Global decision-making in multi-agent decision-making system with dynamically generated disjoint clusters

Applied Soft Computing 40 (2016) 603–615

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Global decision-making in multi-agent decision-making system with dynamically generated disjoint clusters Małgorzata Przybyła-Kasperek ∗ , Alicja Wakulicz-Deja University of Silesia, Institute of Computer Science, Be ¸dzi´ nska 39, 41-200 Sosnowiec, Poland

a r t i c l e

i n f o

Article history: Received 20 February 2013 Received in revised form 12 December 2015 Accepted 13 December 2015 Available online 19 December 2015 Keywords: Decision-making system Global decision Coalition Conflict

a b s t r a c t This paper discusses the issues related to the process of global decision-making on the basis of knowledge which is stored in several local knowledge bases. The approach considered in this paper is very general because we do not assume any additional conditions on the sets of objects or the sets of conditional attributes of local knowledge bases. The paper proposes a new approach to the organization of the structure of multi-agent decision-making system, which operates on the basis of dispersed knowledge. In the presented system, the local knowledge bases will be combined into groups in a dynamic way. We will seek to designate groups of local bases on which the test object is classified to the decision classes in a similar manner. Then, a process of the elimination inconsistencies in the knowledge will be implemented in the created groups. Global decisions will be made by using one of the methods for analysis of conflicts. The paper includes the definition of a multi-agent decision-making system with dynamically generated clusters and a description of a global decision-making process. In addition, the paper presents the results of experiments carried out on data from the UCI repository. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In this paper, the problem of making decisions on the basis of dispersed knowledge stored in many local knowledge bases is examined. In the approach considered, very general assumptions have been adopted. There are several knowledge bases that contain information on the same subject, but that is defined on different sets of attributes. The sets of conditional attributes and the sets of objects in knowledge bases are not necessarily equal, or disjoint. However, the same decision attributes must be present in all of the knowledge bases. In this paper a new approach to the organization of a system structure that uses dispersed knowledge is proposed. The method for creating a coalition of local knowledge bases is the main difference between the previous and the current proposed approach. In earlier papers [33–37,25], the system in which local knowledge bases that had common conditional attributes formed a group was considered, which means that a system with a static structure was previously analyzed. Groups of local knowledge bases that were

∗ Corresponding author. Tel.: +48 32 368 97 57. E-mail address: [email protected] (M. Przybyła-Kasperek). http://dx.doi.org/10.1016/j.asoc.2015.12.016 1568-4946/© 2015 Elsevier B.V. All rights reserved.

created once were used in the decision-making process for all of the test objects. Thus, the structure of groups that were created was completely independent of decisions that were made for the test object based on local knowledge bases. The structure only depends on the conditional attributes of the local knowledge bases. Such an approach is quite rigid. In the approach that is proposed in this paper, an attempt was made to simulate the natural process of creating coalitions. Each time a different test object is considered, new groups of knowledge bases are created that reflect their compatibility in decision making. The new approach is based on the assumption that one group should contain the knowledge bases on the basis of which a similar classification for the test object will be made. As a result, for each test object, the reorganization of local knowledge bases into groups (coalitions) is performed. Such an approach seems to be more natural and obvious. In this paper, it is shown that the proposed dynamic approach provides better results than the static approach. This is also proven using a statistical test that shows that the difference between the classification errors of both approaches is significant. After coalitions are created, the knowledge bases that remain in such a coalition are aggregated. An important problem that occurs when this process is performed is the elimination of inconsistencies in the knowledge that is stored in different knowledge bases. In this paper, the method for the elimination of inconsistencies in the knowledge that was proposed

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in earlier papers [34,35] is used. Local decisions are taken within group based on this aggregated knowledge. Due to the fact that the inference is carried out in groups of knowledge bases, there is a problem of conflict analysis in the proposed system. Two methods of conflict analysis that were proposed in earlier papers [33,34,37] are used in this paper. In the final stage of the global decisionmaking process, a set of global decisions is generated using the method of conflict analysis. From a more technical point of view, the classification problem is under consideration, although not in the classical sense, because we are considering dispersed knowledge. A simple classifier is constructed based on each local knowledge base, which is modeled on the k nearest neighbors method. Local databases are combined into clusters with respect to similarities in the decisions taken. A modified hierarchical agglomeration clustering algorithm is used to build clusters. However, the paper does not focus on the clustering problem. The clustering issue is used in the process of the decisions taken; in other words, in the classification process. Local decisions are taken within the clusters and inconsistencies in the knowledge are eliminated. Then, local decisions are aggregated into global decisions. In order to justify the need for this study, we analyzed an example of a real application of the considered approach. For example, consider four different medical centers that have different apparatuses and diagnostic methods to detect heart problems. Each medical center has different patients, but some patients may be treated at several of the medical centers. Suppose that one of the medical centers in the database stores information about echocardiograms and EKGs. A different medical center only examines the level of cholesterol and performs Doppler tests. Another medical center investigates cholesterol and performs CT Heart Scans and yet another medical center performs echocardiograms, myocardial biopsies and stress tests. Thus, we have the knowledge about the same topic being stored in four different databases. There are different objects in these databases, which can sometimes be shared and there are different attributes that can also be shared. In the previously considered approach, when we used the knowledge from the four medical centers to make a decision for a new patient the following two coalitions always occurred – the first coalition was formed by the first and fourth centers, while the second coalition was formed by the second and third centers. That was because the first and fourth medical centers had the common attribute echocardiograms and the second and third medical centers had the common attribute cholesterol. For each new patient, a coalition tries to reach a common local decision. In this case, compatibility of the decisions within the coalition may not be present, but the members of the coalition must come to an agreement. This approach meant that the voices of some members within the coalition were ignored; those members would perhaps find support among the members of other coalitions. In the new approach, each of the medical centers will first vote on the decision for a new patient. Then the centers that made the same decisions will form a coalition. Different coalitions can be formed for each new patient. In this approach, each member of the coalition can count on the support of the group to which it belongs. The article is organized as follows. In the second section of this paper an overview of papers that are related to the subjects being discussed in this article is included. In the third section a new approach to the structure of a multi-agent decision-making system is proposed. In this section, the definition of a multi-agent decisionmaking system with dynamically generated clusters is given. The fourth section describes the method used for the elimination of inconsistencies in the knowledge. The fifth section describes the methods of conflict analysis. The sixth section gives a description and the results of experiments carried out using some data sets from the UCI repository.

