] proposed the application of CA as a tool
for modelling and understanding social dynamics. They used CA to
illustrate the potential for extended implementation of Schelling and
Sakoda models. Beltran et al. [6] proposed a model that allows the
analysis of the behavior of a small group of agents during a party held
in a closed room. The movements of agents are modelled by changing the
cells on the grid, similarly to CA models. The agent’s decision to move
at a given time is determined by the level of dissatisfaction with the
agent’s current position. Agents move to minimize the level of
dissatisfaction calculated as a function of the discrepancy between the
actual and desired distance between them. This indicator takes into
account concepts of personal and social zone distances derived from
sociology (Hall [24]).
The analytical perspective on group behavior dynamics based on local
interactions also prevails in the observation of pedestrian traffic on
the streets. This approach is applied in modelling evacuation procedures
from public buildings. Differential games are the main method used in
such an analysis of agent mobility. Hoogendoorn and Bovy [25] proposed a
model of pedestrian flow in which the movement of agents is determined
by the respective equations. They are based on physical parameters
modelled as a multimolecular system and empirical pedestrian behavior.
In this way, they determine the particular decisions that influence the
acceleration and direction of movement. Contrary to CA models, here
agents are constantly on the move, and their movement is limited only by
infrastructure parameters like the sidewalk width, pedestrian crossings,
etc. The application of a similar approach to the analysis of crowd
behavior during evacuation is presented by Helbing et al. [26]. In such
circumstances, the movement of the crowd is primarily determined by the
herd behavior. The interactions between agents in the model are mainly
physical, so the movement is described by differential
equations based on precise physical parameters. The authors demonstrated
the applicability of that model to the analysis of facilities safety
design, such as the width of emergency crosspasses, the shape and size
of corridors, etc. Gao et al. [27] proposed a more recent approach in
this field that uses the classic agentbased simulation system to
evaluate urban management strategies. Most MABMs use grids, but recently
one can encounter more often less rigorous approaches. For example,
evacuation models [28] and pedestrian studies in urban spaces [29]
replicate actual physical traffic routes along with relevant constraints
and surrounding objects.
The models described so far demonstrate trends based on analysis of
the agent movement dynamics in a real, geographical, or physical space.
Similar analysis may also be applied in an arbitrary or abstract
universe. For instance, Moya et al. [30] tried to understand the
influence of terrorist attacks on elections in Spain. In the proposal of
Bin and Zhang [31], the movement of agents takes place in the universe
of degrees of loyalty to the group. In this model, the
simulation of group behavior consists in observation of agents’
attitudes towards the company's policy on social or economic incentives.
The migration of agents in an abstract space is analyzed by CA also in
the works of Yu and Helbing [32] and Helbing et al. [33]. The authors
demonstrated a significant impact of agent migration opportunities on
the formation of separate groups of cooperators and defectors (in the
prisoner’s dilemma game) and the selforganization of
cooperative clusters at the macrolevel. Another approach in this trend
is proposed by KowalskaStyczeń & SznajdWeron [34]. The authors
showed how the movements of agents in the space that simulate the
locations of various sources of information affect the efficiency of
persontoperson communication.
2.3. Soft knowledge representation in agentbased modeling
The models discussed in the previous subsections are based on
knowledge represented in strict mathematical formulas and physical
relations that involve crisp calculations. As our method uses soft
knowledge representations, recent developments in this area are covered
in this subsection.
There exists a relatively small, but systematically increasing,
number of studies that include the representation of imprecise or rough
knowledge in general ABM. In an early study, Ma & Nakamori [35]
employed a simple fuzzy linear quantification method to represent and
aggregate data on the properties of objects for Kansei Engineering
purposes. Their ABM involved both objective and subjective information.
The proposal to merge ABM with fuzzy logic was also presented by Iandoli
et al. [36] in the context of individual and collective learning
processes. They represented agents’ opinions as fuzzy variables and
combined them using the ordered weighted averaging operator [37].
Interesting concepts of incorporation of fuzzy relations to ABM in
social simulations such as matchmaking were discussed by Hassan et al.
[38]. Their ideas were further extended to model friendship dynamics in
[39] and lately in [40]. MartínezMiranda and Pavón [41] proposed using
ABM simulations of human behavior to create effective and efficient work
teams. They applied soft knowledge to determine and represent the agent
emotional state and trust with respect to its team members. Other models
regarding the workforce that include fuzzy approaches were also
developed in a different context by Raoufi et al. [42], Kedir et al.
[43], or most recently in [44].
Some of the latest developments that blend soft knowledge with AMB
take advantage of fuzzy cognitive maps. This notable trend is presented,
for example, in the works of Mei et al. [45], Giabbanelli et al. [46],
or Mehryar et al. [47]. Recently, one can also find AMB approaches
focused on decision making that employ 2tuple fuzzy variables [48],
[49], or the fuzzy logic controller (FLC) [50].
While the literature using softknowledgebased modeling in ABM is
growing in size and importance, its application to MABM is scarce. Among
the few works in this area, there are those of Sharma et al. [28] as
well as Yıldız & Çağdaş [29]. Both works present models of moving
agents in real physical spaces, that is, closed rooms and urban spaces,
respectively. In these proposals, the crisp rules and formulae are used
in conjunction with the soft knowledge. The authors applied the same
core idea of the FLC concept [51] to take advantage of the soft
knowledge in some components of their models. In general, the idea of
FLC consists in creating a system of rules that control the behavior of
objects over time. These rules describe the relations between fuzzy
variables that represent the input and output of the mapped system. Such
relations are developed on the basis of expert knowledge and/or
observations of the system behavior. The rules take the form of a set of
specific ifthen phrases. The knowledge represented in this way allows
one to infer the shape of the response to a given set of input
parameters.
Sharma et al. [28] used the FLC framework to generate the agent's
rate of movement toward the exit in the room evacuation process. This
parameter is generated in the model based on the panic level, stress,
physical weight of the agent, and its distance from the exit. The final
agent movement process is modeled by a complex mechanism that combines a
genetic algorithm with a neural network. The resulting movement
direction also takes into account physical constraints in the room.
In turn, Yıldız & Çağdaş [29] proposed a MABM to simulate
pedestrian traffic in urban space. In this approach, the FLC concept was
used to determine the attractiveness of architectural objects in the
environment. The level of attractiveness generated the force of
attraction that acted on moving pedestrian agents. The rules for
controlling this force included fuzzy levels of variables such as
distance, heating, population, and illuminance. The movement of each
agent was also subject to forces resulting from their current geometric
position relative to other agents and possible obstacles. These other
forces are calculated in the model according to simple, strict geometric
and physical rules represented as crisp mathematical formulas.
The described above works take advantage of some soft knowledge
representation in the form of FLC that is one of many components of
MABM, however, they do not involve any LPs. Two features that seem to
particularly distinguish these works from the classical approaches are
the free geometric space of movement and the flexibility in formulating
rules and relationships that define agent movement. Our proposal fits
generally into the trend, as it combines properties of traditional
moving agent models and less formal expert knowledge given in the form
of natural languagelike expressions called LPs. In contrast to using
soft knowledge to only selected relations that affect agent movement
([28], [29]), LPs in our framework control the behavior of agents
completely by generating all virtual forces. Unlike in the classical FLC
approach, where agent's movement parameters are defined at each step,
LPs are a description of the desired state of the whole system. The
virtual forces acting on the agent in our framework are directed to
decrease the distance of the system from this ideal state at each
step.
The concept of LPs was initially introduced by Grobelny [4], [5]. In
principle, an LP is a logical expression to which a particular degree of
truth can be ascribed. The LP can act as a quality criterion in an
appropriately defined system. Grobelny [52], Raoot & Rakshit [53],
and Grobelny & Michalski [54] applied LPs to object layout
optimization. In the Grobelny [55] study, LPs were used to obtain the
hierarchy of objects based on fuzzy pair wise comparisons. The
possibility of using expressions similar to natural language in the form
of LPs enables their application to optimization problems with imprecise
knowledge. The use of LPs with fuzzy set relations along with the
relaxation of certain moving constraints makes our proposal unique and
potentially interesting for further scientific and practical research.
To the best of our knowledge, there are no proposals that include both
linguistic variables, multivalued logic, and LPs in this area. In the
next section, we describe our concept in detail and provide the
mathematical formulas necessary to understand its implementation.
3. The concept of linguistic patterns and virtual force
The approach proposed in this paper involves the implementation of
LPs in the modeling of successdriven agent movements. The following
example illustrates the essential components of the LP application to
MABM. It is inspired by the idea put forward by Beltran and Salas [1]
regarding simulations of party attendees' behavior. In this scenario,
three agents initially found themselves in a rectangular room where the
party takes place. The configuration is presented in Figure 1.
15
18
23
Figure 1. Initial position: 3 agents with distances
presented as a percentage of the maximum distance, i.e., the length of
the longer side of the room. The dimensions of a single cell in a grid
amount to 10% of the maximum distance.
After the official ceremony, the agents begin to move from their
initial random positions. Their movements depend on known relations that
reflect their mutual attitudes. They may be expressed in a way shown in
Table 1.
Table 1. Agents’ mutual attitudes.
1 
× 
Positive Big (PB) 
Negative Big (NB) 
2 
× 
× 
Positive Medium (PM) 
The expressions presented in Table 1 are linguistic exemplifications
of the symmetric Moreno matrix (Klüver and Stoica [22], Moreno [23]).
They simply mean that agents 1 and 2 like each other very much, agents 1
and 3 dislike each other very much, and agents 2 and 3 like each other
moderately. In circumstances such as the said party, the pairs liking
each other would tend to be close to one another, while the resentful
ones would prefer to stay away. Therefore, we may use the following
statements to describe the configuration presented in Figure 1 and its
future dynamics. If I like the person, the distance between us is
small and If I dislike the person, the distance between us is
large. Such statements may be expressed in a slightly more formal
manner as patterns presented in (1) and (2):
P2: 
IF Attitude(i, j) is NEGATIVE, THEN j is
at a LARGE_DISTANCE from i 
(2) 
Both patterns define the attitudedesired distance relationship for
each pair of agents analyzed. The difference between them is related to
the nature of their mutual impact. Pattern (1) corresponds to
attraction, whereas pattern (2) relates to repulsion between agents.
These formulas are examples of LPs that, in the context presented,
constitute the agents’ wellbeing criteria. The levels of such
wellbeing depend on the respective criteria values. They need
to be determined for each agent in a given configuration. The values can
be specified as the degree of truth for a particular pattern. Practical
calculations can be based on the formula for determining the truth of
the implication proposed by Łukasiewicz (Grobelny [55]). If
t(l) and t(r) denote the degree of truth on the left
and right side of the implication, respectively, then the truth value is
computed by formula (3).
The same formula applies to Truth(P2), which is calculated only for
negative attitudes between agents i, j. It may be
noticed that (3) is a kind of generalization of the truth value table
related to a classic implication. There are more such generalizations
possible, e.g., those discussed by Dubois and Prade [56]. Certainly, the
