World Multiconference on Systemics, Cybernetics and Informatics, SCI’2000, Orlando, Florida, USA, July 23-26, 2000, Vol. III, pp. 118-123.
Fuzzy Modeling of Multi-Agent Systems Behavior. Vagueness Minimization. Dan Stefanoiu1, Mihaela Ulieru, Douglas Norrie The University of Calgary Department of Mechanical and Manufacturing Engineering 2500 University Drive NW Calgary, Alberta T2N 1N4, Canada Tel. (+ 403)-220-2991; Fax. (+ 403)-282-8406 @-mails:
[email protected],
[email protected],
[email protected]
rigorous mathematical framework for these interactions and evolution, which approaches agents (in isolation and in group) as dynamical systems. Consider a MAS which evolves transitioning from an initial state through a chain of intermediate states until it reaches its goal in a final state. A main driving force for MAS dynamics during this transition is information exchange among agents. While the MAS evolves through its states towards the goal, its agents associate in groups referred to as clusters [5], each cluster of agents aiming to solve a certain part of the overall task assigned to the MAS. Let us consider now the set of all agents within a MAS. Each possible structure of clusters that covers the agents set points to a partition of this set, if clusters are not overlapping or to a cover, when different clusters contain the same agent. Such a partition/cover will be referred hereafter to as (clustering) configuration. We name plan the succession of all states through which MAS transitions until it reaches its goal. Each MAS state is described by a certain clustering configuration covering the agent set. So, in this context, a plan is a succession of such configurations describing the MAS clustering dynamics on its way towards reaching a goal. One can hardly predict the dynamics of clustering configurations during MAS evolution towards its goal. For this prediction we rely only on uncertain information determined through observations, experiments and/or through simple deductions based on the agents structure (similar to how their PAGE descriptions are determined - see [6]). When facing the uncertainty in the available information about MAS clustering dynamics, two aspects should be considered: vagueness and ambiguity [4]. It is already well known that among the other uncertainty facets, vagueness deals with information that is inconsistent. In the context of MAS, this means that the clear distinction between a possible plan reaching the imposed goal and a plan which, on the contrary, leads the MAS in an opposite direction (i.e. towards a very different goal) is hardly distinguishable. Starting from this information, one can construct only a collection of source-plans (i.e. sets of clustering configurations considered as sources for plans) associated to a specific global goal. Another characteristic property of vagueness in the information about MAS is the non-clarity, i.e. the incapacity to make clear distinctions between an entity and its opposite. The vagueness appears mostly when a manager of the MAS do not
ABSTRACT Holonic manufacturing aims to design standardized, modular manufacturing systems made of interchangeable parts, to enable flexibility and self-organizing capabilities for the production systems. Recent advances in Distributed Artificial Intelligence and Networking Technologies have proven that the theoretical Multi-Agent Systems (MAS) concepts are very suitable for the real life implementation of holonic concepts. Building on our recent results in the design and implementation of holonic reconfiguring architectures, this paper introduces a novel approach to the self-organization of distributed systems. First, by using fuzzy sets and measures theoretical concepts, we construct a mathematical foundation for modeling MAS. Then, by minimizing the vagueness facet of uncertainty in the information dealt with by the MAS, appropriate holonic structures emerge for each particular application. This approach opens new possibilities for the design of any distributed system that needs self-organization as an intrinsic property. Keywords: Multi Agent Systems, Uncertainty, Relations, Shannon Fuzzy Entropy, Holonic Structure.
Fuzzy
1. INTRODUCTION AND PROBLEM STATEMENT This paper aims to define fuzzy models which vaguely outline some structural features of multi-agent systems (MAS) such as associating in clusters and the least uncertain succession of clusters configurations that describes the MAS evolution towards its goal. The knowledge which these fuzzy models reveal about the MAS appears to be very useful as a basis for further, more precise modeling of the multi-agent interaction. Despite the considerable advances in agent research [6], [3], a rigorous mathematical description of agent systems and their interaction is yet to be formulated. Agents can be understood as autonomous problem solvers, in general heterogeneous in nature, who interact with other agents in a given setting, towards generating effective solutions. Thus, interactions and evolution in time towards achieving a preset goal are prime features of each agent and of the MAS overall. Starting from an idea first exposed in [9], in this paper we establish a more
1
On leave from “Politehnica” University of Bucharest, Dept. of Automatic Control and Computer Science, 313 Splaiul Independentei, Sector 6, 77206-Bucharest, Romania.