2. Related work It is difficult to find a solution to the exact same problem that is considered in this paper in the literature, which is a situation in which we have knowledge about the same topic being stored in several different databases. There are different objects in databases being considered, which can sometimes be shared and there are also different attributes that also can be shared. Certainly, a decision attribute is a common element that is present in all of these databases. A similar approach is the multiple model approach [13]. In a multiple classifier system, an ensemble is constructed based on base classifiers, which usually permits more precise predictions than any of the individual base classifiers. The accuracy of an ensemble depends on both the quality of the problem decomposition and the individual accuracies in the base classifiers. One of the methods for decomposition is to use the domain knowledge to decompose the nature of the decisions into a hierarchy of layers [18]. In the papers [19,31,38], an ensemble of feature subsets is considered. In the paper [9], a random subspace technique for building an ensemble is considered. A very important issue is that some form of diversity among the base classifiers must exist in order to improve accuracy [29,32]. The method for generating the final decision also has a significant impact on the efficiency of the ensemble [8]. Examples of the application of this approach can be found in the literature [1,30]. This article describes an approach that is different from all of those mentioned above. The main difference is that we assume that a set of local knowledge bases (base classifiers) that contain information from one domain is determined prior to the process of inference – it is set up in advance. The main aim of this paper is to propose a system in which knowledge bases will be combined into groups (coalitions) in a dynamic way. The negotiations for and the formation of coalitions is an important form of social interaction and it is studied in various branches of the social sciences as well as in computer science. Artificial intelligence researchers have investigated the design of agents with negotiation competence from two main perspectives: a theoretical or formal mathematical perspective and a practical or system-building perspective. The various negotiation models that have been proposed in the literature exhibit different features. A brief overview of various negotiation models that have been proposed in the literature can be found in the paper [15]. Zeng and Sycara [40] proposed a sequential decision-making model, called Bazaar, which is able to learn. The model formalizes the issues of negotiation and incorporates a set of negotiation strategies. Faratin et al. [7] presented a model for bilateral service-oriented negotiation that defines a range of strategies and three groups of concession tactics. Lopes et al. [14] presented a negotiation model that formalizes various problem-solving and concession strategies. Nguyen and Jennings [17] presented a model that handles one-tomany negotiations in service-oriented contexts. A buyer engages in multiple concurrent bilateral negotiations with a set of sellers who are capable of providing a specific service. All of these approaches to the issue of coalition formation that were mentioned above are different from the approach considered in this paper. In the approach considered in the paper, the goal of each agent is a classification based on the knowledge accumulated in the local knowledge base to which the agent has access, whereas in the approaches discussed in the literature, the goals of agents have an extremely diversified nature. In the papers of Zdzisław Pawlak [20,21,23], a different approach to the issue of coalition formation was considered. This model describes a conflict situation in which the agents have decided to analyze the conflict by using a peaceful method. In such a situation the relations of conflict, friendship and neutrality were defined and the method of formulating the coalition as well as the method of calculating the intensity of the conflict were proposed. In this paper,

M. Przybyła-Kasperek, A. Wakulicz-Deja / Applied Soft Computing 40 (2016) 603–615

some issues of conflict analysis and coalition formation that were given in Pawlak’s model are used. Conflict analysis has its roots in game theory where two of the basic methods used to measure the power of agents are the Shapley–Shubik power index and the Banzhaf power index. In the paper [24] of Lech Polkowski and Bolesław Araszkiewicz, the Shapley–Shubik power index was applied in order to analyze a game on the basis of partial data about the game. A number of attempts have been made to solve the conflict. As was mentioned earlier, one of the solutions concerning the analysis of conflicts was suggested by Zdzisław Pawlak [20,23]. Other authors have also investigated this issue [5,27]. The paper [16] that was written by Ryszard Michalski and Janusz Wojtusiak presents different approaches to information fusion. The authors considered a situation in which training examples are split into several subsets, and only the aggregated values of attributes are available for each subset. They introduce methods for preparing training data sets from the available knowledge, and describe how to apply methods of machine learning to this kind of knowledge. In the paper [28], written by Andrzej Skowron, Hui Wang, Arkadiusz Wojna and Jan Bazan, the method for the construction of a series of models under gradually relaxing conditions, which form a hierarchical structure was proposed. This method was developed in order to improve the accuracy of classifications made on the basis of the knowledge stored in one knowledge base. Thus, the approach outlined in [28] is different from that considered in this work, in which the global decisions are taken on the basis of dispersed knowledge stored in pre-specified local knowledge bases. In the papers [2,4,11,12], another (different from that which is considered in this paper) approach to the problem of classification on the basis of several different decision tables can be found. In methods of distributed data mining (DDM), it is assumed that the data are collected and stored in different decision tables that are either horizontally or vertically partitioned. The approach considered in this paper is more general than in the DDM because it is assumed that both the sets of attributes as well as the sets of objects may be different in different decision tables. In the paper [10], compatibility measures for measuring the consensus level in a group decision making (GDM) problem was developed. A decision-aid tool, which provides homogeneous clusters from a set of heterogeneous opinions, was proposed in the paper [3]. In the paper [26], a method for the aggregation of multiple judgments in a case in which there is disagreement within the group of decision makers and the members of the group are either unwilling or unable to revise their judgments is discussed. All of these issues are raised in this paper although in a different context – in the context of distributed knowledge. 3. Structure of a multi-agent system We assume that each local knowledge base is managed by one agent. In this paper, we use the definition of an agent introduced by Zdzisław Pawlak in [20]. 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 3.1. Let Ag be a finite set of agents Ag = {ag1 , . . ., agn }. For any agent ag ∈ Ag, called as the resource agent, has access to resources represented by a decision table Dag : = (Uag , Aag , dag ), where Uag is a finite nonempty set called the universe and elements x ∈ Uag are called the objects; Aag is a finite nonempty set of conditional attributes, which is defined in advance, where each attribute a and V a is a set of attribute a a ∈ Aag is a function a : Uag → Vag ag values, which contain the special signs * and ?. Equation a(x) =* for some x ∈ Uag means that for an object x a value of attribute a

605

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 attribute, it is function d , and V d is called the value set of d . dag : Uag → Vag ag ag



d . In addition, we introduce designation V d = ag ∈ Ag Vag Each resource agent ag ∈ Ag on the basis of the knowledge stored in the decision table Dag can independently determine the value of the decision for a set of conditions defined on the set of attributes Aag . In the proposed system, we will seek to designate homogeneous groups of resource agents. The agents who agree on the classification for a test object into the decision classes will be combined in the group. This is a new approach to the structure of a multi-agent decision-making system. In this section the relation of friendship and the relation of the conflict between agents will be defined, and the process of combining resource agents into clusters, which are groups of agents remaining in the relation of friendship, will be described. Definitions of the relations of friendship and conflict as well as the method for determining the intensity of conflicts were taken from the paper of Zdzisław Pawlak [21]. The clustering process is described below. Let there be given a test object x¯ for which we want to generobject x¯ the values of conditional ate a global decision. Let for the n attributes belonging to the set i=1 Aag i will be defined. Decision ag class Xv from decision table Dag of resource agent ag is defined as follows ag

Xv = {x ∈ Uag : dag (x) = v},

where v ∈ V dag .

In order to determine groups of agents from each decision table of resource agent Dag i , i ∈ {1, . . ., n} and from each decision class Xvag i , v ∈ V dag i , the smallest set containing at least m1 objects is chosen, for which the values of conditional attributes bear the greatest similarity to the test object. The value of the parameter m1 is selected experimentally and in order to determine the subset of relevant objects the Gower similarity measure is used. Definition 3.2. Let the objects x and y be described by a pdimensional set of mixed attributes (quantitative, qualitative and binary) {a1 , . . ., ap }. The Gower similarity measure is defined as follows: s(x, y) =

p s (x, y) i=1 i ,  p ı i=1 i

where si (x, y) denotes the contributions provided by the ith attribute. If ai (x), ai (y) ∈ / { * , ? } then the value of si (x, y) is defined as follows. If the ith attribute is a quantitative attribute then si (x, y) = 1 −

|ai (x) − ai (y)| , the range of values for the ith attribute

if the ith attribute is a qualitative attribute then

si (x, y) =

⎧ ⎨ 1, if ai (x) = ai (y) ,

⎩ 0, else

if the ith attribute is a binary attribute then

si (x, y) =

⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0,

if ai (x) = 1 ∧ ai (y) = 1 (ai (x) = 1 ∧ ai (y) = 0)∨ if

(ai (x) = 0 ∧ ai (y) = 1)

⎪ ⎪ ⎪ ⎪ unknown, if ai (x) = 0 ∧ ai (y) = 0 ⎪ ⎪ ⎩

.

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If ai (x) = ? or ai (y) = ? then the value of si (x, y) is unknown. If there is (ai (x) =* and ai (y) = / ?) or (ai (x) = / ? and ai (y) =*) then if the ith attribute is a quantitative or a qualitative attribute then si (x, y) = 1, if the ith attribute is a binary attribute then

si (x, y) =

⎧ ⎨ 1, if ai (x) =/ 0 ∧ ai (y) =/ 0

.

