=Paper=
{{Paper
|id=Vol-1269/paper208
|storemode=property
|title=Methods of Calculating the Strength of Coalition in a Dispersed Decision Support System with the Stage of Negotiations - a Study of Medical Data
|pdfUrl=https://ceur-ws.org/Vol-1269/paper208.pdf
|volume=Vol-1269
|dblpUrl=https://dblp.org/rec/conf/csp/Przybyla-KasperekW14
}}
==Methods of Calculating the Strength of Coalition in a Dispersed Decision Support System with the Stage of Negotiations - a Study of Medical Data==
https://ceur-ws.org/Vol-1269/paper208.pdf
Methods of calculating the strength of coalition
in a dispersed decision support system
with the stage of negotiations - a study of
medical data
Malgorzata Przybyla-Kasperek and Alicja Wakulicz-Deja
University of Silesia, Institute of Computer Science,
Bȩdzińska 39, 41-200 Sosnowiec, Poland
{malgorzata.przybyla-kasperek,alicja.wakulicz-deja}@us.edu.pl
http://www.us.edu.pl
Abstract. The article discusses the issues related to the decision-making
system using dispersed knowledge. By dispersed knowledge we under-
stand knowledge from one domain but stored in a set of knowledge bases.
This article focuses on the use of dispersed medical data. A dispersed
decision-making system with the negotiation stage was used. The impact
of three different approaches to determining the strength of coalition on
the effectiveness of inference in the system, have been studied.
Keywords: decision support system, dispersed decision-making system,
negotiations, strength of coalition, conflict analysis
1 Introduction
In many hospitals and medical centers large volumes of medical data are col-
lected. Mostly they are stored in a dispersed form, because each hospital has
its own knowledge base. Support the decision-making process in such situations
is a great challenge and a difficult task. Issues concerning the use of dispersed
knowledge was considered by the authors in earlier papers [7–9]. In this paper, we
develop the concepts discussed earlier. We use the system with the negotiation
stage. Our main goal is to investigate the impact on the efficiency of inference
of the methods for determining the strength of coalition.
The concept of distributed decision making (DDM) is widely discussed in
the paper [10]. In DDM methods, it is assumed that the data are collected and
stored in different decision tables representing either horizontally or vertically
partitioned. In this paper, we consider an approach in which both the sets of
conditional attributes and the universe are not necessarily disjoint or equal.
This is more general approach than in DDM and thus we use the term dispersed
knowledge rather than distributed knowledge. The concept of taking a global de-
cision on the basis of local decisions is also used in issues concerning the multiple
model approach [3]. In a multiple classifier system, an ensemble is constructed
2 M. Przybyla-Kasperek, A. Wakulicz-Deja
on the basis of base classifiers. The aim of this approach is to reduce the mis-
classification at the cost of increased computational complexity. Examples of the
application of this approach can be found in the literature [1, 13]. Also in many
other papers [2, 12], the problem of using distributed knowledge is considered.
This paper describes a different approach to the global decision-making process.
We assume that the set of local knowledge bases that contain information from
one domain is pre-specified. The only condition which must be satisfied by the
local knowledge bases is to have common decision attributes.
An approach to the issue of coalition formation was considered in the papers
of Pawlak [4–6]. This model describes a conflict situation in which the agents
have decided to analyze the conflict by using a peaceful method. The articles
provide a definition of the relations - conflict, friendship and neutrality. This
paper is based on the concepts that were provided by Pawlak for the analysis of
conflict and coalition building.
2 The definition of dispersed decision-making system
In a dispersed decision-making system, a set of local knowledge bases that con-
tain the knowledge from the same domain, is available. Local knowledge bases
can be defined very variously, and therefore they can not be easily and without
conflicts combined into a single coherent knowledge base. That’s why, in order
to make a global decision, a hierarchical structure of the system is created. In
this structure the knowledge bases, on the basis of which similar decisions are
taken, are combined into one coalition. It is realized in two steps. At first initial
coalitions are created. Then the negotiation stage is implemented. Below basic
notations are given and a brief overview of the process of structure creating is
described. A more detailed description is given in the papers [8, 9].
We assume that each local knowledge base is managed by one agent. We
use the definition of an agent introduced by Pawlak in [4]. We use two types of
agents. The first is a resource agent. The resource agent has access to its local
knowledge base on the basis of which it can establish the value of a local decision
through the process of inference.