determination of the truth value according to formula (3) requires
t(l) and t(r) to be specified. The theory of fuzzy
sets provides simple and intuitive tools for that purpose (Zadeh [57],
[58]). For example, t(l) for patterns (1) and (2) denotes the
truth value in the linguistic expressions
‘Attitude(i, j) is POSITIVE’ and ‘Attitude(i,
j) is NEGATIVE’. We can treat the linguistic expressions
presented in Table 1 as fuzzy sets (singletons) with appropriate values
of the membership function. Then, these values of the membership
function may be regarded as degrees on truth of the left side of
patterns (1) or (2). An illustration of this approach is given in Figure
2. The linguistic expressions in Table 1 are arranged only on an ordinal
scale and the exact distances between successive items are unknown.
However, the assumption of a proportional increase of the truth value
for successive phrases listed on this scale seems reasonable. It is
especially true and justifiable when additional information on how such
expressions are perceived by humans is not available.
PS/NS PM/NM PB/NB
0.25
0.50
1.00
Attitude(x)
POSITIVE/NEGATIVE(x)
Membership value
Figure 2. Exemplary definitions of truth values for
the patterns (1) and (2) expressions (POSITIVE or NEGATIVE) for three
levels of linguistic expressions (PS – POSITIVE_SMALL, NB –
NEGATIVE_BIG, etc.).
The concept of a possibility proposed by Zadeh [28,29] is a
generalization of a simple, direct assignment of the truth value for
linguistic expressions. This measure allows for the determination of
truth of the LP fulfilment using an expression represented by a fuzzy
set (e.g., a fuzzy number) in a given universe. The aforementioned
criterion may be formally expressed as follows.
If one defines the expression Attitude(a, b) as a
linguistic variable represented by fuzzy sets in the universe of
discourse X = (x_{1}, …, x_{n}),
POSITIVE(x) and A(x) in (4) represent this realization
of the variable as fuzzy sets in this universe, then the truth is
computed according to formula (5):
Formula (5) represents the consistency of two expressions POSITIVE
and A, which are the pattern and a specific realization for a given
pair, respectively. In other words, (5) exhibits the possibility of the
fact that A is POSITIVE. By analogy, we can define the truth value for
the NEGATIVE(x) case. Figure 3 presents exemplary definitions
of LPs from Table 1 expressions. We also illustrate how the calculation
of the possibility measure Truth(l) works for fuzzy
representations of linguistic values. In Zadeh's original proposal [57],
[58], this measure determines the degree of truth of the
fulfillment of a given criterion by specifying its level of magnitude.
The criterion can also be expressed as a linguistic expression.
Figure 3 shows a generalization of the approach presented in Figure 2
where the degree of truth is directly determined. The particular
significance of this extension lies in enabling experts to specify fuzzy
sets that represent linguistic expressions with their concrete spaces of
variability, characteristic for a given context. If we assume that the
linguistic values from Table 1 are defined in the form of fuzzy sets in
the numerical space of 010 ratings (e.g., based on interviews), then
Figure 3 shows how to determine the value of Truth(l) for a
given PM(x). Here, we assume a linear membership function for
POSITIVE(x).
Membership value
Truth(l) = POSS(Attitude(a, b) is
POSITIVE
Attitude(a, b) is PM)
= max_{x} (min_{x}
(POSITIVE(x),
PM(x))
PM(x)
0 5 10
(x)
POSITIVE(x)
PS(x)
0.5
1.0
Figure 3. Graphical illustration of possible
definitions of linguistic variables in an artificial numerical universe
of discourse and calculation of Truth(l) according to equation
(5).
Although Figure 3 defines LPs using fuzzy sets in an artificial
numerical universe, the same could be performed using more objective
information. For example, universe X may represent the number
of interpersonal contacts in a given period, or the proportion of common
views, etc. Then, Truth(l) is simply the value of the
POSITIVE(x) function for a given x value.
The determination of the truth value for the right side of the
patterns t(r) requires that appropriate functions be defined
for the following statements: ‘(j) is at a SMALL_DISTANCE from
i’ and ‘j is at a LARGE_DISTANCE from i’.
Examples of such definitions as fuzzy sets are shown in Figure 4.
0.5
1.0
0 10 15 20
Percent of maximal distance (x)
LARGE_DISTANCE(x)
Membership value
0.75
SMALL_DISTANCE(x)
Figure 4. The pattern expressions LARGE_DISTANCE and
SMALL_DISTANCE are interpreted as fuzzy sets in the universe of the
maximal distance percentages.
In this case, LPs are represented as fuzzy sets defined in the
universe of distances determined as percentages of the maximum distance.
Although the shape of the function is intuitive, it can reflect
objective knowledge of the human perception of distance in a given
context. For example, such an objectified knowledge may refer to the
notions of personal and social spheres (Hall [24], Beltran and Salas
[6]).
The definitions described above allow for the determination of truth
values for patterns (1) and (2). Therefore, the appropriate calculations
can be performed for each agent from the party presented in Figure 1.
Let us consider the situation of agent 1. To assess his
wellbeing, we should use pattern (1) to specify the relation
with agent 2 because the attitude is positive, and pattern (2) to obtain
the relation with agent 3 as the attitude is negative. Based on Table 1,
Figure 2, Figure 4, and using pattern P1 (1) to assess the relationship
with agent 2, we arrive at the following results:
t(l) = 1, because the attitude value is PB (POSITIVE_BIG)
and the degree of truth for the left side of pattern (1) (POSITIVE
Attitude) is 1.
t(r) = 0, because the truth value for the right side of
pattern (2) (SMALL_DISTANCE) for the distance between agents 1 and 2
(greater than 0.2) is 0.
Then,:
Truth_P1(1, 2) = min ((1 – 1 + 0), 1) = 0.
Similarly, using pattern (2) for the assessment of the relation with
agent 3, the results are as follows:
t(l) = 1, because the attitude value is NB and the degree of
truth for NEGATIVE is 1.
t(r) = 0.75, because in Figure 4, the distance of 0.15
between agents 1 and 3 is DISTANCE_LARGE to the degree of 0.75.
Therefore,
Truth_P2(1, 3) = min ((1 – 1 + 0.75), 1) = 0.75.
The total level of satisfaction of agent 1 with the configuration
shown in Figure 1 can be assessed by calculating the average of the
truth values for both patterns. In this example, it is 0.38. Changing
the location of agents may improve this evaluation. Moving agents in
directions that cause an increase in the truth values of the patterns
appears to be reasonable to achieve this goal. It can be assumed that
agent 1 senses a kind of unique virtual attraction or repulsion
to agents 2 and 3 at a given moment. Feelings are proportional to the
truth levels of patterns (1) and (2). Such an approach is a simplified
analogy of the concept of social force described by Helbing
[26]. The vector of the force ($\overrightarrow{VF}$) acting on agent 1 lies
on a straight line that connects agents 1 and 2. It has a length of
VF(1, 2)^{+} = 1 – Truth_P1(1, 2) = 1 and is directed towards
agent 2, which denotes its attractive nature. Let us analyze the force
of magnitude VF(1, 3)^{–} = 1 – Truth_P2(1, 3). In the position
presented in Figure 1, the vector length is 0.25 and lies on the
straight line connecting agents 1 and 3. The direction of this force is
opposite to the position of agent 3. This means that the force has a
repulsive character.
Assuming that the agent's response to the force being felt is motion,
the displacement distance in one step is proportional to the magnitude
of this force. Since the maximum force value between a pair of agents is
1, the appropriate parameter s may define the physical distance
of the displacement. It can be expressed as a percentage of the longer
side of the rectangle. It may also be reasonable to associate the value
of s with a physical limitation of displacement, e.g., possible
speed. Taking into account both forces and assuming s = 0.1,
the agent moves in the direction that is the sum of both vectors, and
ultimately reaches a new position denoted as 1’(←2^{+},
←3^{–}) in Figure 5.
s·( $\overrightarrow{{\mathbf{VF}\left( \mathbf{1,\ 2}
\right)}^{\mathbf{+}}}\mathbf{+}$ $\overrightarrow{{\mathbf{VF}\mathbf{(1,\
3)}}^{\mathbf{–}}}$ )
s·$\overrightarrow{{\mathbf{VF}\left( \mathbf{1,\ 2}
\right)}^{\mathbf{+}}}$
s·$\overrightarrow{{\mathbf{VF}\mathbf{(1,\
3)}}^{\mathbf{–}}}$
3
1’(←2^{+}, ←3^{–})
2
1’(←2^{+})
1’(←3^{–})
1
Figure 5. Vector representation of virtual forces
and the agent 1 movement step in the configuration from Figure 1.
Notation 1’(←2^{+}) depicts the position of agent 1, if only the
attractive force from agent 2 is taken into account;
1’(←3^{–}) denotes the location of agent 1
determined only by the repulsive force from agent 3; 1’(←2^{+},
←3^{–}) is the final position of agent 1 resulting from forces
generated by both agents 2 and 3.
The precise determination of successive positions of the agents
consists of performing simple calculations based on geometrical
dependencies. For example, to obtain the geometric position of agent 1
after acting with an attracting force between agents 1 and 2, i.e. VF(1,
2)^{+} one needs to do the following.
Calculate the distance (DIST) between agents 1 and 2 for their
current coordinates (x_{1}, y_{1}) and
(x_{2}, y_{2}) according to
(6):
Calculate new coordinates for agent 1 (x_{1( ← 2+)}^{′}, y_{1( ← 2+)}^{′}).
Since $\cos(\alpha) = \frac{x_{2} 
x_{1}}{{DIST}_{(1,\ 2)}} = \frac{x_{1\left( \leftarrow 2^{+}
\right)}^{'}  x_{1}}{{s \cdot VF(1,\ 2)}^{+}}$, and $\sin(\alpha) = \frac{y_{2}  y_{1}}{{DIST}_{(1,\
2)}} = \frac{y_{1\left( \leftarrow 2^{+} \right)}^{'}  y_{1}}{{s \cdot
VF(1,\ 2)}^{+}}$, where α is the angle between the positive
horizontal axis (x) and vector $\overrightarrow{{VF(1,2)}^{+}}$, we arrive
at the direct formulas (7):
These computations must be performed for all links of each agent. The
final position in a given step is the vector sum of these partial
displacements. In the same way, the displacement characteristics of each
agent are computed with respect to the appropriate criterion represented
by an LP.
The presented idea of the individual agent behavior can be applied to
all agents. Thus, each of them, at a given moment, can see others and
perform the same calculations. As a result, according to the rules
presented, appropriate displacements take place in consecutive steps. It
eventually generates dynamic movements of agents. Such an approach
derived from the party example may reflect the behavior of small social
groups.
It can also be observed that the distance covered by an agent in a
single step in the simulation depends on the number and strength of
relations with other agents. Then, taking into account the forces coming
from all agents, the displacement of the agent being analyzed in a
single step should not exceed the total value s. Therefore, in
the proposed algorithm (Appendix A), a quotient of s by
n is used, where n is the number of agents.
Once this procedure is completed for each agent in a given step, the
mean truth value for the patterns of all agents is determined by formula
(8). It can be interpreted as a measure of satisfaction for the entire
group of agents in a given configuration.
In the above formula (8), m is the number of agent pairs
with nonnegative attitudes, u – the number of pairs with
negative attitudes, p is the
number of all agent pairs, and m + u = p. In
general, $p = \frac{n^{2}  n}{2}$,
where n is the number of agents. Since for pairs of unrelated
agents t(l) = 0, the truth values for patterns P1 and P2 equal
1. In situations where there are a considerable number of unrelated
pairs of agents, those ones would artificially increase the mean truth.
Therefore, when calculating the mean truth, it is reasonable to take
into account only the pairs of agents for which t(l) > 0. In
this way, our indicator will only refer to agents remaining in any
relation with others. In this case, the value p denotes the
number of only linked agent pairs.
The movement of agents is finished when each VF(i,
j) for each agent amounts to a zero vector or after performing
a specified number of steps. This concept can be described in the form
of a simple algorithm. The pseudocode is provided in Appendix A. In
addition to the principles presented above, the possibility of defining
the radius r to determine the range of vision for all
agents was also introduced. Each agent examines the relations only with
those agents who remain within a distance shorter than r, which
is expressed as a percentage of the maximum distance. In our model
implementation in Delphi, agents are graphically represented by numbered
crosses or squares of equal sizes. The analyzed area is defined as
a rectangle of any size and in any unit. The distances within such an
area are expressed as percentages of the longer side of the area. The
step coefficient s that represents the distance of the agent’s
dislocation in a single step of the simulation in response to VF of
length 1, is also defined as a percentage of that maximal distance.
Given defined patterns and variables, the simulation of the agents'
movements always starts with a random distribution on the plane. After
the random agents’ locations are generated, the distances between each
pair are calculated, allowing the computation of the truth values. The
application tracks the movement of selected agents from their initial
positions to end positions, as shown in Figure 6a, and records the
Mean_truth values for all agents in each step (Figure 6c).