⎩ 0, else

Furthermore ıi =

⎧ ⎨ 1, if the value of si (x, y) is known ,

⎩ 0, else

defined as follows: rank 1 is assigned to the values of the decision attribute which are taken with the maximum level of certainty. Rank 2 is assigned to the values of the decision attribute that have the maximum level of certainty in the set of decisions that have not received the rank 1, 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. The definitions of friendship relation and conflict relation are given next. These definitions are taken from the paper of Zdzisław Pawlak [21]. Relations between agents are defined by their views on the classification of the test object x¯ to the decision class. We define the function vx¯ j for the test object x¯ and each value of the decision attribute vj ∈ V d ; vx¯ j : Ag × Ag → {0, 1}



and if ıi = 0 then si (x, y) = 0.

x¯

For each object from the decision table of resource agent Dag i , the value of the Gower measure of similarity to the test object x¯ is calculated. Then, m1 objects are selected from each decision class. These are objects that carry the greatest similarity to the test object in a given decision class. However, if more than one object from the decision table of the resource agent and decision class Xvag i is as similar to the object x¯ as the m1 th object, it is taken into account the objects closer to the object x¯ than the m1 th object and all objects as similar to the object x¯ as the m1 th object. In the case when card{x ∈ Uag i : d(x) = v} ≤ m1 , then the subset of relevant objects includes all of the objects from decision class Xvag i . The subset of relevant objects is the union of the sets of objects selected from all decision classes. Thus, the subset of relevant objects selected from the decision table Dag i of resource agent agi is defined as follows: rel Uag = i



d v ∈ Vag

v where xm

1

v , x¯ ) and d (x) = v}, {x ∈ Uag i : s(x, x¯ ) ≥ s(xm ag i 1

(1)

i

∈ Uag i is the m1 th object in the sequence of objects

d and ordered in descending valbelonging to decision class v ∈ Vag i ues of the Gower similarity measure to the test object x¯ , and s(x, y) is the Gower measure of similarity between objects x and y. The next stage in the process of generating groups of agents is to determine the vectors of values specifying the classification of the test object made by the agents. So, for each resource agent, the vector that indicates the level of certainty with which the decisions are taken by the agent for the test object is generated. Each coordinate of the vector is determined on the basis of relevant objects that were previously selected from the decision table of the resource agent. Thus, for each resource agent i ∈ {1, . . ., n}, a c-dimensional vector [ ¯ i,1 (¯x), . . .,  ¯ i,c (¯x)] is generated, where the value  ¯ i,j (¯x) means the certainty with which the decision vj ∈ V d , j ∈ {1, . . ., c}, c = card{Vd } is made about the object x¯ by the resource agent agi . ¯ i,j (¯x) is defined as follows: The value 



 ¯ i,j (¯x) =

x, y) rel ∩X ag i s(¯ y ∈ Uag vj i ag

rel ∩ X i } card{Uag vj i

,

i ∈ {1, . . ., n}, j ∈ {1, . . ., c}, (2)

rel is the subset of relevant objects selected where c = card{Vd }, Uag i i is the from the decision table Dag i of resource agent agi and Xvag j

decision class of the decision table of resource agent agi ; s(x, y) is the measure of similarity between objects x and y. Thus, each of the agents is a classifier. According to the three types of classifier outputs given in the paper by Xu et al. [39], the results that are generated by the classifiers are of type three – the measurement level. On the basis of the vector of values defined above, which specify the level of certainty with which decisions for the test object are taken by a given resource agent, a vector of rank assigned to the values of the decision attribute is specified. The vector of rank is

vj (ag i , ag k ) =

0

if ri,j (¯x) = rk,j (¯x)

/ rk,j (¯x) 1 if ri,j (¯x) =

where agi , agk ∈ Ag. Definition 3.3. Agents agi , agk ∈ Ag are in a friendship relation due to the object x and decision class vj ∈ V d , which is written Rv+j (ag i , ag k ), if and only if vxj (ag i , ag k ) = 0. Agents agi , agk ∈ Ag are in a conflict relation due to the object x and decision class vj ∈ V d , which is written Rv−j (ag i , ag k ), if and only if vxj (ag i , ag k ) = 1. We also define the intensity of conflict between agents using a function of distance between agents. We define the distance between agents x¯ for the test object x¯ : x¯ : Ag × Ag → [0, 1]



x¯

 (ag i , ag k ) =

x¯

vj ∈ V d vj (ag i , ag k )

card{V d }

,

where agi , agk ∈ Ag. Definition 3.4. 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 x (agi , agk ) < 0.5. Agents agi , agk ∈ Ag are in a conflict relation due to the object x, which is written R− (agi , agk ), if and only if x (agi , agk ) ≥ 0.5. Using the definitions of the function of distance between agents above, we determine the distance between each pair of resource agents. Then the cluster generation process is initiated as follows. Initially, each resource agent is treated as a separate cluster. These two steps are performed until the stop condition, which is given in the first step, is met. 1. One pair of different clusters is selected (in the very first step a pair of different resource agents) for which the distance reaches a minimum value. If the selected value of the distance is less than 0.5, then agents from the selected pair of clusters are combined into one new cluster. Otherwise, the clustering process is terminated. 2. After defining a new cluster, the value of the distance between the clusters is recalculated. The following method for recalculating the value of the distance is used. Let x : 2Ag × 2Ag → [0, 1], let Di be a cluster formed from the merger of two clusters Di = Di,1 ∪ Di,2 and let it be given a cluster Dj then

⎧ x (Di,1 , Dj ) + x (Di,2 , Dj ) ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ x (Di , Dj ) =

⎪ ⎪ ⎪ max{x (Di,1 , Dj ), x (Di,2 , Dj )} ⎪ ⎪ ⎩

if x (Di,1 , Dj ) < 0.5 and x (Di,2 , Dj ) < 0.5

if x (Di,1 , Dj ) ≥ 0.5 or x (Di,2 , Dj ) ≥ 0.5

M. Przybyła-Kasperek, A. Wakulicz-Deja / Applied Soft Computing 40 (2016) 603–615 Table 1 Vectors that indicate the level of certainty and vectors of rank. Resource agent

ag1 ag2 ag3 ag4 ag5

Table 2 Values of distance function between agents.

Vector that indicate the level of certainty with which the decisions are taken

Vector of rank

[ 0.38, 0.31, 0.31, 0.27, 0.14 ] [ 0.37, 0.5, 0.13, 0.4, 0.37 ] [ 0.4, 0.4, 0.3, 0.2, 0.1 ] [ 0.1, 0.4, 0.3, 0.15, 0.2 ] [ 0.36, 0.36, 0.22, 0.4, 0.51 ]

[1, 2, 2, 3, 4] [3, 1, 4, 2, 3] [1, 1, 2, 3, 4] [5, 1, 2, 4, 3] [3, 3, 4, 2, 1]

The proposed clustering process is similar to the hierarchical agglomeration clustering method. However, the proposed method has a clearly defined stop condition. The stop condition is based on the assumption that one cluster should not contain two resource agents that are in conflict relation due to the test object. A description of the algorithm for the generation of clusters is presented below. Algorithm: We assume that the set of resource agents Ag is given and the test object x¯ is determined.

Resource agents

2. 3. 4. 5.