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know very well, from the start, the system he supposed to manage. For example, this is the case when two or more factories are merging and new structures of clusters are constructed. Another example is issued from the case when the manager is new and did not work with the system before. However, once some more knowledge about the MAS is acquired, the manager is faced with another side of uncertainty: ambiguity. An ambiguous information seems to be consistent, but the facts revealed are mixed such that no distinctions can be made between entities; all seem to be similar and their characteristics, although clearly defined, are either too general or dissonant or confuse or non-specific to make a choice. More specifically, the consistency means here that, from the available information, several possible plans of MAS converging it towards a fixed goal can be constructed. (In general, consistency is a property related to the convergence phenomenon, provided that the convergence be correctly defined. Although it is not very rigorous, here, “convergence” means “reaching a goal”. Naturally, according to this definition, since several plans reaching a goal can be constructed starting from the initial information, it means that the information is consistent.) But we cannot decide which plan to select, because their descriptions seem to be too general, although no contradictions are revealed between different plans. Only the vagueness facet of uncertainty is concerned in this paper. The models based on combined aspects ambiguity and vagueness are described in [8]. Hence, the main hypotheses in our framework are: H1 Although the agents set is a collection of deterministic entities, its behavior could be non deterministic, due to the external perturbations. H2 No prior knowledge about MAS is available, excepting for the general purpose of the system (thus, at least a global goal is known) and the number of agents (denoted by N ), but their PAGE descriptions are not necessarily known. H3 The structures of clustering configurations are observable and their occurrences can be counted during the MAS evolution from the initial state to a final one, for any fixed global goal.
than the number of all possible clusters (which is 2 N ) [8]. The number of all possible covers is even larger. It follows that the number of all possible occurring configurations could be very large. One can refer each Pk to as a source-plan in the sense that it can be a source of configurations for a MAS plan. The main difference between a plan and a source-plan is that, in a plan, the occurrence of the configurations in time is clearly specified, whereas in a source-plan the succession of configurations is, usually, unknown. The only available information about every Pk is the degree of occurrence associated to each of its configurations, Pk ,m , which can be assigned as a number α k ,m ∈ [0,1] . Thus, the corresponding degrees of occurrence are members of a two-dimension family , which, as previously stated, quantifies all {α k ,m } k∈1, K ;m∈1, M k
the available information about MAS. In this framework, we aim to construct a measure of uncertainty, V (from “vagueness”), fuzzy-type, real-valued, defined on the set of all source-plans of A N and optimize it in order to select the least vague source-plan from the family P = {Pk }k ∈1, K : Pk 0 = arg opt V ( Pk ) , where ko ∈ 1, K .
(1)
k ∈1, K
3. THE LEAST VAGUE SOURCE-PLAN The solution proposed here is based on the Theory of fuzzy sets and uses measures of fuzziness [4] to model the vagueness aspect of the uncertainty in the initial information about MAS behavior. Recall that the only available information consists of a family of occurrence degrees for some configurations. The optimization problem stated in Eq. (1) is solved within the next four steps. Step 1: Build a family of fuzzy relations R = {R k }k ∈1, K that are similarity or at least proximity type and a bijection T between P and R :
Starting from this uncertain information, the goal of our research is to provide sound models of MAS, useful in selecting (at least in a coarse manner) the least uncertain (the least vague) source-plan. Thus, the problem considered here can be stated as follows:
T ( Pk ) = R k , ∀k ∈ 1, K .
(2)
Associating agents in clusters is very similar to grouping them into compatibility or equivalence classes, given a (binary) crisp relation between them. The compatibility (reflexivity and symmetry) is achieved in the case of covers (overlapped clusters), whereas the equivalence (compatibility and transitivity) corresponds to partitions. It is, thus, naturally to consider that every clustering configuration covers A N by
• Given a MAS and some vague information about the occurrence of clustering configurations during its evolution towards a goal, construct a (fuzzy) model providing the least uncertain (vague) source-plan.