(a) 
(b) 

(c) 
Figure 6. Agents’ movement patterns from the
configuration shown in Figure 1. The definitions of the truth value for
the patterns correspond to Figures 2 and 4. The grid size is 0.1 (10% of
the maximal length). Parameter s = 0.05 (5% of a maximal
distance) (a) Movements towards the stable
configuration. Locations of agents marked by every 10^{th} step;
(b) The final stable arrangement; (c)
Dynamics of the Mean_truth value in subsequent steps.
The figures presented demonstrate that, in the discussed case, the
system reaches the complete truth for the defined patterns and their
parameters in approximately 45 steps. However, it is easy to conclude a
priori that such a simple implementation of the proposed concept shall
not be considered universal. It can be noted that if there are only
positive values in the attitude matrix (Table 1), then the optimal
solution would result in more or less dense clusters. Such a behavior is
observed regardless of the level of these positive values, and the final
configuration depends on the definition of the distance adopted in P1.
An alternative optimum result would involve all agents located at one
point. As it is considered to be rather unlikely behavior, the proposed
framework and its application were supplemented with the possibility of
introducing additional, reasonable patterns and virtual forces they
generate. The concepts and basic properties of our model are further
discussed on the basis of several illustrative examples.
4. Impact of model parameters on agent behavior
The presented framework provides flexibility in formulating both
input data and agent behavior rules. The influence of key parameters on
the formation of the resultant configurations of agents can be
conveniently analyzed using examples. It is convenient to select
problems in which reasonable or optimal stable layouts can be predicted
in terms of accepted criteria or patterns. For this purpose, two test
matrices with specific structures were created to represent the
relations between agents.
It was assumed that for each pair of agents, the degree of mutual
positive relationship reflected the number of daily personal contacts
such as emails, phone calls, conversations, etc. The number of
interactions ranged from 0 to 10. It was also assumed that the truth
degree of t(l) for P1 is defined as shown in Figure 7. It is
linearly dependent on the number of contacts (Figure 7a), or the truth
of any number of contacts larger than 0 is taken as 1. It should also be
noted that in this structural context, the relations and their degree of
truth need not be strictly dependent on mutual personal attitudes but
rather on the more substantive cooperation of the agents.
0.5
1.0
0 5 10
Number of personal contacts (x)
POSITIVE(x)
Membership value 
0.5
1.0
0 1 5 10
Number of personal contacts (x)
POSITIVE(x)
Membership value 
(a) 
(b) 
Figure 7. Two definitions of the degree of truth in
relations between agents.
The two exemplary structures are shown in Figure 8. Degrees of truth
t(l), presented in green, were calculated according to the
function in Figure 7a for randomly assigned contact numbers. When the
function in Figure 7b is used, the truth value will be equal to 1.
Analogously to the party example, one can analyze the spatial behavior
of agents initially placed at random locations, in particular, their
final stable configurations. By maximizing the truth value of pattern
(1), we should obtain structures in which pairs of agents with stronger
relationships are closer to each other. One can also notice that similar
criterion is also used to search for optimal solutions in facility
layout problems (Grobelny & Michalski [54]).


(a) 
(b) 
Figure 8. Two exemplary structures of t(l)
relations with easily predictive solutions.
(a) Circular arrangement; (b) Grid
arrangement.
The dynamics of agent movement and the resulting layout configuration
for the P1 pattern will be shaped by the perception of SMALL_DISTANCE.
Figure 9 shows a simple formulation of the truth t(r) for the
SMALL_DISTANCE expression in the universe of Percent of maximal
distance.
afuzzy bfuzzy
0.5
1.0
Percent of maximal distance (x)
SMALL_DISTANCE(x)
Membership value
Figure 9. Schematic definition of a SMALL_DISTANCE
expression in the universe of Percent of maximal distance.
This definition makes it possible to reset the strength of attraction
for each pair of agents if the distance between them is less than the
afuzzy value. Consequently, using this value, the model can
adopt the concepts of social and personal proximity proposed by Hall
[24]. They were initially implemented in the context of MABM by Beltran
et al. [6]. These concepts take into account the dissatisfaction of
people resulting from strangers crossing a certain distance in
interpersonal relationships. They can be understood as a small
protective sphere that an organism tries to keep between itself and
others (Hall [24]). Depending on the context, culture etc., the radius
of this comfort zone is estimated to be 1.5–4 feet (personal distance)
and 4–12 feet (social distance).
A series of simulations of agent behavior were run with the
SMALL_DISTANCE definition from Figure 9 and setting
afuzzy = 0.05, bfuzzy = 0.1. The relationships between
agents were taken from the structures in Figures 8a and 8b. We applied
a linear function to assess the value of t(l) (Figure 7a), and
set the value of s between 0.01 and 0.1. The highest values for
Mean_truth in the resulting configurations were obtained for s
= 0.05, yielding layouts similar to those shown in Figure 10. All agents
were clustered in a small area around the center of the plane such that
the distances between them were essentially proportional to the
t(l) values that represent their relations (Figure 8a).