For each resource agent agi , i ∈ {1, . . ., n}, a c-dimensional ¯ i,1 (¯x), . . .,  ¯ i,c (¯x)] is generated vector [ For each resource agent agi , i ∈ {1, . . ., n}, the vector of rank [ri,1 (¯x), . . ., ri,c (¯x)] is defined The value x¯ (ag i , ag k ) is determined for each pair of agents agi , agk ∈ Ag Each agent is assigned to a separate cluster When there is more than one cluster, the following is executed: (a) Look for the pair of clusters D, D with the minimum value of x¯ (D, D) (b) If the minimum value of function x¯ is less than 0.5 Merge the selected pair of clusters and recalculate the value of the distance between the new cluster and the other clusters (c) Else Stop the loop

Note that the complexity of computing the algorithm above is dependent on the number of resource agents and the number of objects in their decision tables. The first and the fifth step of the algorithm above is the most time-consuming. The sets of relevant objects have to be defined in the first step in order to determine the values of vectors, which involve counting the value of the similarity measures for each object from the decision tables to the test object. Assuming that each of resource agents has access to the decision table containing N objects, in order to calculate the values of similarity measure we have the complexity O(card{Ag} · N). Then, in order to select m1 relevant objects, the similarity values have to be compared m1 times, thus the complexity is equal to O(card{Ag} · N · m1 ). As is well known, in a general case, the complexity of agglomerative clustering is polynomial of degree three. Thus, the fifth step of the algorithm has a complexity of O(card{Ag}3 ). In most cases, the number of resource agents card{Ag} and the value m1 are much smaller than the number of objects in decision tables N. Thus, the size of the data sets of the resource agents has the greatest impact on the complexity. In order to explain the process of clustering in detail, consider the following example. Example 3.1. Consider a set of agents consisting of five resource agents Ag = {ag1 , ag2 , ag3 , ag4 , ag5 }. We assume that V d = {v1 , v2 , v3 , v4 , v5 }. Each resource agent has access to a decision table Dag i = (Uag i , Aag i , dag i ), i ∈ {1, . . ., 5}, based on which the vectors [ ¯ i,1 (¯x), . . .,  ¯ i,5 (¯x)] and [ri,1 (¯x), . . ., ri,5 (¯x)], i ∈ {1, . . ., 5} are defined. Vectors that indicate the level of certainty with which the decisions are taken by the resource agents for the test object and the corresponding vectors of rank are given in Table 1. The value

ag1

ag1 ag2 ag3 ag4 ag5

ag2

ag3

ag4

ag5

1

1 5 4 5

4 5 3 5 3 5

1 2 5

1 1

Table 3 Values of distance function between agents, stage 1. Clusters {ag1 , ag3 }

{ag1 , ag3 } {ag2 } max



1,

4 5



{ag4 } = 1 max

4 3 , 5

5

{ag5 } =

4 5

max {1, 1} = 1 2 5

3 5

{ag2 } {ag4 } {ag5 }

1

Table 4 Values of distance function between agents, stage 2. Clusters

1.

607

{ag1 , ag3 }

{ag1 , ag3 }

{ag2 , ag5 } max {1, 1} = 1

{ag2 , ag5 }

{ag4 } 4 5

max

3  5

,1

=1

{ag4 }

of function of distance between agents is calculated for each pair of resource agents on the basis of the vectors of rank according to the definition given above. These values are given in Table 2. These values are presented in the upper triangular matrix. The minimum value of the distance function is equal to 15 and it is taken for a pair of agents (ag1 , ag3 ). Thus, in the first stage of the process of clustering, we combine these two agents into one cluster and recalculate the value of the distance function. The calculation results are given in Table 3. Again, the minimum value of the distance function is determined. This time it is equal to 25 and is taken for a pair of clusters ({ag2 }, {ag5 }). Thus, in the second stage, we combine the two clusters into one cluster and recalculate the value of the distance function. The calculation results are given in Table 4. Again, the minimum value of the distance function is determined. This time it is equal to 45 and is greater than 0.5, which means that we terminate the clustering process. Finally, we obtain the following set of clusters {{ag1 , ag3 }, {ag2 , ag5 }, {ag4 }}. After the completion of the clustering process, a synthesis agent is defined for each cluster that contains at least two resource agents. If a single resource agent forms cluster, it becomes the synthesis agent. Definition 3.5. For each cluster a superordinate agent is defined, which is called a synthesis agent, asj , where j – number of cluster. The synthesis agent, asj , has access to knowledge that is the result of the process of inference carried out by the resource agents that belong to its subordinate group. As is a finite set of synthesis agents. Definition 3.6. By the multi-agent decision-making system with dynamically generated clusters we mean dyn

WSDAg = Ag, {Dag : ag ∈ Ag}, {Asx : x is a classified object}, {ıx : x is a classified object}

(3)

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

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x, ıx : Asx → 2Ag is a injective function which each synthesis agent assign a cluster generated due to classification of the object x. We will end this section with one final remark. In preliminary experiments, the possibility of a clustering process based on the Euclidean distance of vectors, which indicates the level of certainty with which the decisions are taken, was also considered. The efficiency of the inference of such a system was worse than the efficiency of the inference of the system that uses the method of dynamically generate clusters described above. 4. Elimination of inconsistencies in the knowledge After creating the clusters of agents that classify the test object in a similar way, the aggregation of knowledge that is accessible to the agents from one cluster is carried out. The process of determining the global decision is made on the basis of the aggregated knowledge of the clusters. An important problem that occurs during the aggregation of knowledge of agents from one cluster is to eliminate any inconsistencies in the knowledge stored in different knowledge bases. We understand an inconsistency of knowledge to be a situation in which conflicting decisions are made on the basis of two different knowledge bases that have common attributes and for the same values for common attributes using logical implications. The approximated method of aggregation of decision tables was proposed and described in detail in the papers [34,35]. This method for the elimination of any inconsistencies in the knowledge will be implemented for resource agents belonging to one cluster. The essence of the approximated method of the aggregation of decision tables is to create a set of new decision tables on the basis of the subsets of objects that are relevant to the test object from decision tables of resource agents. The number of tables created is equal to the number of clusters identified in the multi-agent decision-making system. In this method, each synthesis agent has access to resources that are given in the form of a decision table that was formed by the aggregation of relevant objects form the decision tables of resource agents that belong to its subordinate group. The subset of relevant objects is defined as follows: the smallest set containing at least m2 objects is chosen for which the value of conditional attributes bears the greatest similarity (we use Gower similarity measure) to the test object from the decision table of a resource agent and from each decision class. The parameter m2 , which must be determined in this method, is selected experimentally. The subset of relevant objects is the union of the sets of objects selected from all decision classes. Thus, the subset of relevant objects selected from the decision table Dag of resource agent ag is defined as follows: rel Uag

=

 d v ∈ Vag

v , x¯ ) and d (x) = v}, {x ∈ Uag : s(x, x¯ ) ≥ s(xm ag 2

(4)

v ∈ U where xm ag is the m2 th object in the sequence of objects 2

d and ordered in descending valbelonging to decision class v ∈ Vag ues of the Gower similarity measure to the test object x¯ , and s(x, y) is the Gower measure of similarity between objects x and y. Objects of the aggregated decision table are constructed by combining objects from the decision tables of the resource agents that belong to a subordinate group, but only those objects are combined for which the values of the decision attribute and common conditional attributes are equal. The aggregated decision tables of synthesis agents are constructed so that they do not contain any inconsistent knowledge. A detailed discussion of the approximated method of aggregation of decision tables can be found in the papers [34,35]. In order to explain the approximated method of aggregation of decision tables in detail, consider the following example.