2. MATHEMATICAL STATEMENT OF PROBLEM
such classes. The corresponding (unique) crisp relation can be described by the statement: two agents are related if they belong to the same cluster. Express by “ aRk ,m b ” and
Denote by A N = {a n }n∈1, N the set of N ≥ 1 agents that belong to a MAS. Based only on the initial uncertain information, one can construct a family P = {Pk }k ∈1, K ,
“ a¬Rk ,m b ” the facts that a and b are, respectively are not in
containing K ≥ 1 source-plans, for a preset global goal. Each source-plan is expressed as a collection of M k ≥ 1 clustering
the relation Rk ,m (where a,b ∈ A N , k ∈ 1, K and m ∈1, M k ).
configurations: Pk = {Pk ,m }m∈1, M , ∀k ∈1, K (covering A N ),
N × N matrix H k ,m ∈ ℜ N × N - the characteristic matrix -
The relation Rk ,m can also be described by means of a
k
possible to occur during the MAS evolution towards its goal. Notice that the number of all possible partitions (nonoverlapped clusters) covering A N increases faster with N
whose elements ( H k ,m [i, j ] ) are only 0 or 1, depending on whether the agents are or not in the same cluster. (Here, ℜ points to the real numbers set.) Thus:
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Step 2: Construct a measure of fuzziness over the above fuzzy relations. This measure will be used to select the “minimally fuzzy” relation of the set R = {R k }k ∈1, K .
def 1 , a i R k ,m a j H k , m [i , j ] = , ∀i, j ∈ 1, N . (3) 0 , a i ¬Rk ,m a j This matrix is symmetric (obviously, if aRk ,m b , then
One very important class consists of measures that evaluate “the fuzziness” of a fuzzy set by taking into consideration both the set and its (fuzzy) complement. From this large class, we have selected the Shannon measure, constructed starting from the multi-dimensional Shannon’s function:
bRk,m a ) and with unitary diagonal (since every agent is in the same cluster with itself). It allows us to completely specify only the configuration Pk ,m , as proves the following result: Theorem 1. Let
P = { A1 , A2 , ..., AM }
S : [0,1] M → ℜ (6) + M def ( x1 ,..., x M ) a S ( x ) = − ∑ [x m log 2 x m + (1 − x m ) log 2 (1 − x m )] m =1 It has a unique maximum (equal by M , for x m = 1 / 2 ,
be a clustering
configuration of the agents set A N (where Am is a cluster, M
∀m ∈ 1, M ): A N = U Am . Construct the following matrix H ∈ {0,1} N × N :
m =1
∀m ∈ 1, M
def 1 , ∃ m ∈ 1, M so that {a , a } ⊆ A i j m , ∀i, j ∈1, N . H [i , j ] = 0 , otherwise Then P is uniquely determined by H .
) and 2 M null minima (in apexes of hyper-cube
[0,1] M ). For example, if M = 2 , this function generates the
surface depicted in Figure 1. In general, a hyper-surface is generated in Euclidean space ℜ M , but all its minima are null,
This result shows, in fact, that the relation Rk ,m defined by the statement of agents inclusion to the same cluster is uniquely assigned to the configuration Pk ,m (no other configuration can be described by Rk ,m ). Thus, each crisp relation Rk ,m can uniquely be associated to the degree of occurrence assigned to its configuration: α k ,m . Consider now a source-plan Pk , for an arbitrarily fixed k ∈ 1, K and extract freely a configuration Pk ,m ( m ∈ 1, M k ).
The assigned degree of occurrence is then α k ,m , whereas the characteristic matrix of the associated crisp relation is H k ,m (Eq. (3)). In this context, one can construct first the fuzzy relation α k ,m Rk ,m (defined on A N × A N ), whose
as mentioned. Figure 1. Shannon’s function for M = 2 .
membership matrix is expressed as α k ,m H k ,m . Then, the
When the argument of function (6) is a probability distribution, it is referred to as Shannon entropy [4]. When the argument is a membership function defining a fuzzy set, it is refereed to as Shannon fuzzy entropy [10]. Denote the fuzzy entropy by S µ . Then, according to Eq. (4) and (5), S µ is
fuzzy relation R k uniquely associated to the source plan Pk is simply the fuzzy union of all above fuzzy relations: Rk =
Mk
U α k , m R k ,m .