(a) 
(b) 

(c) 
Figure 10. Agent configuration in experiment 1 with
t(l) values for the relations (afuzzy = 0.05,
bfuzzy = 0.1, s = 0.05). Mean_truth = 1 (left and
right). (a) Circular link structure;
(b) Grid link structure; (c)
Mean_truth value dynamics.
Note that in the resulting configuration in Figure 10a, the
anticipated and desirable proximity of the agents’ locations is
maintained (Figure 8a). However, there are also some close accidental
neighborhoods, for example, between unrelated agents 3 and 9 or 5 and 1.
For the relationships in Figure 8b, the resulting layouts (Figure 10b)
are even denser. This is because the length of the personal/social
distance represented in pattern P1 by afuzzy only affects the
distance of related agents. Since agents defined as mutually
neutral do not interact with each other during their movements,
it seems reasonable to propose an additional pattern that incorporates
the notion of social or personal distance for them. We propose to
construct this LP in the following way (9):
The scheme to determine the truth value of PERSONAL/SOCIAL_DISTANCE
is defined in Figure 11. As in the example shown above, the membership
function reflects the degree of truth of achieving the personal
distance with respect to the distance between a pair of agents.
Likewise in pattern P2, it generates a corresponding repulsive
virtual force, proportional to the deficit of truth in pattern P3
(9).
0.5
1.0
pd/sd
Percent of maximal distance(x)
PERSONAL/SOCIAL_DISTANCE(x)
Membership value
Figure 11. Schematic definition of
PERSONAL/SOCIAL_DISTANCE (pd/sd).
In the developed implementation, afuzzy and pd/sd
can be set independently. The afuzzy parameter is equivalent to
personal distance, but it is only applicable to agent pairs that are in
a relationship according to the P1 or P2 patterns. The P1 pattern
generates an attractive force (P2 – a repulsive force) only until a pair
of agents reaches a distance with the value of afuzzy. In
contrast, pd/sd creates pattern P3 that is applicable
only to pairs that are not connected by any relation. P3 generates a
repulsive force until the unrelated agent pair reaches a distance of
pd/sd.
Another simulation experiment conducted included the third pattern
(9). In this modified approach, pd/sd was set to be
three times greater than afuzzy. It reflects the assumption
that unrelated agents should rather remain at a social distance that is
about three times the personal distance. The value of
pd/sd set here was inspired by the concepts of
personal and social distances presented in the work of Hall [24]. The
authors claim that the social distance of approximately three times
greater than the personal one is statistically desirable. It applies to
people who are not in a relationship with each other.
The resulting layouts were qualitatively different from previous
simulations. Figure 12 demonstrates these configurations of agents that
achieved complete satisfaction with Mean_truth = 1. The
simulation involved pattern P3 with pd/sd = 0.15. It
started with the same random agent layouts as in previous experiments
shown in Figure 10. All other parameters were unchanged. The final
Mean_truth value for pattern (P3) is recorded and can be included in the
overall Mean_truth evaluation. Figures 12c and 12d illustrate the
dynamics of Mean_truth values for patterns P1 and P3.


(a) 
(b) 


(c) 
(d) 
Figure 12. Final layouts for agents with parameters
from Figure 10 with added pattern P3. (a) Circular link
structure; (b) Grid link structure; (c)
(d) Mean truth value dynamics for the patterns P1 and P3 for
the circular and grid link structure, respectively.
The Mean_truth value for P3 can be treated as a separate measure or
quality criterion of a given configuration. It can also be appropriately
combined with Mean_truth for P1 and/or P2. The definition of truth shown
in Figure 7b allows us to evaluate attitudes or relations in binary
terms (1 – 0, true – false). The described experiments
(afuzzy = 0.05, bfuzzy = 0.1, pd/sd
= 0.15, s = 0.1) demonstrate the surprising reasonable dynamics
of the presented approach. The final layouts obtained for the tested
relationship structures (Figure 8 with t(l) = 1 for each link)
are shown in Figures 13a and 13b.


(a) 
(b) 


(c) 
(d) 
Figure 13. Best final layouts of two cases analyzed
in Figure 8 with all truths of links t(l) = 1.
(a) Circular link structure; (b) Grid
link structure; (c) (d) Mean truth value dynamics for
patterns P1 and P3 for circular and grid link structure,
respectively.
In both examples, the resulting layouts preserve the desired
structure of agent proximity and form very regular patterns. The
dynamics of changes in truth values for patterns P1 and P3 are shown in
Figures 13c and 13d. Both approaches similarly achieve a stable
configuration within approximately 25 simulation steps, reaching full
truth for pattern P1 and a truth value of 0.98 for pattern P3.
5. Segregation examples
The proposed method of modelling the movement of agents based on
a party example may seem specific and of limited practical use.
However, its flexibility, resulting from the freedom to define patterns
and their parameters, gives many more possibilities. For example, it
seems that the proposed method can be applied to model the dynamics of
many social processes determined by mutual attitudes of agents. For this
purpose, Sakoda [1] proposed a social interaction checkboard model,
which allows us to analyze the movement of agents on a regular grid. In
the Sakoda model, members of two groups live on a checkerboard. They
have positive, neutral, or negative attitudes toward each other, called
valences. These values are defined by integers. Individuals have the
opportunity to move to empty cells in their 3×3 neighborhood. If there
are no empty cells, an individual can jump over one cell. Migration is
always local or is allowed only within certain limits. Individual
i uses the migration option to move to locations where (10) is
maximized.
In formula (10), V_{ij} denotes the valence of
individual j for individual i, P is the set
of all individuals when not all cells are occupied, d is the
Euclidean distance between i and j, and w
determines how strongly the valences are discounted by distance. In this
model, all agents can see each other. Sakoda's world is an 8×8
checkerboard occupied by two groups, each with 6 members. Members of one
group are represented as squares, and members of the other group are
represented as crosses. Sakoda analyzes different combinations of
attitudes. He calls one of them segregation, another suspicion. These
attitudes are shown in Table 2.
Schelling [2], [3] proposed a different approach to segregation. The
concept of his model differs from Sakoda’s approach in that agents act
only on the basis of local observations that involve at most eight
nearest neighbors. The number of neighbors to whom the agent has a
positive attitude determines the satisfaction with a given location. The
agent may decide to move to the nearest free cell on the grid if the
number of positive neighbors at the new location is greater
than this number at its current location. The relations for this
segregation model are included in the last column of Table 2.
Table 2. Attitude combinations for suspicion and
segregation models.
Squares 
Crosses 
Squares 
Crosses 
Squares 
Crosses 
Squares 
0 
 1 
1 
1 
1 
0 
Crosses 
1 
0 
1 
1 
0 
1 
The solutions of the original research of Sakoda and Shelling are
schematically shown in Figure 14. Analysis of the above models from the
perspective of LPs and virtual forces required the determination of
appropriate model parameters. In the proposed implementation, it is
necessary to determine the appropriate agent visibility, that
is, the distance within the range of which patterns and generated forces
are exercised. Such an approach is a type of local environment
estimation in regular gridbased models. In this approach, each agent
can analyze patterns and is subjected to forces only inside a circle
with a radius equal to the visibility range. Of course, the
corresponding truth values for patterns are calculated only with respect
to this range. Patterns P1 and P2 are, respectively, involved instead of
formula (10).


(a) 
(b) 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 

+ 
□ 
□ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 

+ 
+ 
□ 
□ 
+ 


+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 


+ 
□ 
□ 
□ 
□ 
□ 
□ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 

+ 




□ 
□ 
□ 
□ 
□ 
+ 
+ 
+ 



+ 

+ 
+ 

□ 
□ 
□ 
□ 
□ 
□ 
□ 
□ 
□ 
□ 
□ 
+ 
+ 


□ 


□ 

□ 
□ 
□ 

□ 
□ 
□ 



+ 
□ 


□ 






□ 




+ 
□ 
□ 






□ 

□ 
□ 
□ 

+ 
+ 

□ 


□ 
□ 
□ 
□ 

□ 


□ 
□ 
+ 
+ 


□ 

□ 


□ 
□ 

□ 

□ 
□ 



□ 
□ 


□ 

□ 

□ 
□ 
□ 


+ 
+ 

(c) 
Figure 14. Final configurations of Sakoda and
Schelling models. (a) Sakoda suspicion;
(b) Sakoda segregation; (c) Schelling
segregation.
To examine the behavior of agents whose attitudes are
defined similarly to the Sakoda and Schelling models, a value +1 or –1
in Table 2 was assumed to indicate compliance with patterns P1 and P2,
respectively, to the degree of 1 (t(l) = 1).
0.5
1.0
5
Percent of maximal distance (x)
LARGE_DISTANCE(x)
Membership value
75
20
SMALL_DISTANCE(x) 
0.5
1.0
5
Percent of maximal distance (x)
PERSONAL/SOCIAL_DISTANCE(x)
Membership value 
(a) 
(b) 
Figure 15. Definitions of distances used in
segregation experiments.
Figure 15 shows the corresponding fuzzy definitions of distances used
in the analyzed patterns. There were 10 experiments conducted for the
matrix corresponding to the suspicion and segregation models. The
experiments involved 36 agents. The purpose of this study was to test
the behavior of agents with attitudes corresponding to Sakoda models
with four different visibility ranges (r = 0.1, 0.3, 0.5, and
1). For each r, the displacement process started with the same
random layout of agents. The resulting configurations with the highest
Mean_truth(P2) values are shown in the screenshot excerpts displayed in
Figure 16.