Definition 1. We call ag in Ag a resource agent if he has access to resources
represented by a decision table Dag := (Uag , Aag , dag ), where Uag is a set called
a
the universe; Aag is a set of conditional attributes, and Vag is a set of attribute
a values that contain the special signs * and ?. Equation a(x) = ∗ for some
x ∈ Uag means that for an object x, the value of attribute a has no influence
on the value of the decision attribute, while the equation a(x) =? means that
the value of attribute a for object x is unknown; dag is referred to as a decision
d
attribute and Vag is called the value set of dag .
Each resource agent ag ∈ Ag can independently determine the value of the
decision for a test object for which the values on the set of attributes Aag are
defined. In order to identify agents who make consistent decisions, for each agent
a vector representing the classification of the test object made by the agents is
Methods of calculating the strength of coalition 3
determined. This vector will be defined on the basis of certain relevant objects.
That is the objects from the decision tables of agents that carry the greatest
similarity to the test object. From decision table of resource agent Dag , ag ∈
Ag and from each decision class Xvag , v ∈ V dag , the smallest set containing at
least m1 objects for which the values of conditional attributes bear the greatest
similarity to the test object is chosen. The value of the parameter m1 is selected
experimentally. Then for each resource agent i ∈ {1, . . . , n} and the test object x,
a c-dimensional vector [µ̄i,1 (x), P
. . . , µ̄i,c (x)] is generated, where the value µ̄i,j (x)
rel ∩X agi s(x,y)
y∈Uag vj
is defined as follows: µ̄i,j (x) = i
card{Uagrel ∩X agi } , i ∈ {1, . . . , n}, j ∈ {1, . . . , c},
vj
i
d rel
where c = card{V }, Uag i
is the subset of relevant objects selected from the
decision table Dagi of resource agent agi and Xvag j
i
is the decision class of the
decision table of resource agent agi ; and s(x, y) is the measure of similarity
between objects x and y. In the experimental part of this paper the Gower
similarity measure [9] was used. This measure enables the analysis of data sets
that have qualitative, quantitative and binary attributes. The value of µ̄i,j (x)
is equal to the average value of the similarity of the test object to the relevant
objects of agent agi , belonging to the decision class vj . On the basis of the
vector of values defined above, a vector of the rank is specified. The vector
of rank is defined as follows: rank 1 is assigned to the values of the decision
attribute that are taken with the maximum level of certainty. Rank 2 is assigned
to the next most certain decisions, etc. Proceeding in this way for each resource
agent agi , i ∈ {1, . . . , n}, the vector of rank [ri,1 (x), . . . , ri,c (x)] will be defined.
In order to create clusters of agents, relations between the agents are defined.
The definitions of friendship, conflict and neutrality relation are given next.
Relations between agents are defined by their views on the classification of the
test object x to the decision class. We define the function φxvj for the test object
x and each value of the decision attribute vj ∈ V d ; φxvj : Ag × Ag → {0, 1}
x 0 if ri,j (x) = rk,j (x)
φvj (agi , agk ) = where agi , agk ∈ Ag. We also define the
1 if ri,j (x) 6= rk,j (x)
intensity of conflict between agents using a function of the distance between
agents. We define the distance between P agents ρx for the test object x: ρx :
vj ∈V d
φx
v (agi ,agk )
Ag × Ag → [0, 1], ρx (agi , agk ) = j
card{V d }
, where agi , agk ∈ Ag.
Definition 2. Let p be a real number, which belongs to the interval [0, 0.5).
We say that agents agi , agk ∈ Ag are in a friendship relation due to the object
x, which is written R+ (agi , agk ), if and only if ρx (agi , agk ) < 0.5 − p. Agents
agi , agk ∈ Ag are in a conflict relation due to the object x, which is written
R− (agi , agk ), if and only if ρx (agi , agk ) > 0.5 + p. Agents agi , agk ∈ Ag are in a
neutrality relation due to the object x, which is written R0 (agi , agk ), if and only
if 0.5 − p ≤ ρx (agi , agk ) ≤ 0.5 + p.
By using the relations defined above we can create groups of resource agents,
which are not in conflict relation. The first step involves the creation of groups
of agents remaining in the friendship relation.
4 M. Przybyla-Kasperek, A. Wakulicz-Deja
Definition 3. Let Ag be the set of resource agents. The initial cluster due to the
classification of object x is the maximum, due to the inclusion relation, subset
of resource agents X ⊆ Ag such that ∀agi ,agk ∈X R+ (agi , agk ).