Table 5 The subsets of relevant objects from decision tables of resource agents. rel Uag 1

a

b

c

dAg

x1ag 1 x2ag 1

1 0

0 0

0 1

0 1

rel Uag 3

b

e

g

dAg

x1ag 3 x2ag 3

0 0

0 1

0 1

0 1

rel Uag 2

a

e

f

dAg

x1ag 2 x2ag 2 x3ag 2

1 0 0

1 0 1

1 1 0

0 0 1

Table 6 The decision table of a synthesis agent. Uas

a

b

c

e

f

g

d

Component objects

x1as x2as x3as x4as

1 1 0 0

0 0 0 0

0 0 ? 1

1 0 0 1

1 ? 1 0

? 0 0 1

0 0 0 1

x1ag 1 , x1ag 2 x1ag 1 , x1ag 3 x2ag 2 , x1ag 3 x2ag 1 , x3ag 2 , x2ag 3

Example 4.2. Suppose that in a multi-agent system there is a cluster consisting of three agents whose decision tables are as follows: Dag 1 = (Uag 1 , {a, b, c}, dag 1 ), Dag 2 = (Uag 2 , {a, e, f }, dag 2 ), Dag 3 = (Uag 3 , {b, e, g}, dag 3 ); and the subsets of relevant objects are presented in Table 5. Therefore, we obtain the decision table of synthesis agent as presented in Table 6. The first object in decision table resulted from the merger of the object x1ag 1 and the object x1ag 2 . Since in decision table Dag 3 there is no relevant object that would have the value of a decision attribute and the values of common conditional attributes b and e equal the values of these attributes for the constructed object, so the value of the first object from decision table of synthesis agent of attribute g is equal to ?. Similarly, the second object from decision table of synthesis agent resulted from the merger of the object x1ag 1 and the object x1ag 3 . This time, in decision table Dag 2 there is no relevant object which would have the value of the decision attribute and the values of common conditional attributes a and e equal to the values of these attributes for the constructed object, so the value of the second object from decision table of synthesis agent of attribute f is equal to ?. Other objects were constructed in the same way. 5. Conflict analysis Conflict analysis is implemented after the completion of the process of the elimination inconsistencies in the knowledge because then the synthesis agents have access to the knowledge on the basis of which they can independently establish the value of a local decision to just one cluster. Conflicts between agents are understood as situations in which the synthesis agents take contradictory decisions for a given set of conditions on the basis of available knowledge. Two methods for resolving conflict analysis are briefly described below. These methods will be used in this paper: • the method of weighted voting, • the method of a density-based algorithm. These methods allow the analysis of conflicts and enable a set of global decisions to be generated. In the case of the method of a density-based algorithm, the generated set will contain not only the value of the decisions that have the greatest support of the

M. Przybyła-Kasperek, A. Wakulicz-Deja / Applied Soft Computing 40 (2016) 603–615

knowledge stored in local knowledge bases, but also those for which the support is relatively high. After the completion of the process of the elimination of any inconsistencies in the knowledge, c-dimensional vector of values [j,1 (¯x), . . ., j,c (¯x)] is generated for each cluster j ∈ {1, . . ., card{As}}, where c is the number of all of the decision classes. The value j,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 j,i (¯x) is equal to the maximum value of the Gower similarity measure of objects from the decision class vi of the decision table of synthesis agent asj to the test object x¯ . In the method of weighted voting, each synthesis agent votes for different values of the decision with the voting power equal to the value of the coordinate of the vector corresponding to a given decision. The set of global decisions generated using the method of weighted voting is defined by selecting the decisions that are taken with the maximum level of certainty. So the set of global decisions generated using the method of weighted voting is defined as follows:



card{As}

dˆ

dyn WSD Ag

(¯x) = {vi :

j=1

where dˆ

WSD

dyn Ag

j,i (¯x) =



card{As}

max

w ∈ {1,...,c}

j,w (¯x)},

609

Table 7 Data set summary. Data set

# The training set

Soybean 307 Vehicle Silhouettes 592 Landsat Satellite 4435

# The test set

# Conditional attributes

# Decision classes

376 254 1000

35 18 36

19 4 6

• Soybean data set, which is associated with the analysis of the external characteristics of soybeans in order to diagnose pathogens affecting any impairments in the processes of plant life, • Vehicle Silhouettes data set from the project “StatLog”, which is related to vehicle type recognition based on a set of features describing the figure, • Landsat Satellite data set from the project “StatLog”, which refers to the identification of satellite images. In order to determine the efficiency of the inference of the multiagent decision-making system with respect to the data analyzed, each data set was divided into two disjoint subsets:

(5)

j=1

(¯x) denotes the set of global decisions taken by the dyn

multi-agent decision-making system WSDAg for the test object x¯ . In the method of a density-based algorithm, the algorithm DBSCAN [6] from the literature was used. A density-based algorithm searches for densely situated points. In the situation that is considered in this paper, a set of elements that is grouped is the set of decision classes and the distance between two values of the decision is dependent on the level of certainty with which the decision is made. Using the DBSCAN algorithm with the Euclidean distance, we determine the group of decisions that have similar support among the synthesis agents. When groups of decision values are established, i.e. decisions having similar support of the synthesis agents, the cluster to which the decision that has the biggest support of all synthesis agents is chosen and it is interpreted as the set of global decisions taken by the multi-agent decision-making system dyn WSDAg for the test object x¯ . A detailed discussion of the method of a density-based algorithm can be found in [34,37].

6. Results of computational experiments The algorithm for performing the method of taking the global decisions discussed is implemented in C#. On the basis of a loaded *.txt file containing data on a number of resource agents as well as the decision tables of agents, the algorithm dynamically generates clusters for the test object and then implements the approximated method of the aggregation of decision tables and one of methods of conflict analysis and generates a set of global decisions for the test object stored in a separate *.txt file. The aim of the experiments was to examine the quality of the classification made by the multi-agent decision-making system with dynamically generated clusters. An additional objective was to compare the effectiveness of this system with the multi-agent decision-making system in which clusters are defined statically. This means that the clusters are generated only once, and they are the same for each test object. Such a system was proposed in the papers [34,35,37]. Resource agents taking decisions on the basis of common conditional attributes form one cluster in this system. The following data, which are in the UCI repository, were used for the experiments:

• a training set that included objects on the basis of which the multi-agent decision-making system would be trained to make global decisions for a given object, • a test set that was used to verify the efficiency of the inference of the multi-agent decision-making system by generating a set of global decisions for the objects belonging to the test set. Table 7 gives a numerical summary of the data sets. The proposed multi-agent decision-making system allows global decisions on the basis of distributed knowledge that is stored in several decision tables to be made. To fully explore the possibilities and to evaluate the efficiency of inference of the multi-agent decision-making system, it is necessary to provide the knowledge stored in the form of a set of decision tables. Therefore, each of the training sets was divided into a set of decision tables. Divisions with a different number of decision tables were considered. For each of the data sets used, the multi-agent decision-making system with five different versions (with 3, 5, 7, 9 and 11 resource agents) was considered. Note that the division of the data set was not made in order to improve the quality of the decisions taken by the multi-agent decision-making system, but in order to store the knowledge in a distributed form. The division of the data set was performed only once. The division of the data set into the decision tables of the resource agents was carried out as follows. In the first step, the cardinality of a 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. These values were defined by the authors. Then the conditional attributes were assigned to the decision tables so that the conditions that were defined earlier were met, and each conditional attribute that appeared in the data set was included in at least one set of conditional attributes of the decision tables. The decision attribute in the decision tables is the same as the decision attribute in the data set. Each universe of the decision tables included all of the objects from the data set. Of course, the knowledge about the properties of objects described by conditional attributes belonging to the predefined set of attributes of this table and the decision attribute is stored in a given decision table. The knowledge stored in the data set was distributed and saved in several decision tables of resource agents in this way. The cardinalities of the sets of conditional attributes of the decision tables in the multi-agent decision-making systems are given in Table 8.