(4)
m =1
expressed by:
If the max fuzzy union is employed in Eq. (4), the N × N membership matrix defining R k is expressed by: def
{
}
M k = max • α k ,m H k ,m , m∈1, M k
N
S µ (R k ) = −
(5)
i =1 j =1 N
−
N
∑∑ M k [i, j ] log 2 M k [i, j] −
(7)
N
∑∑ [1 − M k [i, j]]log 2 [1 − M k [i, j]], i =1 j =1
where “ max • ” stands for max operator applied locally, on matrices elements and not globally, on matrices. Notice that the crisp relation expressed by H k ,m is not an
where M k [i, j ] is the current element of matrix M k (Eq. (5)). Obviously, this function also has a unique maximum and all minima null, with respect to variables M k [i, j ] , its dimension
α -cut (as defined in [4]), but an α -sharp-cut of R k , because it contains only those elements of the fuzzy relation with the membership grades exactly equal to α k ,m and not ≥ α k , m .
being M = N 2 . Two main reasons motivate this choice. First, S µ helps us
Given that all crisp relations are at least compatibility type, it is easy to prove that the fuzzy relations {R k }k∈1, K are at least
make a direct connection between “how fuzzy” is a set and “how much uncertainty” it contains. Thus, since S µ computes
proximity type (fuzzy reflexive and symmetric). They could be similarity relations (proximity and fuzzy transitive [4]), but it is also possible that they do not be fuzzy transitive, even if all crisp relations are equivalence ones. Similarity is however important, because it can reveal the holonic behaviour of MAS [1], [5].
the quantity of information of an informational entity, say a fuzzy set, as the estimated uncertainty that the entity contains, the minimally fuzzy sets will subsequently contain the minimally uncertain information. (Notice, however, that only the vagueness facet of the uncertainty is measured here. Ambiguity requires more sophisticated measures [8].) Secondly, the “total ignorance” (or uncertain) information is
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pointed out by the unique maximum of S µ , whereas multiple
R k0 is a similarity relation) and compatibility source-plans
minimum points (actually, the apexes of the hyper-cube) belong to the “perfect knowledge zone” (as less uncertain information as possible). Between “total ignorance” (which, interestingly, is unique) and “perfect knowledge zone” (which is always multiple) there are many intermediate points associated to different degrees of uncertainty in the available knowledge about the entity. Moreover, a force driving towards knowledge can be determined by computing the gradient of Shannon fuzzy entropy. It is interesting to remark that the amplitude of this force (its norm), expressed as:
(when R k0 is a proximity relation).
∇S µ (R k ) =
N
N
∑∑ log 2 i =1 j =1
1 − M k [i , j ] M k [i , j ]
2
,
When the associated fuzzy relation R k0 is a similarity one, then an interesting property of the MAS is revealed: clusters are associated in order to form new clusters, as in a “clusters within clusters” holonic-like paradigm [1], [5]. Moreover, the (unique) similarity relation Qk0 can be constructed starting from the proximity relation R k0 , by computing its transitive closure, following the procedure described in [4]. The new relation Qk0 is similarity one. This
(8)
was proven by means of the following result: Theorem 2. Let Q and R be two binary fuzzy relations and M Q , respectively M R their N × N membership matrices. Denote by C the composition between Q and R , that is: C = Q o R . Then M C = M Q o M R (fuzzy product) and:
increases very fast in the vicinity of any “perfect knowledge” point. Step 3: Construct the required measure of uncertainty, V , by composing S µ (the fuzzy entropy, Eq. (7)) with the map T
1.If Q and R are reflexive relations, then C is reflexive. 2. If Q and R are symmetric relations, then C is symmetric. 3.If Q = R and R is a transitive relation, then C is transitive.