After the first stage of clusters creating we obtain a set of initial clusters and
a set of agents who are not included in any cluster. In the second group there
are agents who remained undecided. By undecided agents we mean those who
are in the neutrality relation with agents belonging to the initial cluster. In the
second step the negotiations issues are applied, and agents who are neutral are
join to an existing coalition. But now some concessions are accepted. We assume
that during the negotiation, agents put the greatest emphasis on compatibility
of ranks assigned to the decisions with the highest ranks. That is the decisions
that are most significant for the agent. Compatibility of ranks assigned to less
meaningful decision is omitted. Now we will proceed to the formal description
of the second stage of cluster creating process.
We define the P
function φxG for the test object x; φxG : Ag × Ag → [0, ∞)
v ∈Sign |ri,l (x)−rj,l (x)|
φxG (agi , agj ) = l i,j
card{Signi,j } where agi , agj ∈ Ag and Signi,j ⊆ V d
is the set of significant decision values for the pair of agents agi , agj . In the set
Signi,j there are the values of the decision, which the agent agi or agent agj
gave the highest rank.
During the negotiation stage, the intensity of the conflict between the two
groups of agents is determined by using the generalized distance. The generalized
distance between agents ρxG for the test object x is defined as follows; ρxG :
2Ag × 2Ag → [0, ∞)
0 if card{X ∪ Y } ≤ 1
ρxG (X, Y ) =
X
φxG (ag, ag ′ )
ag,ag ′ ∈X∪Y
else
card{X ∪ Y } · (card{X ∪ Y } − 1)
where X, Y ⊆ Ag. The value of the generalized distance function for two sets of
agents X and Y is equal to the average value of the function φxG for each pair
of agents ag, ag ′ belonging to the set X ∪ Y . This value can be interpreted as
the average difference of the ranks assigned to significant decisions within the
combined group of agents consisting of the sets X and Y .
For each agent ag that has not been included to any initial clusters, the
generalized distance value is determined for this agent and all initial clusters,
with which the agent ag is not in a conflict relation and for this agent and other
agents without coalition, with which the agent ag is not in a conflict relation.
Then the agent ag is included to all initial clusters, for which the generalized
distance does not exceed a certain threshold, which is set by the system’s user.
Also agents without coalition, for which the value of the generalized distance
function does not exceed the threshold, are combined into a new cluster. The
value of the threshold is selected experimentally.
Methods of calculating the strength of coalition 5
After completion of the second stage of the process of clustering we get the
final form of clusters. As was mentioned above, the proposed decision-making
system has a hierarchical structure. The resource agents that are connected into
clusters are located at the lowest level of the hierarchy. For each cluster that
contains at least two resource agents, a superordinate agent is defined, which is
called a synthesis agent, asj , where j- number of cluster.
The definition of a dispersed decision-making system is given next.
Definition 4. By a dispersed decision-making system (multi-agent system) with
dyn
dynamically generated clusters we mean W SDAg = Ag, {Dag : ag ∈ Ag}, {Asx :
x is a classified object}, {δx : x is a classified object} where Ag is a finite set of
resource agents; {Dag : ag ∈ Ag} is a set of decision tables of resource agents;
Asx is a finite set of synthesis agents defined for clusters dynamically generated
for the test object x, δx : Asx → 2Ag is a injective function that each synthesis
agent assigns a cluster generated due to classification of the object x.
3 The strength of coalition and conflict analysis
On the basis of the knowledge of agents from one cluster, local decisions are
taken. An important problem that occurs when taking a global decision is to
eliminate inconsistencies in the knowledge stored in different knowledge bases.
This problem stems from the fact that the system has the general assumptions
and we do not require that the sets of conditional attributes of decision tables
are disjoint. In previous papers some methods of elimination inconsistencies in
the knowledge have been proposed [9]. In this paper, one of these methods - the
approximated method of the aggregation of decision tables, will be used. In this
method for every cluster, a kind of combined information is determined. Each
synthesis agent has access to aggregated decision table. Object of this table are
constructed by combining relevant object from decision tables of the resource
agents that belong to one cluster.
After the completion of the process of the elimination of any inconsistencies in
the knowledge, a c-dimensional vector of values [µj,1 (x), . . . , µj,c (x)] is generated
for each cluster j ∈ {1, . . . , card{As}}, where c is the number of all of the decision
classes. The value µj,i (x) determines the level of certainty with which the decision
vi is taken by agents for a given test object x belonging to the cluster j. The
vector of values assigned to the cluster is defined as follows. The value µj,i (x)
is equal to the maximum value of the similarity measure of objects from the
decision class vi of the decision table of synthesis agent asj to the test object x.
Because the synthesis agents can take contradictory decisions for a given set
of conditions, conflict analysis methods must be used. The method of a density-
based algorithm, which was described in the paper [9], is used. In this method the
generated set of global decisions will contain not only the value of the decisions
that have the greatest support of knowledge stored in local knowledge bases,
but also those for which the support is relatively high. Below, three different
approaches to determining the global decisions are described. These approaches
are used together with the density-based method.