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Table 8 The cardinalities of sets of conditional attributes. Data set, # Ag

Aag 1

Aag 2

Aag 3

Aag 4

Aag 5

Aag 6

Aag 7

Aag 8

Aag 9

Aag 10

Aag 11

Soybean, 3 Soybean, 5 Soybean, 7 Soybean, 9 Soybean, 11

13 11 8 5 4

15 5 9 5 4

11 9 7 5 4

– 6 5 6 4

– 6 5 5 4

– – 3 5 6

– – 4 5 4

– – – 5 4

– – – 4 4

– – – – 4

– – – – 4

7 5 4 3 2

9 5 4 3 2

6 5 4 3 2

– 4 4 3 2

– 4 4 3 2

– – 3 2 3

– – 3 2 2

– – – 2 2

– – – 2 2

– – – – 2

– – – – 3

10 8 5 5 4

16 8 7 5 5

15 9 6 6 4

– 8 6 6 4

– 9 6 6 3

– – 6 5 3

– – 7 4 4

– – – 3 4

– – – 3 4

– – – – 4

– – – – 4

Vehicle, 3 Vehicle, 5 Vehicle, 7 Vehicle, 9 Vehicle, 11 Satellite, 3 Satellite, 5 Satellite, 7 Satellite, 9 Satellite, 11

The measures of determining the quality of the classification are: • estimator of classification error on a test set independent of the training set e=



1 card{Utest }

I(d(x) ∈ / dˆ

WSD

x ∈ Utest

where I(d(x) ∈ / dˆ

WSD

I(d(x) ∈ / dˆ

WSD

dyn Ag

dyn Ag

dyn Ag

(x)),

(6)

(x)) = 1, when d(xi ) ∈ / dˆ

WSD

(x)) = 0, when d(x) ∈ dˆ

WSD

stored in a decision table Dtest = (Utest ,



dyn Ag

dyn Ag

(x) and

(x); the test set is ˆ

ag ∈ Ag Aag , d); dWSDdyn (x) is Ag

a set of global decisions generated by the multi-agent decisiondyn making system with dynamically generated clusters WSDAg for the test object x, • estimator of classification ambiguity error on a test set independent of the training set eONE =

1 card{Utest }



WSD

x ∈ Utest

/ dˆ where I(d(x) =

WSD

/ dˆ I(d(x) =

WSD

dyn Ag

I(d(x) = / dˆ

dyn Ag

dyn Ag

(x)),

(7)

(x)) = 1, when {d(x)} = / dˆ

WSD

(x)) = 0, when {d(x)} = dˆ

WSD

dyn Ag

dyn Ag

(x) and

(x);

• the average size of the global decisions sets generated for a test set d¯

dyn WSD Ag

=

1 card{Utest }



x ∈ Utest

card{dˆ

dyn WSD Ag

(x)}.

(8)

Experiments were carried out in accordance with the following plan: 1. In the first step, the optimal value of the parameter m1 , which affects the form of clusters and the optimal value of the parameter m2 of the approximated method of aggregation of the decision tables were selected. The parameter m1 determines 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. The parameter m2 determines the number of relevant objects that are selected from each decision class of the decision table of the resource agent, and then are used to design the decision table of the synthesis agents. For small and medium data sets, values of parameter m1 , m2 belonging to the set {1, . . ., 10} were used, while for the large data set, the Landsat Satellite data set, values from the set {1,

. . ., 5} was used. In order to determine the optimal values of the parameters, a multi-agent decision-making system with dynamically generated clusters and the algorithm that would give an approximated method of the aggregation of decision tables with weighted voting, was used. Obviously, the larger the value of the parameters m1 , m2 , the longer the execution time of the algorithm, as more objects are used in the process of generating clusters and in the process of the aggregation of decision tables. Increased complexity does not always translate into an improvement in the efficiency of the inference of a multi-agent decision-making system. Thus, 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. 2. In the second step, parameters ε and MinPts of the method of a density-based algorithm were optimized using the approximated method of the aggregation of decision tables. A multi-agent decision-making system with dynamically generated clusters and the approximated method of the aggregation of decision tables with the optimal values of the parameters m1 , m2 , which was established in the first step of experiments were used. The value of parameter MinPts was established on the basis of preliminary experiments. Then, the value of ε was optimized for the chosen value for MinPts parameter. This process was carried out by performing a series of experiments with different values of the parameter ε that were increased from 0 to the threshold point. Then a graph was created on which the points with coordinates (d¯ dyn , e) are marked in increasing order of value ε. The WSD

Ag

marked points 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 that point you can see a significant decrease in the value of 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 the parameter ε. 6.1. Experiments with Soybean data set The results of experiments with the Soybean data set are presented in Table 9. In the table the following information is given: • the name of multi-agent decision-making system with dynamically generated clusters (System), • the algorithm’s symbol (Algorithm), the algorithms are identified by a code consisting of two characters. Particular pieces of code represent: a method of elimination inconsistencies in the knowledge and method of conflicts analysis, which are

M. Przybyła-Kasperek, A. Wakulicz-Deja / Applied Soft Computing 40 (2016) 603–615 Table 9 Summary of experiments results with the Soybean data set, dynamically generated clusters. Algorithm

e

eONE

d¯

WSDAg1 m1 = 1

dyn

A(1)G(0.0072 ; 2) A(1)G(0.00645 ; 2)

0.026 0.064

0.319 0.287

WSDAg2 m1 = 4

dyn

A(1)G(0.01575 ; 2) A(1)G(0.013 ; 2)

0.019 0.043

dyn

A(1)G(0.01875 ; 2) A(1)G(0.0135 ; 2)

WSDAg4 m1 = 1

dyn

dyn

System

WSDAg3 m1 = 6

WSDAg5 m1 = 1

Table 11 Summary of experiments results with the Vehicle Silhouettes data set, dynamically generated clusters. Algorithm

e

eONE

d¯

dyn

A(2)G(0.0051 ; 2) A(2)G(0.0018 ; 2)

0.138 0.209

0.476 0.358

1.528 1.224

0.06 0.06

dyn

A(10)G(0.0069 ; 2) A(10)G(0.003 ; 2)

0.181 0.244

0.516 0.421

1.528 1.236

0.11 0.11

dyn

A(10)G(0.0087 ; 2) A(10)G(0.00435 ; 2)

0.130 0.205

0.492 0.398

1.500 1.260

0.24 0.24

WSDAg4 m1 = 1

dyn

A(4)G(0.00615 ; 2) A(4)G(0.0033 ; 2)

0.177 0.260

0.528 0.429

1.516 1.244

0.10 0.10

dyn

A(5)G(0.00615 ; 2) A(5)G(0.003 ; 2)

0.138 0.197

0.516 0.398

1.512 1.232

0.22 0.22

t

System

2.401 2.082

0.03 0.03

WSDAg1 m1 = 2

0.311 0.285

2.059 1.545

0.05 0.05

WSDAg2 m1 = 1

0.016 0.019

0.306 0.279

2.008 1.598

0.07 0.07

WSDAg3 m1 = 6

A(1)G(0.006 ; 2)

0.043

0.261

1.529

0.06

A(4)G(0.005 ; 2)

0.037

0.327

1.838

4.22

WSDAg5 m1 = 4

WSD

dyn Ag

Table 10 Summary of experiments results with the Soybean data set, statically-defined clusters.

611

WSD

dyn Ag

t

Table 12 Summary of experiments results with the Vehicle Silhouettes data set, staticallydefined clusters.