(given by Eq. (2)): V = S µ o T . Since T is a bijection, the optimization problem stated in Eq. (1) becomes: Pk 0 = T −1 (arg min S µ ( R k )) , where k o ∈ 1, K .
(9)
k ∈1, K
According to Step 2, Pk0 is the least fuzzy (minimally
This result allows us to reveal the potential holonic behavior even for MAS that seems to evolve in a non holonic manner. When R k0 is only a proximity relation, tolerance
fuzzy), that is the least uncertain source-plan from the family and the most attracted by the knowledge zone. (Hereafter, not only Pk0 , but also the optimum fuzzy relation R k0 is useful
(compatibility) classes can be constructed as collections of eventually overlapping clusters (covers). This time, the fact that clusters could be overlapped (i.e. one or more agents can belong to different clusters simultaneously) reveals the capacity of some agents to play multiple roles by being involved in several tasks at the same time.
when constructing another least uncertain source-plans.) Step 4: Once one pair ( Pk 0 , R k 0 ) has been selected by solving problem (9) (multiple choices are possible since multiple global minima are available; a genetic algorithm could be used in this aim [2]), a corresponding least vague source-plan should be pointed out. Two choices are possible: a) List all the clustering configurations of Pk0 :
4. VAGUENESS MINIMIZATION PROCEDURE START 1. Initial data: a. the number of agents: N ; b. the collection of K source-plans: P = {Pk } ; k∈1, K
Pk 0 = {Pk 0 ,1 , Pk 0 ,2 ,K, Pk 0 , M k } . 0
b) Construct another source-plans starting not from Pk0 , but from R k0 .
c. the occurrence degrees: {α k ,m }k∈1, K ;m∈1, M .
There is a reason for the second option. Usually, the initial available information about MAS is so vague that it is impossible to construct even consistent source-plans. This is the case, for example, when all we can set are the degrees of occurrence corresponding to clusters created only by couples of agents. It is suitable to point out at least a consistent sourceplan to solve the problem. The main idea in constructing different source-plans is to evaluate the α -cuts of R k0 and to arrange them in decreasing
k
2. For k ∈ 1, K : 2.1. For m ∈ 1, M k : 2.1.1. Construct
the
characteristic
matrix
H k ,m
associated to configuration Pk ,m . 2.1.2. Multiply H k ,m by α k ,m ( H k ,m ← α k ,m H k ,m ). 2.2. Construct the membership matrix of fuzzy relation R k by: M k = max • {H k ,m } .
order of membership grades. This ordering is the unique we can specify starting from the initial information about MAS. Since the time dimension of MAS evolution was not taken into consideration when constructing the model, no time ordering criterion is yet available. Thus, basically, plans are not constructible with this model. However, it is possible that a plan be coincident with the source-plan generated in this manner (especially when the relation is a similarity one). Two categories of source-plans can be generated using the α -cuts of R k0 : equivalence or holonic source-plans (when