6 M. Przybyla-Kasperek, A. Wakulicz-Deja
The approache without the strength of cluster
Each j-th synthesis agent votes for different decision values with the voting power
equal to the value of the coordinate of the vector [µj,1 (x), . . . , µj,c (x)].
The approache with the strength of cluster
In this paper, a modification of the method of calculating the vectors that as-
signed to clusters, is proposed. This modification consists in taking into account
the strength of the cluster, which is expressed by the number of its component
agents. In the proposed method of creating the system’s structure, inseparable
clusters are generated. Thus, one agent may be included in many clusters. This
means that the partial participation of the agent in the creation of the cluster
should be considered. Thus, in the first stage of the process of determining the
strength of cluster, a membership of each agent in the clusters is calculated.
For each resource agent ag ∈ Ag and given test object x a coefficient of agent’s
membership in clusters is defined mxag = card{as∈As1x :ag∈δx (as)} . The value of the
agent’s membership in clusters is inversely proportional to the number of clus-
ters to which the agent belongs. Then, for each cluster the sum of the agent’s
membership in clusters is calculated. Thus, for each synthesis agent P as ∈ Asx the
strength of cluster subordinate to the agent as is determined ag∈δx (as) mxag .
The vector
P assigned to j-th cluster j ∈ {1, . . . , card{As}} is multiplied by a
x
ag∈δx (asj ) mag
h i
scalar card{Ag} · µ j,1 (x), . . . , µ j,c (x) . In this way the vectors are calcu-
lated in proportion to the strength of the clusters. Thanks to such transforma-
tion, the large clusters have a greater impact on the decisions, while the impact
of small clusters decreases.
The approache with the strength of cluster and the diversity of agents
In this modification of the method of calculating the vectors that assigned to
clusters, in addition to the agent’s membership in clusters, it is also taken into
account the variability of the vector values [µ̄i,1 (x), . . . , µ̄i,c (x)] assigned to the
i-th resource agent. This is realized in the following way. For each resource
agent agi ∈ Ag, the q standard deviation of the vector values is determined as
Pc 2
k=1 µ̄i,k (x)
x 1
Pc
follows SDag i
= c · j=1 c − µ̄i,j (x) . Then for each resource
x
x SDag
agent a coefficient is calculated mvag = mxagi · P . Analogously to the
i
x
i ag ′ ∈Ag SDag ′
x
P
previous approach for each synthesis agent as ∈ Asx the value ag∈δx (as) mvag
is determined and the vector assigned to j-th cluster is multiplied by this value.
In this way the vectors are recalculated in proportion not only to the strength of
the clusters but also to the decisiveness of agents. Thanks to such transformation,
the agents who make decisions with the same degree of certainty have less impact
on the global decisions, while the impact of agents, which are sure of decisions
taken, increases.
4 Results of experiments
The experiments were performed on data sets from medical domain. The follow-
ing data were used: Audiology (Standardized), Lymphography, Primary Tumor.
Methods of calculating the strength of coalition 7
These data are available in the UCI repository. Audiology was obtained from
the Baylor College of Medicine, Houston, Texas. In this data set, on the basis of
values of 69 attributes, a decision is taken what is the cause of hearing problems
(one of 24 possibilities). Lymphography and Primary Tumor was obtained from
the University Medical Centre, Institute of Oncology, Ljubljana, Yugoslavia (M.
Zwitter and M. Soklic provided this data). Lymphography is a medical imaging
technique in which a radiocontrast agent is injected, and then an X-ray picture
is taken to visualize structures of the lymphatic system. This test method gives
great service especially in the evaluation of cancer stage of the lymphatic sys-
tem. In the Primary Tumor data set, on the basis of values of attributes such as
histologic-type, supraclavicular etc. a decision is taken where (of 22 organs) the
cancer cells are located. In order to determine the efficiency of inference each
data set was divided into two disjoint subsets: a training set and a test set. A
numerical summary of the data sets is as follows: Audiology: # The training set
- 200; # The test set - 26; # Conditional - 69; # Decision - 24; Lymphography:
# The training set - 104; # The test set - 44; # Conditional - 18; # Decision - 4;
Primary Tumor : # The training set - 237; # The test set - 102; # Conditional -
17; # Decision - 22. These data can be considered as quite challenging, because
on the basis of a small number of examples, the proper decision value for the test
object, must be assigned, form a large set of decision values. This difficulty has
been confirmed by the experimental results presented in the paper [11]. Our goal
is not only to consider the case in which medical data are collected in a dispersed
form but also improve the efficiency of inference obtained by other methods. The
issue of the use of dispersed medical data is a very important and real problem.