System

Algorithm

e

eONE

d¯ WSDAg

t

System

Algorithm

e

eONE

d¯ WSDAg

t

WSDAg1 WSDAg2

A(1)G(0.0025 ; 2) A(1)G(0.00575 ; 2)

0.027 0.035

0.295 0.327

2.005 1.968

0.03 0.02

WSDAg1

A(20)G(0.00225 ; 2)

0.106

0.492

1.524

0.32

0.093 0.008

0.242 0.306

1.335 1.968

0.02 0.03

WSDAg2

A(20)G(0.00125 ; 2) A(17)G(0.0035 ; 2)

0.177 0.154

0.378 0.508

1.236 1.535

0.32 0.16

WSDAg3

A(1)G(0.0035 ; 2) A(1)G(0.00575 ; 2)

0.024 0.058

0.274 0.319

1.559 1.755

0.03 0.03

WSDAg3

A(17)G(0.0015 ; 2) A(1)G(0.00525 ; 2)

0.244 0.161

0.406 0.543

1.224 1.520

0.16 0.04

WSDAg4

A(1)G(0.00375 ; 2) A(1)G(0.0108 ; 2) A(1)G(0.0128 ; 2) A(1)G(0.0084 ; 2)

0.029 0.090

0.309 0.253

1.755 1.298

0.03 0.03

WSDAg4

A(1)G(0.0025 ; 2) A(1)G(0.00375 ; 2)

0.240 0.189

0.457 0.504

1.260 1.512

0.04 0.04

A(1)G(0.002 ; 2) A(1)G(0.00475 ; 2) A(1)G(0.0025 ; 2)

0.272 0.146 0.213

0.453 0.500 0.406

1.248 1.512 1.256

0.04 0.04 0.04

WSDAg5

WSDAg5

implemented by the algorithm. The following designations were adopted: – A(m) – the approximated method of aggregation of decision tables, – W – the method of weighted voting, – G(ε, MinPts) – the method of a density-based algorithm, • the three measures discussed earlier e, eONE , d¯ dyn , WSD

Ag

• the time t needed to analyse a test set expressed in minutes. The results of experiments carried out with the Soybean data set and all of the structures of a multi-agent decision-making system with statically-defined clusters were presented in the papers [36,25]. This means that the clusters are generated only once and that they are the same for each test object. For comparison, Table 10 shows the results presented in the papers [36,25]. Designations used in Table 10 are the same as in Table 9. Based on the results of experiments given in Tables 9 and 10, the following conclusions can be drawn: • better results for a multi-agent decision-making system with dynamically generated clusters were obtained for system versions with 5 and 9 resource agents than for a multi-agent decision-making system with statically-defined clusters; • the same value of the estimator of classification error as that for a multi-agent decision-making system with statically-defined clusters was obtained for a multi-agent decision-making system with 3 resource agents when clusters were generated dynamically, but with a higher value of the average size of the global decision sets; • a lower value of the estimator of classification error was obtained for a multi-agent decision-making system with 7 resource agents

and the average size of the global decision sets equal to 1.6 approximately when the clusters were generated dynamically. However, a higher value of the estimator of classification error was obtained for the average size of the global decision sets approximately equal to 2 when clusters were generated dynamically; • worse results were obtained for a multi-agent decision-making system with 11 resource agents when clusters were generated dynamically than for a multi-agent decision-making system with statically-defined clusters. 6.2. Experiments with Vehicle Silhouettes data set The results of experiments with the Vehicle Silhouettes data set are presented in Table 11. For comparison, Table 12 shows the results of experiments carried out using the Vehicle Silhouettes data set and all of the structures of a multi-agent decision-making system with statically-defined clusters. These experiments were presented in the papers [36,25]. Designations used in Tables 11 and 12 are the same as in Table 9. Based on the results of the experiments given in Tables 11 and 12, the following conclusions can be drawn: • much better results were obtained for a multi-agent decisionmaking system with 7, 9 and 11 resource agents when clusters are generated dynamically than for the multi-agent decision-making system with statically defined clusters; • worse results were obtained for a multi-agent decision-making system with 3 and 5 resource agents when clusters are generated

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Table 13 Summary of experiments results with the Landsat Satellite data set, dynamically generated clusters. Algorithm

e

eONE

d¯

A(5)G(0.00225 ; 2)

0.033

0.330

WSDAg2 m1 = 1

dyn

A(3)G(0.0053 ; 2) A(3)G(0.0022 ; 2)

0.012 0.047

WSDAg3 m1 = 2

dyn

A(3)G(0.0057 ; 2) A(3)G(0.0029 ; 2)

WSDAg4 m1 = 2

dyn

dyn

System dyn

WSDAg1 m1 = 1

WSDAg5 m1 = 1

Table 15 Optimization of the parameters m1 , m2 for the Soybean data set with the use of 10-fold cross-validation method.

t

System

The optimal value of the parameter m1

The optimal value of the parameter m2

1.736

6.00

0.402 0.2151

1.724 1.235

6.30 6.30

WSDAg1 WSDAg2 WSDAg3 WSDAg4 WSDAg5

4 2 6 8 2

2 1 1 1 3

0.015 0.046

0.401 0.245

1.720 1.263

7.25 7.25

A(4)G(0.0064 ; 2) A(4)G(0.0032 ; 2)

0.013 0.047

0.378 0.236

1.644 1.252

8.30 8.30

A(3)G(0.01 ; 2) A(3)G(0.0042 ; 2)

0.011 0.041

0.425 0.232

1.764 1.255

7.50 7.50

WSD

dyn Ag

Table 14 Summary of experiments results with the Landsat Satellite data set, staticallydefined clusters. System

Algorithm

e

eONE

d¯ WSDAg

t

WSDAg1

A(10)G(0.0014 ; 2)

0.042

0.401

1.757

4.15

WSDAg2

A(1)G(0.0031 ; 2) A(1)G(0.0012 ; 2)

0.012 0.047

0.396 0.221

1.744 1.230

3.08 3.08

WSDAg3

A(7)G(0.0046 ; 2) A(7)G(0.0022 ; 2)

0.012 0.047

0.381 0.232

1.716 1.255

4.55 4.55

WSDAg4

A(10)G(0.0057 ; 2) A(10)G(0.00285 ; 2)

0.020 0.052

0.348 0.216

1.698 1.229

6.32 6.32

WSDAg5

A(1)G(0.00585 ; 2) A(1)G(0.00285 ; 2)

0.010 0.048

0.400 0.230

1.733 1.259

3.25 3.25

dynamically than for the multi-agent decision-making system with statically-defined clusters. 6.3. Experiments with Landsat Satellite data set The results of experiments with the Landsat Satellite data set are presented in Table 13. For comparison, Table 14 shows the results of experiments carried out using the Landsat Satellite data set and all of the structures of a multi-agent decision-making system with statically-defined clusters. These experiments were presented in the papers [36,25]. Designations used in Tables 13 and 14 are the same as in Table 9. Based on the results of experiments given in Tables 13 and 14, it can be concluded that for all of the types of multi-agent decision-making systems that were considered, the application of dynamically generated clusters improved the efficiency of inference in terms of the value of the estimator of the classification error, or the value of the average size of the global decision sets. 6.4. The use of a cross-validation test In this paper we considered the issue of decision-making on the basis of distributed knowledge. In the paper the multiagent decision-making system with a dynamic structure was proposed and a series of tests using the train-and-test method and distributed data sets were conducted. Distributed data sets are sets in which data is stored in many local knowledge bases. As mentioned earlier, in the system considered in the paper, very general assumptions have been adopted; the sets of conditional attributes and the sets of objects of different local knowledge bases do not need to be equal or disjoint. In addition, there are no stored object identifiers in the

local knowledge bases. Such general assumptions mean that it is not possible to check whether the same objects are stored in different local knowledge bases and it is not possible to identify which objects are the same, which means that using the k-fold crossvalidation method is impossible when preserving the generality. More specifically, this is due to the assumption that the knowledge is stored in the form of a number in the local knowledge bases (which are given in advance), while the test set is stored in a single table, in which a set of conditional attributes is equal to the sum of the sets of the conditional attributes of local knowledge bases. When we want to use the cross-validation method, a test sample would have to be drawn from each decision table. Under the adopted assumptions, it was not possible to identify whether the same objects were drawn from different local knowledge bases. Thus, it was not possible to generate a decision table from the selected set of test samples. However, due to the completeness of the test, an attempt was made to use the k-fold cross-validation method. In order to do this, it was necessary to derogate from the generality of the assumptions. Firstly, it was assumed that the same objects were stored in all of the local knowledge bases, and that every object had an assigned identifier. A test sample was selected as follows. First, a set of object identifiers was drawn up. Then the same set of objects with the selected identifiers was chosen from each local knowledge base. In this way, it was possible to generate one decision table from the selected set of test samples. Other objects, which were not drawn from local knowledge base in the k-fold cross-validation, formed the training set. Tests were carried out for all three data sets: Soybean, Vehicle Silhouettes and Landsat Satellite using the 10-fold cross-validation method. For each data set, the tests began with the optimization of the parameter m1 , which influences the form of clusters and the optimization of the parameter m2 of the approximated method of the aggregation of decision tables. In the next step, experiments to determine the optimal values of parameters ε and MinPts of the method using a density-based algorithm were performed. The process of the optimization of the parameters proceeded in the same way as the optimization process discussed in the previous part of this paper; the only difference was that this time the 10-fold cross-validation method was used. A detailed description of the parameter optimization process will be omitted; optimal parameter values for each data set is presented in Tables 15, 17 and 19. In Tables 16, 18 and 20, the results of experiments with the data sets that were considered (Soybean, Vehicle Silhouettes and Landsat Satellite) and the multiagent decision-making system with dynamically generated disjoint clusters using 10-fold cross-validation are presented. Comparing the results obtained by the train-and-test method (Tables 9, 11 and 13) with the results obtained using the 10-fold cross-validation method (Tables 16, 18 and 20), it can be concluded that both methods produce very similar test results. This means that the results obtained by the train-and-test method have been confirmed and it can be used to compare the effectiveness