(a) 
(b) 


(c) 
(d) 


(e) 
(f) 


(g) 
(h) 
Figure 16. Suspicion (left column:
(a), (c), (e),
(g)) and segregation (right: (b),
(d), (f), (h))
experiment results for different neighborhood ranges r.
(a) r = 0.1, Mean_truth(P2) = 0.98,
Mean_truth(P3) = 1; (b) r = 0.1,
Mean_truth(P2) = 0.99; Mean_truth(P3) = 1; (c)
r = 0.3, Mean_truth(P2) = 0.99; Mean_truth(P3) = 0.99;
(d) r = 0.3, Mean_truth(P2) = 0.99;
Mean_truth(P3) = 0.96; (e) r = 0.5,
Mean_truth(P2) = 0.99; Mean_truth(P3) = 0.97; (f)
r = 0.5, Mean_truth(P2) = 0.99; Mean_truth(P3) = 0.93;
(g) r = 1, Mean_truth(P2) = 0.91;
Mean_truth(P3) = 0.94; (h) r = 1,
Mean_truth(P2) = 0.98; Mean_truth(P3) = 0.86.
A direct comparison of the results obtained in this experiment with
the model results in Figure 14 is not possible, but some qualitative
observations can be made. In general, agents clustered into groups with
individuals neutral towards each other. Assuming that agent mobility is
based on global observations (r = 1), the resulting model will
predict the formation of clearly isolated groups, similar to the
original solutions in a regular grid. These groups are somewhat more
concentrated in the case of segregation, which is the result of
attraction forces generated by the LP used in this case. Figure 17
presents the simulation results for Schelling’s model (Table 2, column
3). Simulations were performed for the same values of r as for
the Sakoda models. Since the configuration in all runs for r =
0.5 is the same as for r = 1, it is not presented in the
figure.
Figure 17. Stable configurations of the Schelling
model for different values of r. (a) r = 0.1,
Mean_truth(P1) = 0.98, Mean_truth(P3) = 1; (b) r = 0.2,
Mean_truth(P1) = 0.97; Mean_truth(P3) = 1; (c) r = 0.3,
Mean_truth(P1) = 0.97; Mean_truth(P3) = 1; (d) r = 0.5 and
r = 1, Mean_truth(P1) = 0.97; Mean_truth(P3) = 1.
Although, as in the previous example, a direct and precise comparison
of the results of our approach with those obtained by Schelling on a
regular grid (Figure 14) is not possible, a qualitative analysis is
feasible. Assuming that the mobility of agents is based on local
observations (r = 0.1, 0.2, and 0.3), the resulting
configuration predicts the formation of increasingly isolated groups.
Similarly as in the original, regular grid model. In each case, the
model reaches a stable configuration in 40–70 steps. Figure 18
demonstrates the dynamics of the mean truth value in the types of
relations studied, under the assumption that all agents can see each
other (r = 1).
Figure 18. Mean truth values dynamics (P1 or P2) for
the examples studied.
In the experiments presented above, the agents moved according to the
established patterns with fixed parameters. The fundamental element that
determines the achievement of a stable configuration is the assumed
perception of the distance defined in each pattern. To qualitatively
illustrate the importance of the assumed perceived distance parameters,
the effect of a radical change in the parameter pd/sd
on the stable configuration shape is shown in Figure 19. As mentioned
above, the parameter pd/sd defines an acceptable
distance between individuals in different social situations. In this
sense, it can be implemented directly in the party scenario described
above.
The analyzed segregation models are related to social groups
migrations, where a direct interpretation of pd/sd in
its literal sense is not applicable. However, it can generally be
assumed that the relation between the afuzzy and
pd/sd parameters describes to some extent the unified
usthem attitudes in the analyzed group of virtual agents.
Using Schelling’s model of relations (Table 2, column 3), simulations
were performed for 72 agents. They started from the same random layout
and situation where agents do not tolerate the proximity of
strangers.
The pd/sd value was assumed to be 4 times greater
than afuzzy. In the second experiment, the opposite was
assumed: the pd/sd value was twice lower than afuzzy.
Such a relation may reflect the actual curiosity or openness of some
social groups. Figure 19 shows the resulting stable virtual community
configurations, assuming that agents can see 20% of the environment with
(r = 0.2) and 100% (r = 1), respectively.
Figure 19. Schelling’s model simulation results for
afuzzy = 0.05, bfuzzy = 0.5 for two different levels
of pd/sd {0.2, 0.025} and r {0.2, 1}.
(a) r = 0.2, Mean_truth(P1) = 0.96,
Mean_truth(P3) = 1; (b) r = 1,
Mean_truth(P1) = 0.97; Mean_truth(P3) = 1;
(c) r = 0.2, Mean_truth(P1) = 0.96;
Mean_truth(P3) = 1; (d) r = 1,
Mean_truth(P1) = 0.96; Mean_truth(P3) = 1.
(e) Mean_truth dynamics of P1 and P3 for configurations
presented in (a), (b),
(c), (d) with r = 1 and
pd/sd = 0.2.
Although it might not be noticed at first glance in the figure
presented, the structures resulting from different values of
pd/sd are qualitatively entirely different. This
effect of the pd/sd parameter on the results consists
in a clear variation in the homogeneity of the groups formed. In parts
(a) and (b) of Figure 19, where high values of pd/sd
were applied, the groups are highly homogeneous. They consist
exclusively of squares or crosses. In parts (c) and (d) of Figure 19,
which depict results obtained with considerably lower value of the
pd/sd parameters, mixed groups are formed. An
individual group includes both squares and crosses.
The dynamics of the mean truth value (Figure 19e) for P1 in the case
where r = 1, suggests that a stable configuration is obtained
in approximately 30 steps and is similar to that presented in Figure 18
for 36 agents.
6. Comparison with scattered plots
The examples analyzed so far lead to stable agent configurations that
maximized the average truth value for specified patterns with certain
parameters. Since they take the form of scattered plots, it
seems interesting to compare the properties of these configurations with
the results of Drezner’s classical approach to the construction of
suboptimal scattered plots for facility layouts [7]. As the author
writes, they may be applied in various areas: ‘They proved to be
very useful, for example, in the work of architects. Drawings of
facilities scattered on a plane may be a useful benchmark for them in
urban planning, locating industrial plants, etc.’.
The starting point for the construction of scattered plots proposed
in [7] is the definition of the objective function to minimize,
specified according to (11).
In formula (11), c_{ij} denotes the link (intensity
of interaction) between facilities i and j, while
d_{ij} denotes the distance between them. The proposed
heuristic is very effective and is based on the properties of
eigenvectors and the eigenvalues of matrices. Namely, if in formula (11)
d_{ij} is replaced by
d_{ij}^{2} , which Drezner considers
‘intuitively reasonable’, we have formula (12).
Equation (12) has its optimal solution in a straight line. The
x coordinates of the solution are successive elements of the
eigenvector associated with the second smallest eigenvalue of matrix
S where s_{ij} = –
c_{ij} for i = j and
s_{ii} =$\sum_{j =
1}^{n}c_{ij}$ for all i. The y coordinates are
elements of the eigenvector associated with the third smallest
eigenvalue of matrix S. Thus, the algorithm for
obtaining a suboptimal solution is relatively simple. Having the set of
links c_{ij}, it is enough to: (i) construct the matrix
S; (ii) calculate all eigenvalues and associated
eigenvectors for S; (iii) select two eigenvectors
associated with the second and third lowest eigenvalues, treat them as
the coordinates x and y of the solution on a plane,
and calculate objective function values.
The implementation of the algorithm described above allows for a
comparative analysis of examples taken from Drezner’s work that
demonstrate the potential of the LPbased approach in relation to the
eigenvector approach. Table 3 presents three cases of searching for
optimal scattered plots in terms of the objective function (12). In the
matrix of relations between agents, pairs that should be close to each
other are denoted by 1, and by zeros otherwise.
In Figure 20, the first row shows the suboptimal scattered plots
obtained using the Drezner approach. Agent configurations for the
defined examples were based on agent relations expressed as the truth
value t(l) using the proposed Drezner method. The respective
patterns (1) and (3) were applied, assuming the distance definition
consistent with Figure 9 where afuzzy = 0.05, bfuzzy =
0.1, and pd/sd = 0.05 is taken from the graph in
Figure 11. The best configurations resulting from 10 trials are
presented in the second row in Figure 20.
Table 3. Drezner’s examples for scattered plots.
Drezner1 
Drezner2 
Drezner3 
1 
8 10 15 
5 6 10 12 17 18 
7 8 9 
2 
6 14 15 
6 7 12 13 14 16 
4 6 8 
3 
11 18 19 
8 9 15 16 
10 13 16 
4 
6 10 13 
9 11 15 17 
2 15 17 
5 
10 16 18 
1 11 17 18 
9 18 19 
6 
2 4 14 17 
1 2 10 12 13 19 
2 5 11 
7 
9 13 19 
2 13 14 
1 12 14 
8 
1 15 16 
3 14 16 
1 2 18 
9 
7 12 17 
3 4 12 15 16 17 
1 10 11 
10 
1 4 5 16 
1 6 18 19 
3 9 12 
11 
3 13 18 
4 5 17 
6 9 19 
12 
9 14 17 
1 2 6 9 16 17 
7 10 16 
13 
4 7 11 19 
2 6 7 19 
3 15 17 
14 
2 6 12 
2 7 8 16 
7 17 18 19 
15 
1 2 8 
3 4 9 
4 13 16 
16 
5 8 10 
2 3 8 9 12 14 
3 12 15 
17 
6 9 12 
1 4 5 9 11 12 
4 13 14 
18 
3 5 11 
1 5 10 
5 8 14 
19 
3 7 13 
6 10 13 
5 11 14 