Each hospital or medical center collects data sets, but sets are separable, differ-
ent for each hospital. The possibility to use knowledge from the sets collected
separately and containing information from one domain is very important in real
life area. This approach should significantly improve the efficiency of inference.
At this moment, the authors do not have access to distributed real data sets.
But in the future it is planned to conduct tests on real data. To test the capa-
bilities of the dispersed decision-making system, we must provide knowledge in
a dispersed form. Therefore, each of the training sets was divided into a set of
decision tables. Divisions with a different number of decision tables were consid-
ered. For each of the data sets used, a dispersed decision-making system with
five different versions (with 3, 5, 7, 9 and 11 resource agents) was considered.
dyn
For these systems, we used the following designations: W SDAg1 - 3 resource
dyn dyn dyn
agents; W SDAg2 - 5 resource agents; W SDAg3 - 7 resource agents; W SDAg4 -
dyn
9 resource agents; W SDAg5 - 11 resource agents. Note that the division of the
data set was not made in order to improve the quality of the decisions taken by
the decision-making system, but in order to store the knowledge in a distributed
form. Influence of dispersion of knowledge on the effectiveness of inference was
not tested. The method of dispersing knowledge was specified by the authors in
the following way. The cardinality of the set of conditional attributes in each
decision table of a resource agent was determined and the number of common
conditional attributes of the decision tables was defined. Then, the conditional
8 M. Przybyla-Kasperek, A. Wakulicz-Deja
attributes were randomly assigned to the decision tables so that the conditions
that had been defined earlier were met and each conditional attribute that ap-
pears in the data set is included in at least one set of the conditional attributes
of the decision tables. The measures of determining the quality of the classifi-
cation are: estimator of classification error e in which an object is considered
to be properly classified if the decision class used for the object belonged to
the set of global decisions generated by the system; estimator of classification
ambiguity error eON E in which object is considered to be properly classified if
only one, correct value of the decision was generated to this object; the average
size of the global decisions sets dW SDdyn generated for a test set. In the descrip-
Ag
tion of the results of experiments for clarity some designations for algorithms
have been adopted: A(m2 ) - the approximated method of the aggregation of
decision tables; W - the method of weighted voting; G(ε, M inP ts) - the method
of a density-based algorithm. During experiments influence of the parameter p,
which occurs in the definition 2 of friendship, conflict and neutrality relations,
on the effectiveness of inference of a dispersed decision-making system was an-
alyzed. Four different values of the parameter p were examined. The analyzed
values are p = 0.05, p = 0.1, p = 0.2, p = 0.3. In tables presented below the best
results, obtained for values of the parameter p, are given. Also the optimal pa-
rameter values of m1 , m2 , ε were selected. The parameter m1 and m2 determine
the number of relevant objects that are selected from each decision class of the
decision table of the resource agent, and then are used in the process of cluster
generation or to design the decision table of the synthesis agents. In order to
identify the optimum, the values form 1 to 10 were used. Then, for each system,
the minimum value of the parameters m1 and m2 was chosen, which allowed the
lowest value of estimator of classification error on a test set to be reached. The
value of parameter ε of the method of a density-based algorithm was optimized
by performing a series of experiments with different values of parameter ε that
were increased from 0 to the threshold point. Then a graph was created on which
the points with coordinates (dW SDdyn , e) are marked in an increasing order of
Ag
value ε. Then, the points were marked on the graphs indicate those that had
the greatest improvement in the efficiency of inference. These points satisfy the
following conditions: on the left of the point you can see a significant decrease
in the value of the estimator of classification error and on the right of the point
there is a slight decrease in the value of this estimator with an increase in the
value of parameter ε. The results of the experiments with the Audiology data
set are presented in Table 1, with the Lymphography data set are presented in
Table 2 and with the Primary Tumor data set are presented in Table 3. The
tables show the results for three different approaches presented in this paper:
– dispersed decision-making system with the stage of negotiations and without
calculating the strength of cluster,
– dispersed decision-making system with the stage of negotiations and with
calculating the strength of cluster,
– dispersed decision-making system with the stage of negotiations and with
calculating the strength of cluster and the diversity of agents.