M. Przybyła-Kasperek, A. Wakulicz-Deja / Applied Soft Computing 40 (2016) 603–615 Table 16 Summary of experiments results for the Soybean data set with the use of 10-fold cross-validation method. Dynamically generated inseparable clusters

613

Table 20 Summary of experiments results for the Landsat Satellite data set with the use of 10-fold cross-validation method. Dynamically generated inseparable clusters

System

Algorithm

e

eONE

d¯

t

System

Algorithm

e

eONE

d¯

WSDAg1 m1 = 4

A(2)G(0.0072 ; 2)

0.009

0.275

1.703

0.05

WSDAg1 m1 = 5

A(1)G(0.0001 ; 2)

0.093

0.310

2.050

6.10

WSDAg2 m1 = 2

A(1)G(0.01225 ; 2) A(1)G(0.00725 ; 2)

0.006 0.010

0.294 0.241

1.988 1.493

0.04 0.04

WSDAg2 m1 = 5

A(3)G(0.0051 ; 2) A(3)G(0.0022;2)

0.014 0.044

0.391 0.212

1.719 1.245

7.10 7.10

WSDAg3 m1 = 6

A(1)G(0.014 ; 2) A(1)G(0.01075 ; 2)

0.006 0.013

0.306 0.269

1.774 1.588

0.07 0.07

WSDAg3 m1 = 2

A(2)G(0.0057 ; 2) A(2)G(0.0029 ; 2)

0.012 0.042

0.390 0.241

1.725 1.275

7.25 7.25

WSDAg4 m1 = 8

A(1)G(0.01475 ; 2)

0.018

0.290

1.559

0.10

WSDAg4 m1 = 2

A(1)G(0.006 ; 2) A(1)G(0.0031 ; 2)

0.020 0.047

0.362 0.232

1.601 1.245

8.15 8.15

WSDAg5 m1 = 2

A(3)G(0.01875 ; 2)

0.012

0.379

1.791

1.35

WSDAg5 m1 = 1

A(1)G(0.0094 ; 2) A(1)G(0.0044 ; 2)

0.012 0.041

0.406 0.231

1.751 1.259

7.40 7.40

WSD

dyn Ag

Table 17 Optimization of the parameters m1 , m2 for the Vehicle Silhouettes data set with the use of 10-fold cross-validation method. System

The optimal value of the parameter m1

The optimal value of the parameter m2

WSDAg1 WSDAg2 WSDAg3 WSDAg4 WSDAg5

4 4 2 1 2

2 10 1 4 3

Table 18 Summary of experiments results for the Vehicle Silhouettes data set with the use of 10-fold cross-validation method. Dynamically generated inseparable clusters System

Algorithm

e

eONE

d¯

WSDAg1 m1 = 4

A(2)G(0.00495 ; 2) A(2)G(0.00135 ; 2)

0.138 0.236

0.480 0.369

1.531 1.229

0.08 0.08

WSDAg2 m1 = 4

A(10)G(0.0066 ; 2) A(10)G(0.00315 ; 2)

0.142 0.219

0.515 0.405

1.529 1.230

0.14 0.14

WSDAg3 m1 = 2

A(1)G(0.00885 ; 2) A(1)G(0.00435 ; 2)

0.124 0.188

0.502 0.398

1.505 1.257

0.13 0.13

WSDAg4 m1 = 1

A(4)G(0.00555 ; 2) A(4)G(0.00255 ; 2)

0.140 0.219

0.523 0.413

1.507 1.235

0.10 0.10

WSDAg5 m1 = 2

A(3)G(0.00525 ; 2) A(3)G(0.0027 ; 2)

0.155 0.243

0.523 0.418

1.511 1.227

0.20 0.20

WSD

dyn Ag

t

Table 19 Optimization of the parameters m1 , m2 for the Landsat Satellite data set with the use of 10-fold cross-validation method. System

The optimal value of the parameter m1

The optimal value of the parameter m2

WSDAg1 WSDAg2 WSDAg3 WSDAg4 WSDAg5

5 5 2 2 1

1 3 2 1 1

of inference in the proposed decision-making system with the efficiency of other global decision-making systems. 7. Conclusion In this paper a new approach to the organization of the structure of a multi-agent decision-making system that operates on the

WSD

dyn Ag

t

basis of dispersed knowledge is proposed. In earlier papers, the multi-agent decision-making system in which resource agents take decisions on the basis of common conditional attributes form one cluster was considered. The new approach is based on the assumption that one cluster should contain resource agents who agree on the classification of a test object into the decision classes. In the process of global decision-making, the knowledge of resource agents that remain in such a coalition is aggregated. On the basis of aggregated knowledge, local decisions are taken within clusters. In the last stage of the global decision-making process a set of global decisions is generated with the use of the method of conflict analysis. In this paper, the efficiency of inference of the multi-agent decision-making system with dynamically generated clusters with the efficiency of inference of the system proposed in the earlier papers were compared. For the overwhelming majority of the types of multi-agent decision-making systems considered, the application of dynamically generated clusters improved the efficiency of inference. The best results were obtained for the Vehicle Silhouettes and Landsat Satellite data sets; slightly worse results were obtained for the Soybean data set. It should also be noted that the greater the number of resource agents in a multi-agent decision-making system, the greater the expected improvement in the efficiency of inference when dynamically generated clusters are applied. Additionally, a comparative box-whiskers chart for the values of the classification error obtained during the experiments was created. The comparison chart is presented in Fig. 1, which includes the classification error values obtained using a multi-agent decisionmaking system with dynamically generated disjoint clusters as well as one using the system with statically defined clusters. The graph clearly shows that the average value of the classification error observed for the system with dynamically generated disjoint clusters is lower than the average value of the classification error for the system with statically defined clusters. In order to justify that the differences between the classification errors for the two approaches are significant, a statistical test was carried out. At the beginning, the data obtained for all of the decision systems being analyzed were considered (five different versions with 3, 5, 7, 9 and 11 resource agents). However, it was not possible to reject the null hypothesis – that the means of two populations are equal for these data. After analyzing the results of experiments, it was found that the proposed approach that has dynamically generated disjoint clusters does not cause a significant improvement in the case of a small number of resource agents. Thus, a statistical test was performed for the data that was obtained for the decision systems with 5, 7, 9 and 11 resource agents. The t-test was used because we have two dependent groups, quantitative scales, it was tested that we cannot reject the hypothesis of normal distribution of the difference of variables and

614

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Fig. 1. Comparison of system with dynamically generated disjoint clusters and system with statically defined clusters.

group size is less than 30. The t-test confirmed that the differences between the classification error for the two approaches being considered are significant with a level of p < 0.05.

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