(a) 
(b) 
(c) 



(d) 
(e) 
(f) 



(g) 
(h) 
(i) 
Figure 20. Configurations obtained for Drezner
examples with different parameters. The first row, that is,
(a), (b), (c)
presents scattered plots obtained by Drezner’s approach. The distances
in these plots are defined relatively and the physical scale is
irrelevant. Hence there is no reference to the experimental area (no
grid). However, to make comparisons easier, we have applied the same
scale as the one used in the second row. The second row, that is,
(d), (e), (f) shows
stable configurations for afuzzy = 0.05, bfuzzy = 0.1,
and pd/sd = 0.05. In the third row, i.e.,
(g), (h), (i) there
are configurations obtained for afuzzy = 0.01,
bfuzzy = 0.1, and pd/sd = 0.3. The scale here
is different, which results from applied LP parameter values. The values
of f are calculated according to formula (11).
(a) f = 0.060 (b)
f = 0.112; (c) f = 0.075;
(d) f = 0.083, Mean_truth(P1) = 1;
Mean_truth(P3) = 0.99; (e) f = 0.122,
Mean_truth(P1) = 1; Mean_truth(P3) = 1;
(f) f = 0.096, Mean_truth(P1) = 1;
Mean_truth(P3) = 0.98; (g) f = 0.060,
Mean_truth(P1) = 0.4; Mean_truth(P3) = 0.92;
(h) f = 0.111, Mean_truth(P1) = 0.19;
Mean_truth(P3) = 0.85; (i) f = 0.074,
Mean_truth(P1) = 0.28; Mean_truth(P3) = 0.88.
In these configurations, the mean truth values for the applied
patterns and the value of f were calculated according to
Drezner’s method. Analyzing the results from the perspective of the
f function, the plots obtained using the LP approach (row 2)
are worse than those obtained using eigenvectors. However, the
configurations shown in the second row are similar to those in the first
row in terms of the neighborhood structure of the agents. They are
characterized by high average truth values of the P1 and P3 patterns. In
addition, they show a very uniform distribution of agents.
This is undoubtedly due to the similar perception of mutual personal
/ social distance in this model. In practical applications for objects
layout purposes, these distances can reflect, for example, the size of
objects or the required dimensions of space allocated for agents’
activity. The eigenvector approach does not take into account the sizes
of objects in question.
The last row in Figure 20 provides interesting results obtained in
the LP approach. To obtain scatter plots for this row, a series of pilot
experiments were conducted. They aimed to establish such parameters for
patterns P1 and P3 that were expected to act similarly to equation (11)
in computing the objective function.
It is worth noting that the value of f decreases as the
denominator increases. Thus, in principle, the best configuration of
agents is the one in which related objects are close together and
unrelated objects remain distant. In successive trials, the distance
definitions for given patterns, (the afuzzy value from Figure
12) gradually decreased, while the pd/sd value (Figure
14) – increased. The configurations presented in the third row of
Figure 20 (the best of 10 trials) were obtained for afuzzy =
0.01, bfuzzy = 0.1, and pd/sd = 0.3.
As can be seen from the graphs presented in row 3, the resulting
configurations are very similar in quality to those obtained using
eigenvectors. Furthermore, the assessment of their function f
was analogous and even slightly better for examples 2 and 3. However,
obtaining a high mean truth value for the P1 pattern is not possible due
to the distance definitions adopted in these simulations. The
pd/sd and afuzzy parameters would generally
result in the layout of objects over a larger area than in the second
row presented in Figure 20 and the lower Mean_truth(P1).
7. Discussion and possible applications
The simulation examples presented were designed for illustrative
purposes of the proposed methodology. The analyzed models of
qualitatively known optimal configurations (Figures 8, 12, 13)
reproduced them in repeated simulations. For the bivariate evaluations
t(l) (Figure 13), they are even surprising and suggest that
some analytical solutions exist and could be found for these cases. The
properties of the configurations obtained in the migration models
studied (Figures 17, 19) are generally consistent with the results from
the Sakoda and Schelling models. The observed differences arise
logically from the assumptions made in our approach. Analysis of
Drezner’s [7] scattered plot examples (Figure 20) indicates that our LP
concept gives much more flexibility in modelling this type of problem,
which allows it to be used in specific practical implementations. For
example, it is possible to include the dimensions of objects, which is
impossible in the classical Drezner optimization method.
In general, the incorporation of knowledgebased LPs has expanded the
modelling possibilities and moved beyond treating agents as individuals.
As a result, the approach can be applied in completely different
contexts, and it allows for modelling practical issues other than social
group migration. For example, the freedom to define LPs makes it
possible to obtain scatter plots for facility layout problems. In this
area, it may be interesting to analyze the desired locations and
neighborhoods of the collaborating human team members. The results of
such modelling can be used, for example, to design their arrangements in
open office spaces. Furthermore, scattered plots may facilitate the
determination of the desired arrangement of greenfielddesigned factory
components, or production systems. In the latter cases, agents can be
interpreted as interacting buildings and/or machines.
Importantly, relationships and variables are specified for each task
using formulations defined by expert knowledge in terms similar to
natural language. Overall, in any application problem, one has to create
such linguistic patterns that reflect logical relationships and the
desired state of the examined system in reality. Since often such a
formal description by classic mathematical formulae may be difficult due
to the information uncertainty, the fuzzy sets and linguistic patterns
appear to be well fitted to this job. The determination of the
appropriate patterns can be obtained, for instance, by finding a
consensus between the knowledge of different experts within the given
field or in concrete situations. For example, such a compromise for the
arrangement of production machines within the factory layout may be
expressed by a linguistic pattern of the following form: ‘IF
Transport_between_machines(i, j) IS FREQUENT THEN
Distance_between_machines(i, j) IS SMALL’. Experts
should define the fuzzy set membership functions for FREQUENT and SMALL,
based on their knowledge, experience, and available data. The proposed
linguistic expression corresponds to the logical economic requirement of
arranging objects to minimize transportation costs.
The features described above and the flexibility of our approach also
have some negative consequences. It provides a relatively large number
of degrees of freedom in model construction and analysis. Therefore, the
properties of this methodology require further detailed research, in
particular on the sensitivity of the model to changes in various
parameters, LP definitions, etc.
There are many possibilities to improve and extend the method
proposed in this work. For example, by broadening the concept and its
implementation to include additional evaluation criteria defined by LPs.
These could, for example, involve simultaneous consideration of safety
recommendations in the design of production halls, interaction between
facilities, social or cultural preferences between groups of workers, or
aesthetic evaluation of design solutions. Such a multicriteria approach
would allow modeling, analyzing, and searching for solutions in much
more complex systems. As far as the implementation of the proposed
methodology is concerned, the potential extension should allow movement
parameters to be set for individual agents or their defined groups. An
addition of this type would even more strongly increase the flexibility
of the proposed approach in modeling practical issues from different
areas. In the future, the corresponding computer implementations of our
proposal should take into account the ability to independently and
freely construct LPs and define linguistic variables.
The examples presented in this paper suggest that the LPbased
approach may be an interesting perspective to analyze the dynamics of
interconnected agents with different mutual attitudes in various
contexts. A unique feature of this approach is the ability to define
mutual relations and behavioral rules using expressions similar to those