Methods of calculating the strength of coalition 9
The results for one approach discussed in earlier paper [7] - dynamically gener-
ated disjoint clusters, are also presented in Tables. In this approach only friend-
ship and conflict relations are defined. In the process of generating clusters the
negotiations did not occur and agents in friendship relation are connected in
clusters. In tables 1, 2 and 3 the best results in terms of the measures e and
dW SDdyn are bold. Summarizing the results presented in tables 1, 2 and 3: six
Ag
times the best results were achieved using the approach with the stage of negoti-
ations and without calculating the strength of cluster; nine times the best results
were achieved using the approach with the stage of negotiations and calculating
the strength of cluster; sixteen times the best results were achieved using the
approach with the stage of negotiation, calculating the strength of cluster and
the diversity of agents; and only four times the best results were achieved using
the approach with dynamically generated disjoint clusters. Thus, for dispersed
medical data, which were tested, the best approach was the approach with the
stage of negotiation, the strength of cluster and the diversity of agents.
5 Conclusion
In this article the impact of three different approaches to determining the strength
of coalition on the effectiveness of inference in the system with dispersed knowl-
edge have been studied. In the first approach each of the coalition was equally
strong, regardless of the number of agents in the coalition. In the second ap-
proach the strength of coalition was determined by the size of the coalition. In
the third approach, in addition to the size of the coalition, also the diversity of
the decisions made by the individual agents were taken into account. In the ex-
periments dispersed medical data have been used. Based on the presented results
of experiments it can be concluded that the best approach is the third approach.
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10 M. Przybyla-Kasperek, A. Wakulicz-Deja
Table 1. Experiments results with the Audiology data set
The stage of negotiation, without the strength of cluster
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 1, p = 0.3 A(1)G(0.00069; 2) 0.154 0.462 1.615 0.01
m1 = 1, p = 0.3 A(1)G(0.000015; 2) 0.192 0.423 1.308 0.01
W SDAg2 m1 = 1, p = 0.2 A(2)G(0.00099; 2) 0.077 0.462 1.808 0.01
m1 = 1, p = 0.2 A(2)G(0.00078; 2) 0.154 0.385 1.308 0.01
W SDAg3 m1 = 5, p = 0.1 A(1)G(0.002844; 2) 0.077 0.346 1.462 0.01
m1 = 1, p = 0.05 A(1)G(0.00195; 2) 0.154 0.385 1.308 0.01
W SDAg4 m1 = 1, p = 0.1 A(2)G(0.001248; 2) 0.115 0.385 1.462 0.05
m1 = 1, p = 0.05 A(1)G(0.000725; 2) 0.154 0.346 1.231 0.05
W SDAg5 m1 = 2, p = 0.05 A(2)G(0.0036; 2) 0.115 0.462 1.577 0.12
m1 = 5, p = 0.2 A(1)G(0.00249; 2) 0.192 0.423 1.308 0.14
The stage of negotiation, with the strength of cluster
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 2, p = 0.3 A(1)G(0.0003; 2) 0.154 0.500 1.500 0.01
m1 = 1, p = 0.3 A(1)G(0.000011; 2) 0.192 0.423 1.308 0.01
W SDAg2 m1 = 1, p = 0.05 A(1)G(0.0006; 2) 0.115 0.500 1.654 0.01