m∈1, M k
2.3. Compute the entropy/vagueness of R k (by using the symmetry): N
S µ (R k ) = − 2
N
∑ ∑ M k [i, j ] log 2 M k [i, j] − i =1 j =i +1 N
−2
N
∑ ∑ [1 − M k [i, j ]]log 2 [1 − M k [i, j]] i =1 j =i +1
3. Solve the problem: R k 0 = arg min S µ (R k ) . k ∈1, K
4. The least vague (genuine) source-plan is Pk0 .
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agents ( a1 , a5 , a 6 and a 7 )2. Then the procedure starts from very parsimonious information about clusters created by couples of agents. Every degree of occurrence is associated with only a pair of agents. Although this information is vague enough, compatibility and holonic plans are still constructed. One constructs first the corresponding fuzzy relation R between the 7 agents. The corresponding membership symmetric matrix M R is depicted below:
5. Other alternatives desired? Yes 5.1. Construct Qk0 - the transitive closure of R k0 : 5.1.1. Set M kR = M k . 0
{ (
0
)}
5.1.2. Compute: M kQ = max • M kR , M kR o M kR . 0
0
0
0
5.1.3. If M kR ≠ M kQ , replace the matrix M kR by M kQ 0
0
0
0
( M kR ← M kQ ) and jump to step 5.1.2. 0
1.0000 0.5402 0.6343 0.1877 0.3424 0.2001 0.4863
0
M kQ 0
Otherwise,
is the membership matrix of
the transitive closure Qk0 . 5.2. If M k ≠ M kQ , then R k0 is just a proximity relation 0 0
and generates a compatibility source-plan PkC , by
0.5402 1.0000 0.3561 0.4797 0.7651 0.2092 0.2794
0.6343 0.3561 1.0000 0.4780 0.1389 0.6858 0.6414
0.1877 0.4797 0.4780 1.0000 0.3191 0.4888 0.3347
0.3424 0.7651 0.1389 0.3191 1.0000 0.2784 0.4291
0.2001 0.2092 0.6858 0.4888 0.2784 1.0000 0.1666
0.4863 0.2794 0.6414 0.3347 0.4291 0.1666 1.0000
0
computing its α -cuts. 5.3. Qk0 is a similarity relation and generates a holonic
A first interesting aspect is revealed by looking on the 4-th line: the manager is tempted to work in association with the executive a 2 (0.4797) rather than with a 3 (0.4780), but he is
source-plan PkH , by computing the corresponding
also oriented to solve problems by itself, with the resource a 6 (0.4888). The unit on the main diagonal is due to the fact that every agent is in the same cluster with itself. Then, since this relation is only proximity one, its transitive cover Q is generated. As mentioned, Q is a similarity relation between agents. In membership matrix M Q only 6 non-unitary largest occurrence degrees remained (and the smallest 15 vanished):
0
α -cuts. No STOP Several remarks about this procedure are necessary. First of all, notice that some optimizations are possible, such as computing the max and min operators inside the loops 2.1. and, respectively, 2. Secondly, the source-plans Pk0 , PkC and PkH 0
are different, in general. Moreover,
PkC 0
1.0000 0.5402 0.6343 0.4888 0.5402 0.6343 0.6343
0
could contain less
configurations than Pk0 and PkH even less than PkC . Usually, 0
0
the small degrees of occurrence disappear not only by computing the max union, but also due to the transitive closure procedure. This reveals how the very uncertain configurations (probably due to subjective observations) are removed. These configurations should also be removed from Pk0 .
0.5402 1.0000 0.5402 0.4888 0.7651 0.5402 0.5402
0.6343 0.5402 1.0000 0.4888 0.5402 0.6858 0.6414
0.4888 0.4888 0.4888 1.0000 0.4888 0.4888 0.4888
0.5402 0.7651 0.5402 0.4888 1.0000 0.5402 0.5402
0.6343 0.5402 0.6858 0.4888 0.5402 1.0000 0.6414
0.6343 0.5402 0.6414 0.4888 0.5402 0.6414 1.0000
Two source-plans (different from the initial one) could be generated: one emerging from R and including (tolerance) covers (with overlapped clusters) and another from Q , including partitions (with disjoint clusters). But the holonic source-plan is the most interesting, since a real plan could be proposed, by ordering its configurations in decreasing order of the 6 occurrence degrees above. This plan is represented in Figure 2. The simulations showed how the final holonic behavior emerged. Moreover the nature of the agents (i.e. if they are resource or manager’s interface agents, etc.) has been identified without any prior knowledge about the agents type. The holonic behavior can be observed for the equivalence source-plan: clusters associate together in order to form larger clusters and, finally, the whole agents set is grouped in one single cluster. In our case study, a possible holonic plan could be the following (starting from the corresponding holonic sourceplan). First, the manager states a goal. Immediately, the executives a 2 and a 3 reach for resources a5 , and,
The main hypotheses H1, H2 and H3 could be used to preset the initial data, by using a System Identification stimulation technique [7]. Thus, since non-deterministic external perturbations affect the MAS behavior, acting like a “black box”, the system is stimulated by imposing a class of goals as large as possible. Every goal is imposed several times in order to observe the clustering configurations. They are each time memorized and counted. Thus, each configuration is characterized by an occurrence rate (the number of its occurrences over the total number of all configurations occurrence for that goal), which, actually, is the occurrence degree. The number of all source-plans associated to the same goal is, in fact, the number of stimulating experiments. 5. SIMULATION RESULTS Since the optimization step in the previous procedure is well known, we skip it here. One focuses only on generation of holonic source-plan, starting from an initial source plan, assumed to be the least vague. Consider a MAS consisting of N = 7 agents: one manager ( a 4 ), two executive agents ( a 2 and a 3 ) and four resource
respectively, a 6 . The executive a 3 realizes that he needs more resources and he starts to use both a1 and a 7 . The next step shows that the two executives associate together (including their resources) in order to reach the goal. The manager 2
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Notice that the nature of these agents are initially unknown. It was revealed only after the vagueness minimization procedure has been run.