found in natural language. They model imprecisely defined behavioral
rules for agents. The rules of multivalued logic and fuzzy sets applied
to such modelling are a generalization of traditional bivalued logic and
sets. Thus, our models can also operate on exact data (physical
properties) and/or combine various types of data and relations. The
implemented model of agent behavior, although evaluated and validated
rather qualitatively, is promising. In all cases, a stable configuration
of the agents was achieved relatively quickly.
The interaction rules in our approach are simple, the same for each
agent, and the number of agents is significant. According to the
classification proposed by Kliemt [59], this type of model belongs to
the thin group. At the other extreme of the mentioned
classification are thick models. This type of modeling consists
in reproducing, as accurately as possible, the knowledge about
characteristics and behavior of a comparatively small group of diverse
agents as, for example, in [60], [61]. In these works, the
characteristics and rules of agent behavior were constructed based on
multidisciplinary knowledge of consumer behavior, social psychology,
marketing, and organizational culture. Since our LPbased proposal
facilitates flexible design of patterns using multiple, natural
languagelike expressions, it allows one to easily take the knowledge of
multiple experts from different fields and encapsulate it in a single
approach. Therefore, the design of thick models seems to be
another interesting challenge and a direction for further exploration of
the possibilities offered by our approach.
8. Conclusions
In this research, we showed how the concept of LPs combined with
expert knowledge can be used to model the dynamics of social groups.
This approach belongs to the domain of agentbased modeling that
involves migration.
In our proposal, linguistic phrases similar to natural language
define ideal properties of the examined system. They are the basis for
generating virtual forces that govern the movements of the individual
agent. The development of agent behavior rules results from logical
sentences and the methodology to determine their degree of truth. Such
an approach allows us to construct flexible simulation models.
In this paper, we not only describe our idea in detail, but also
illustrate its capabilities and properties by simple examples and a
series of simulation experiments. They include problems of known
solutions with stable final configurations that were analyzed. The
analyses show that qualitative results of classical ideas from previous
works can be obtained without original limitations such as moving on a
grid. These models, as shown in the examples, also use the paradigm of
inferring the behavior of the dynamics of the entire social system based
on the interactions between its members.
We also validated our approach by applying and comparing it with a
suboptimal scattered plot generation method proposed by Drezner. The
simulations performed for classic problems showed the convergence of
agent dynamic behavior in our method with the solutions provided by the
Drezner approach.
As was broadly discussed in the previous section, the presented
method can be potentially widely used in a variety of situations, and
the proposed framework can be easily extended to model other types of
complex systems.
References
[1] J. M. Sakoda, “The checkerboard model of social interaction,”
The Journal of Mathematical Sociology, vol. 1, no. 1, pp.
119–132, Jan. 1971, doi: 10.1080/0022250X.1971.9989791.
[2] T. C. Schelling, “Models of Segregation,” The American
Economic Review, vol. 59, no. 2, pp. 488–493, 1969.
[3] T. C. Schelling, “Dynamic models of segregation,” The Journal
of Mathematical Sociology, vol. 1, no. 2, pp. 143–186, Jul. 1971,
doi: 10.1080/0022250X.1971.9989794.
[4] J. Grobelny, “The fuzzy approach to facilities layout problems,”
Fuzzy Sets and Systems, vol. 23, no. 2, pp. 175–190, Aug. 1987,
doi: 10.1016/01650114(87)900571.
[5] J. Grobelny, “The ‘linguistic pattern’ method for a workstation
layout analysis,” International Journal of Production Research,
vol. 26, no. 11, pp. 1779–1798, Nov. 1988, doi:
10.1080/00207548808947991.
[6] F. S. Beltran, L. Salas, and V. Quera, “Spatial Behavior in
Groups: an AgentBased Approach,” Journal of Artificial Societies
and Social Simulation, vol. 9, no. 3, pp. 1–5, 2006.
[7] Z. Drezner, “A Heuristic Procedure for the Layout of a Large
Number of Facilities,” Management Science, vol. 33, no. 7, pp.
907–915, Jul. 1987, doi: 10.1287/mnsc.33.7.907.
[8] I. N. Lymperopoulos and G. D. Ioannou, “Understanding and
modeling the complex dynamics of the online social networks: a scalable
conceptual approach,” Evolving Systems, vol. 7, no. 3, pp.
207–232, Sep. 2016, doi: 10.1007/s1253001691459.
[9] R. Mercuur, V. Dignum, and C. M. Jonker, “Integrating Social
Practice Theory in AgentBased Models: A Review of Theories and Agents,”
IEEE Transactions on Computational Social Systems, vol. 7, no.
5, pp. 1131–1145, Oct. 2020, doi: 10.1109/TCSS.2020.3007930.
[10] J. Castro et al., “A review of agentbased modeling of
climateenergy policy,” WIREs Climate Change, vol. 11, no. 4,
p. e647, 2020, doi: 10.1002/wcc.647.
[11] R. de A. Silva and R. T. V. Braga, “Simulating
SystemsofSystems With AgentBased Modeling: A Systematic Literature
Review,” IEEE Systems Journal, vol. 14, no. 3, pp. 3609–3617,
Sep. 2020, doi: 10.1109/JSYST.2020.2980896.
[12] A. Micolier, P. Loubet, F. Taillandier, and G. Sonnemann, “To
what extent can agentbased modelling enhance a life cycle assessment?
Answers based on a literature review,” Journal of Cleaner
Production, vol. 239, p. 118123, Dec. 2019, doi:
10.1016/j.jclepro.2019.118123.
[13] S. M. Reia, A. C. Amado, and J. F. Fontanari, “Agentbased
models of collective intelligence,” Physics of Life Reviews,
vol. 31, pp. 320–331, Dec. 2019, doi: 10.1016/j.plrev.2018.10.004.
[14] K. Rakić, M. Rosić, and I. Boljat, “A Survey of AgentBased
Modelling and Simulation Tools for Educational Purpose,” Tehnički
vjesnik, vol. 27, no. 3, pp. 1014–1020, Jun. 2020, doi:
10.17559/TV20190517110455.
[15] C. Santos, M. J. P. Neto, and M. Neves, “The Influence of
Infographics in Accessing Information: Multidimensionality in Visual
Representation and Configuration of Different Media,” in Advances in
Ergonomics in Design, 2019, pp. 497–508.
[16] P. Hansen, X. Liu, and G. M. Morrison, “Agentbased modelling
and sociotechnical energy transitions: A systematic literature review,”
Energy Research & Social Science, vol. 49, pp. 41–52, Mar.
2019, doi: 10.1016/j.erss.2018.10.021.
[17] H. R. Nasrinpour, M. R. Friesen, and R. D. McLeod, “Agent Based
Modelling and West Nile Virus: A Survey,” J. Med. Biol. Eng.,
vol. 39, no. 2, pp. 178–183, Apr. 2019, doi:
10.1007/s4084601803968.
[18] S. Flaxman et al., “Estimating the effects of
nonpharmaceutical interventions on COVID19 in Europe,”
Nature, vol. 584, no. 7820, pp. 257–261, Aug. 2020, doi:
10.1038/s4158602024057.
[19] L. Wynants et al., “Prediction models for diagnosis and
prognosis of covid19: systematic review and critical appraisal,”
BMJ, vol. 369, p. m1328, Apr. 2020, doi: 10.1136/bmj.m1328.
[20] M. J. North, “A theoretical formalism for analyzing agentbased
models,” Complex Adapt Syst Model, vol. 2, no. 1, p. 3, May
2014, doi: 10.1186/2194320623.
[21] R. Hegselmann and A. Flache, “Understanding Complex Social
Dynamics,” Journal of Artificial Societies and Social
Simulation, vol. 1, no. 3, 1998, Accessed: Oct. 30, 2021. [Online].
Available: https://www.jasss.org/1/3/1.html
[22] J. ; S. Klüver, “Simulations of Group Dynamics with Different
Models,” Journal of Artificial Societies and Social Simulation,
vol. 6, no. 4, Oct. 2003, Accessed: Oct. 30, 2021. [Online]. Available:
https://www.jasss.org/6/4/8.html
[23] J. L. Moreno, Who shall survive? A new approach to the
problem of human interrelations. Washington: Nervous and Mental
Disease Pub. Co., 1934.
[24] E. T. Hall, The hidden dimension. Garden City, N.Y.:
Doubleday, 1966.
[25] S. Hoogendoorn and P. H.L. Bovy, “Simulation of pedestrian flows
by optimal control and differential games,” Optim. Control Appl.
Meth., vol. 24, no. 3, pp. 153–172, May 2003, doi:
10.1002/oca.727.
[26] D. Helbing, I. Farkas, and T. Vicsek, “Simulating dynamical
features of escape panic,” Nature, vol. 407, no. 6803, pp.
487–490, Sep. 2000, doi: 10.1038/35035023.
[27] L. Gao, B. Durnota, Y. Ding, and H. Dai, “An agentbased
simulation system for evaluating gridding urban management strategies,”
KnowledgeBased Systems, vol. 26, pp. 174–184, Feb. 2012, doi:
10.1016/j.knosys.2011.07.018.
[28] S. Sharma, K. Ogunlana, D. Scribner, and J. Grynovicki,