m1 = 1, p = 0.2 A(2)G(0.00042; 2) 0.154 0.423 1.346 0.01
W SDAg3 m1 = 8, p = 0.05 A(2)G(0.00045; 2) 0.077 0.385 1.538 0.01
m1 = 2, p = 0.3 A(2)G(0.000148; 2) 0.115 0.385 1.308 0.01
W SDAg4 m1 = 2, p = 0.3 A(2)G(0.000364; 2) 0.038 0.500 1.808 0.05
m1 = 2, p = 0.3 A(2)G(0.000284; 2) 0.077 0.346 1.385 0.05
W SDAg5 m1 = 3, p = 0.3 A(3)G(0.000306; 2) 0.077 0.462 1.731 0.29
m1 = 3, p = 0.3 A(3)G(0.00011; 2) 0.154 0.346 1.239 0.29
The stage of negotiation, with the strength of cluster and the diversity of agents
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 1, p = 0.3 A(1)G(0.00018; 2) 0.154 0.500 1.885 0.01
m1 = 1, p = 0.3 A(1)G(0.000012; 2) 0.192 0.423 1.308 0.01
W SDAg2 m1 = 1, p = 0.2 A(2)G(0.000456; 2) 0.115 0.423 1.385 0.01
W SDAg3 m1 = 5, p = 0.3 A(1)G(0.000588; 2) 0.077 0.500 1.808 0.01
m1 = 5, p = 0.2 A(1)G(0.000339; 2) 0.154 0.385 1.346 0.01
W SDAg4 m1 = 3, p = 0.2 A(2)G(0.000486; 2) 0.038 0.385 1.462 0.02
m1 = 2, p = 0.2 A(1)G(0.000276; 2) 0.077 0.346 1.308 0.02
W SDAg5 m1 = 3, p = 0.3 A(3)G(0.000687; 2) 0.038 0.615 2.077 0.28
m1 = 3, p = 0.3 A(3)G(0.000195; 2) 0.077 0.423 1.385 0.28
Dynamically generated disjoint clusters - results presented in the paper [7]
System Algorytm e eON E dW SDAg t
W SDAg1 A(1)G(0.00179; 2) 0.154 0.538 1.808 0.01
m1 = 2 A(1)G(0.0009; 2) 0.192 0.500 1.385 0.01
W SDAg2 A(1)G(0.00219; 1) 0.154 0.577 1.808 0.01
m1 = 1 A(1)G(0.00111; 2) 0.231 0.462 1.308 0.01
W SDAg3 A(1)G(0.00288; 2) 0.077 0.346 1.423 0.01
m1 = 8 A(1)G(0.00207; 2) 0.115 0.346 1.308 0.01
W SDAg4 , m1 = 4 A(1)G(0.002775; 2) 0.154 0.385 1.346 0.01
W SDAg5 A(3)G(0.00642; 2) 0.115 0.538 1.885 0.01
m1 = 3 A(3)G(0.0009; 2) 0.269 0.423 1.269 0.01
Methods of calculating the strength of coalition 11
Table 2. Experiments results with the Lymphography data set
The stage of negotiation, without the strength of cluster
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 2, p = 0.05 A(1)G(0.03; 2) 0.091 0.568 1.477 0.01
m1 = 2, p = 0.05 A(1)G(0.0076; 2) 0.114 0.227 1.114 0.01
W SDAg2 m1 = 2, p = 0.3 A(3)G(0.0204; 2) 0.114 0.523 1.409 0.01
m1 = 2, p = 0.3 A(3)G(0.0128; 2) 0.136 0.318 1.182 0.01
W SDAg3 m1 = 1, p = 0.05 A(1)G(0.0515; 2) 0.114 0.523 1.409 0.01
m1 = 1, p = 0.05 A(1)G(0.0005; 2) 0.159 0.273 1.114 0.01
W SDAg4 m1 = 1, p = 0.05 A(1)G(0.0625; 2) 0.114 0.591 1.477 0.01
m1 = 1, p = 0.05 A(1)G(0.052; 2) 0.136 0.500 1.364 0.01
W SDAg5 m1 = 5, p = 0.3 A(2)G(0.058; 2) 0.159 0.568 1.409 0.07
m1 = 5, p = 0.3 A(2)G(0.0292; 2) 0.182 0.545 1.364 0.07
The stage of negotiation, with the strength of cluster
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 2, p = 0.05 A(1)G(0.0196; 2) 0.091 0.477 1.386 0.01
m1 = 2, p = 0.05 A(1)G(0.004; 2) 0.136 0.273 1.136 0.01
W SDAg2 m1 = 2, p = 0.3 A(3)G(0.014; 2) 0.091 0.568 1.477 0.01
m1 = 2, p = 0.3 A(3)G(0.0044; 2) 0.136 0.227 1.091 0.01
W SDAg3 m1 = 1, p = 0.1 A(1)G(0.0144; 2) 0.114 0.568 1.455 0.01
m1 = 1, p = 0.1 A(1)G(0.0002; 2) 0.159 0.273 1.114 0.01
W SDAg4 m1 = 1, p = 0.05 A(1)G(0.0148; 2) 0.091 0.568 1.477 0.01
m1 = 1, p = 0.05 A(1)G(0.0116; 2) 0.136 0.477 1.341 0.01
W SDAg5 m1 = 2, p = 0.05 A(1)G(0.0126; 2) 0.159 0.636 1.477 0.07
m1 = 2, p = 0.05 A(1)G(0.0092; 2) 0.182 0.500 1.318 0.07
The stage of negotiation, with the strength of cluster and the diversity of agents
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 2, p = 0.3 A(1)G(0.0184; 2) 0.045 0.500 1.455 0.01
m1 = 2, p = 0.1 A(1)G(0.0124; 2) 0.091 0.318 1.227 0.01
W SDAg2 m1 = 2, p = 0.3 A(3)G(0.0128; 2) 0.091 0.568 1.477 0.01
m1 = 2, p = 0.3 A(3)G(0.0044; 2) 0.136 0.227 1.091 0.01
W SDAg3 m1 = 3, p = 0.3 A(1)G(0.014; 2) 0.136 0.523 1.386 0.01
m1 = 3, p = 0.3 A(1)G(0.0002; 2) 0.159 0.273 1.114 0.01
W SDAg4 m1 = 1, p = 0.05 A(1)G(0.0196; 2) 0.114 0.568 1.455 0.01
m1 = 1, p = 0.05 A(1)G(0.0156; 2) 0.136 0.523 1.386 0.01
W SDAg5 m1 = 2, p = 0.05 A(1)G(0.0128; 2) 0.159 0.614 1.477 0.07
m1 = 2, p = 0.05 A(1)G(0.0004; 2) 0.205 0.455 1.250 0.07
Dynamically generated disjoint clusters - results presented in the paper [7]
System Algorithm e eON E dW SDdyn t
Ag
W SDAg1 A(1)G(0.0624; 2) 0.091 0.591 1.545 0.01
m1 = 2 A(1)G(0.0092; 2) 0.182 0.295 1.159 0.01
W SDAg2 A(1)G(0.0775; 2) 0.136 0.636 1.500 0.01
m1 = 2 A(1)G(0.029; 2) 0.159 0.364 1.205 0.01
W SDAg3 A(1)G(0.0858; 2) 0.136 0.591 1.455 0.01
m1 = 2 A(1)G(0.0006; 2) 0.159 0.273 1.114 0.01
W SDAg4 , m1 = 2 A(1)G(0.0702; 2) 0.136 0.455 1.318 0.01
W SDAg5 A(1)G(0.084; 2) 0.159 0.614 1.477 0.01
m1 = 1 A(1)G(0.0672; 2) 0.182 0.545 1.364 0.01
12 M. Przybyla-Kasperek, A. Wakulicz-Deja
Table 3. Experiments results with the Primary Tumor data set
The stage of negotiation, without the strength of cluster
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 5, p = 0.05 A(1)G(0.00546; 2) 0.373 0.814 3.020 0.01
W SDAg2 m1 = 3, p = 0.1 A(2)G(0.00001; 2) 0.343 0.814 3.029 0.02
W SDAg3 m1 = 2, p = 0.05 A(1)G(0.00001; 2) 0.373 0.902 3.745 0.02
W SDAg4 m1 = 4, p = 0.1 A(2)G(0.00001; 2) 0.353 0.882 3.686 0.05
W SDAg5 m1 = 2, p = 0.2 A(3)G(0.00001; 2) 0.314 0.892 4.245 0.36
The stage of negotiation, with the strength of cluster
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 4, p = 0.05 A(3)G(0.00273; 2) 0.382 0.843 3.441 0.01
W SDAg2 m1 = 5, p = 0.05 A(2)G(0.00003; 2) 0.333 0.833 3.049 0.02
W SDAg3 m1 = 2, p = 0.05 A(1)G(0.00001; 2) 0.353 0.892 3.804 0.02
W SDAg4 m1 = 3, p = 0.05 A(3)G(0.00001; 2) 0.353 0.892 3.676 0.05
W SDAg5 m1 = 2, p = 0.2 A(2)G(0.00001; 2) 0.314 0.922 4.294 0.36
The stage of negotiation, with the strength of cluster and the diversity of agents
System Parameters Algorithm e eON E dW SDdyn t
Ag
W SDAg1 m1 = 1, p = 0.3 A(2)G(0.00291; 2) 0.363 0.882 3.167 0.01
W SDAg2 m1 = 3, p = 0.05 A(2)G(0.00003; 2) 0.333 0.833 3.049 0.02
W SDAg3 m1 = 5, p = 0.3 A(1)G(0.00003; 2) 0.353 0.892 3.784 0.02
W SDAg4 m1 = 3, p = 0.3 A(3)G(0.00222; 2) 0.333 0.902 3.784 0.05
W SDAg5 m1 = 2, p = 0.2 A(3)G(0.00003; 2) 0.314 0.912 4.245 0.36
Dynamically generated disjoint clusters - results presented in the paper [7]
System Algorithm e eON E dW SDdyn t
Ag
W SDAg1 , m1 = 5 A(2)G(0.00549; 2) 0.373 0.814 3.020 0.01
W SDAg2 , m1 = 17 A(3)G(0.0003; 2) 0.353 0.814 2.990 0.02
W SDAg3 , m1 = 5 A(5)G(0.00573; 2) 0.373 0.912 3.755 0.02
W SDAg4 , m1 = 4 A(3)G(0.0063; 2) 0.343 0.902 3.667 0.03
W SDAg5 , m1 = 6 A(1)G(0.0003; 2) 0.333 0.941 4.294 0.02
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