“Modeling human behavior during emergency evacuation using intelligent
agents: A multiagent simulation approach,” Inf Syst Front,
vol. 20, no. 4, pp. 741–757, Aug. 2018, doi:
10.1007/s107960179791x.
[29] B. Yıldız and G. Çağdaş, “Fuzzy logic in agentbased modeling of
user movement in urban space: Definition and application to a case study
of a square,” Building and Environment, vol. 169, p. 106597,
Feb. 2020, doi: 10.1016/j.buildenv.2019.106597.
[30] I. Moya, M. Chica, J. L. SáezLozano, and Ó. Cordón, “An
agentbased model for understanding the influence of the 11M terrorist
attacks on the 2004 Spanish elections,” KnowledgeBased
Systems, vol. 123, pp. 200–216, May 2017, doi:
10.1016/j.knosys.2017.02.015.
[31] H. Bin and D. Zhang, “CellularAutomata Based Qualitative
Simulation for Nonprofit Group Behavior,” Journal of Artificial
Societies and Social Simulation, vol. 10, no. 1, 2007, Accessed:
Oct. 30, 2021. [Online]. Available:
https://www.jasss.org/10/1/1.html
[32] W. Yu and D. Helbing, “Game Theoretical Interactions of Moving
Agents,” in Simulating Complex Systems by Cellular Automata, J.
Kroc, P. M. A. Sloot, and A. G. Hoekstra, Eds. Berlin, Heidelberg:
Springer, 2010, pp. 219–239. doi: 10.1007/9783642122033_10.
[33] D. Helbing, W. Yu, and H. Rauhut, “SelfOrganization and
Emergence in Social Systems: Modeling the Coevolution of Social
Environments and Cooperative Behavior,” The Journal of Mathematical
Sociology, vol. 35, no. 1–3, pp. 177–208, Jan. 2011, doi:
10.1080/0022250X.2010.532258.
[34] A. KowalskaStyczeń and K. SznajdWeron, “Access to information
in word of mouth marketing within a cellular automata model,”
Advances in Complex Systems, vol. 15, no. 8, p. 1250080,
2012.
[35] T. Ma and Y. Nakamori, “Agentbased simulation for Kansei
Engineering: Testing a fuzzy linear quantification method in an
artificial world,” J. Syst. Sci. Syst. Eng., vol. 16, no. 3,
pp. 308–322, Sep. 2007, doi: 10.1007/s1151800750555.
[36] L. Iandoli, E. Marchione, C. Ponsiglione, and G. Zollo,
“Computing ambiguity in complex systems with fuzzy logic,” Fuzzy
Economic Review, vol. XIV, pp. 67–90, Jan. 2009, doi:
10.25102/fer.2009.01.04.
[37] R. R. Yager, “Families of OWA operators,” Fuzzy Sets and
Systems, vol. 59, no. 2, pp. 125–148, Oct. 1993, doi:
10.1016/01650114(93)90194M.
[38] S. Hassan, L. Garmendia, and J. Pavon, “Introducing uncertainty
into social simulation: using fuzzy logic for agentbased modelling,”
International Journal of Reasoningbased Intelligent Systems,
Aug. 2010, Accessed: Jan. 18, 2022. [Online]. Available:
https://www.inderscienceonline.com/doi/abs/10.1504/IJRIS.2010.034907
[39] S. Hassan, M. Salgado, and J. Pavón, “Friendship Dynamics:
Modelling Social Relationships through a Fuzzy AgentBased Simulation,”
Discrete Dynamics in Nature and Society, vol. 2011, p. e765640,
Sep. 2011, doi: 10.1155/2011/765640.
[40] S. Anand, I. Sharma, and R. Goyal, “Extended Understanding of
Dyadic Friendship Using Fuzzy Measures: a Simulation Approach,” IEEE
Access, vol. 5, pp. 21179–21192, 2017, doi:
10.1109/ACCESS.2017.2758395.
[41] J. MartínezMiranda and J. Pavón, “Modeling the influence of
trust on work team performance,” SIMULATION, vol. 88, no. 4,
pp. 408–436, Apr. 2012, doi: 10.1177/0037549711404714.
[42] M. Raoufi and A. Robinson Fayek, “Fuzzy AgentBased Modeling of
Construction Crew Motivation and Performance,” J. Comput. Civ.
Eng., vol. 32, no. 5, p. 04018035, Sep. 2018, doi:
10.1061/(ASCE)CP.19435487.0000777.
[43] N. S. Kedir, M. Raoufi, and A. R. Fayek, “Fuzzy AgentBased
Multicriteria DecisionMaking Model for Analyzing Construction Crew
Performance,” Journal of Management in Engineering, vol. 36,
no. 5, p. 04020053, Sep. 2020, doi:
10.1061/(ASCE)ME.19435479.0000815.
[44] M. Raoufi and A. R. Fayek, “Hybrid fuzzy Monte Carlo agentbased
modeling of workforce motivation and performance in construction,”
Construction Innovation, vol. 21, no. 3, pp. 398–416, Jan.
2021, doi: 10.1108/CI0320200045.
[45] S. Mei et al., “Individual Decision Making Can Drive
Epidemics: A Fuzzy Cognitive Map Study,” IEEE Transactions on Fuzzy
Systems, vol. 22, no. 2, pp. 264–273, Apr. 2014, doi:
10.1109/TFUZZ.2013.2251638.
[46] P. J. Giabbanelli, S. A. Gray, and P. Aminpour, “Combining fuzzy
cognitive maps with agentbased modeling: Frameworks and pitfalls of a
powerful hybrid modeling approach to understand humanenvironment
interactions,” Environmental Modelling & Software, vol. 95,
pp. 320–325, Oct. 2017, doi: 10.1016/j.envsoft.2017.06.040.
[47] S. Mehryar, R. Sliuzas, N. Schwarz, A. Sharifi, and M. van
Maarseveen, “From individual Fuzzy Cognitive Maps to Agent Based Models:
Modeling multifactorial and multistakeholder decisionmaking for water
scarcity,” Journal of Environmental Management, vol. 250, p.
109482, Nov. 2019, doi: 10.1016/j.jenvman.2019.109482.
[48] J. GiráldezCru, M. Chica, O. Cordón, and F. Herrera, “Modeling
agentbased consumers decisionmaking with 2tuple fuzzy linguistic
perceptions,” International Journal of Intelligent Systems,
vol. 35, no. 2, pp. 283–299, 2020, doi: 10.1002/int.22211.
[49] J. GiráldezCru, M. Chica, and O. Cordón, “A framework of
opinion dynamics using fuzzy linguistic 2tuples,” KnowledgeBased
Systems, vol. 233, p. 107559, Dec. 2021, doi:
10.1016/j.knosys.2021.107559.
[50] A. E. BadilloMárquez, A. A. AguilarLasserre, M. A.
MirandaAckerman, O. O. SandovalGonzález, D. VillanuevaVásquez, and R.
PosadaGómez, “An AgentBased ModelDriven Decision Support System for
Assessment of Agricultural Vulnerability of Sugarcane Facing Climatic
Change,” Mathematics, vol. 9, no. 23, Art. no. 23, Jan. 2021,
doi: 10.3390/math9233061.
[51] E. H. Mamdani and S. Assilian, “An experiment in linguistic
synthesis with a fuzzy logic controller,” International Journal of
ManMachine Studies, vol. 7, no. 1, pp. 1–13, Jan. 1975, doi:
10.1016/S00207373(75)800022.
[52] J. Grobelny, “On one possible ‘fuzzy’ approach to facilities
layout problems,” International Journal of Production Research,
vol. 25, no. 8, p. 1123, Aug. 1987.
[53] A. D. Raoot and A. Rakshit, “A ‘linguistic pattern’ approach for
multiple criteria facility layout problems,” International Journal
of Production Research, vol. 31, no. 1, pp. 203–222, Jan. 1993,
doi: 10.1080/00207549308956721.
[54] J. Grobelny and R. Michalski, “A novel version of simulated
annealing based on linguistic patterns for solving facility layout
problems,” KnowledgeBased Systems, vol. 124, pp. 55–69, May
2017, doi: 10.1016/j.knosys.2017.03.001.
[55] J. Grobelny, “Fuzzybased linguistic patterns as a tool for the
flexible assessment of a priority vector obtained by pairwise
comparisons,” Fuzzy Sets and Systems, vol. 296, pp. 1–20, Aug.
2016, doi: 10.1016/j.fss.2015.05.012.
[56] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and
Applications. New York: Academic Press, 1980.
[57] L. A. Zadeh, “Fuzzy sets as a basis for a theory of
possibility,” Fuzzy Sets and Systems, vol. 1, no. 1, pp. 3–28,
Jan. 1978, doi: 10.1016/01650114(78)900295.
[58] L. A. Zadeh, “PRUF—a meaning representation language for natural
languages,” International Journal of ManMachine Studies, vol.
10, no. 4, pp. 395–460, Jul. 1978, doi:
10.1016/S00207373(78)800030.
[59] H. Kliemt, “Simulation and Rational Practice,” in Modelling
and Simulation in the Social Sciences from the Philosophy of Science
Point of View, R. Hegselmann, U. Mueller, and K. G. Troitzsch, Eds.
Dordrecht: Springer Netherlands, 1996, pp. 13–27. doi:
10.1007/9789401586863_2.
[60] T. J. Norman et al., “Agentbased formation of virtual
organisations,” KnowledgeBased Systems, vol. 17, no. 2–4, pp.
103–111, May 2004, doi: 10.1016/j.knosys.2004.03.005.
[61] O. Roozmand, N. GhasemAghaee, G. J. Hofstede, M. A.
Nematbakhsh, A. Baraani, and T. Verwaart, “Agentbased modeling of
consumer decision making process based on power distance and
personality,” KnowledgeBased Systems, vol. 24, no. 7, pp.
1075–1095, Oct. 2011, doi: 10.1016/j.knosys.2011.05.001.
Appendix A
The proposed model pseudocode:
Define input data:
Links matrix L_{n×n}
Patterns definitions and relationships of calculating t(l)
and t(r)
Determination of the (rectangular) area
Arrange randomly n agents in the available area
Determine maximal number of steps in simulation t and values
of s and r
count = 1,
Repeat
For i = 1 to n do
Begin
VF = 0
For j = 1 (and j <> i) to n
do
Begin
If Distance(i, j) <= r then
Begin
For each Pattern do
Begin
Determine pattern Truth(i, j)
Determine vector VF(i, j)
VF := VF + VF(i, j)
End
End(If)
Move agent i according to VF adjusted by
s/n
End
Calculate Mean_truth values for agent i (for all
patterns)
End
Calculate and write Mean_truth values for all patterns and all
agents
count = count + 1
Until count = t + 1