associates to them only in the final phase, when the goal is reached. The Shannon fuzzy entropy (Eq. (7)) has the value S µ (R ) = 36.6784 (of maximum 49 ) for this relation, whereas
elementary crisp relation can be defined in a less general manner and the model should be more accurate. The model also depends on the measure of fuzziness used to minimize the uncertainties of the initial information about the system. The Shannon fuzzy entropy is not the unique measure to be used. Another measures could be employed, such as: Minkowski index of fuzziness, Klir general measure of fuzziness [4] and so forth. This approach is opening new directions for further works that may constitute better contributions to the difficult task of MAS modeling. For example, a comparative study between different measures of fuzziness could be developed starting from a real application. Besides, a more complex analysis could be developed, by focussing on the ambiguity facet of the uncertainty (this analysis is performed in [8]). A third envisaging direction is aiming to take into consideration the time as a supplementary coordinate since the MAS evolution develops in time (actually, the time dimension has been already considered in [8] as well). But all these models are useful when the information about the system seems very blurred and we want to give it a little more clarity or specificity. Once a certain degree of specificity is reached, they probably should be replaced by more accurate models.
the amplitude of force towards knowledge (Eq. (8)) is, in this case, ∇S µ (R ) = 12.2942 . a1
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Figure 2. Holonic self-organization case study. 7. REFERENCES 6. CONCLUDING REMARKS
[1] Christensen J.H. – Holonic Manufacturing Systems: Initial Architecture and Standard Directions – First European Conf. on HMS, Hanover, Germany, 1994. [2] Goldberg D.E. – Simple Genetic Algorithms, University of Michigan, Dept. of Civil Engineering, Ann. Arbor MI, 1982. [3] O’Hare G.M.P., Jennings N.R. – Foundations on Distributed Artificial Intelligence, John Wiley & Sons Interscience, 1996. [4] Klir G.J., Folger T.A. - Fuzzy sets, Uncertainty, and Information, Prentice Hall, 1984. [5] Maturana F., Norrie D.H. - Multi-Agent Coordination Using Dynamic Virtual Clustering in a Distributed Manufacturing System, Proceedings of Fifth Industrial Engineering Research Conference (IERC5), Institute of Industrial Engineering, Minneapolis, May 18-20, 1996, pp. 473-478. [6] Russell S., Norvig P. - Artificial Intelligence – A Modern Approach, Prentice Hall, 1995. [7] Söderström T., Stoica P. – System Identification, Prentice Hall, London, 1989. [8] Stefanoiu D., Ulieru M., Norrie D. – Fuzzy models for Multi-Agent Systems: Vagueness and Ambiguity Minimization, submitted to IEEE Transactions on Fuzzy Systems, 2000. [9] Subramanian R., Ulieru M. – An Approach to the Modeling of Multi-Agent Systems as Fuzzy Dynamical Systems, Intl. Conference on System Research, Informatics and Cybernetics, Baden-Baden, Germany, August, 1999. [10] Zimmermann H.J. - Fuzzy Set Theory And Its Applications, Kluwer Academic, 1991.
In this paper, we were concerned with the problem of MAS modeling starting from a very parsimonious information about its goals and behavior. The Theory of fuzzy sets and measures seemed to be a very suitable tool in construction of such models. But, the fuzzy model constructed here is not unique. Several possible extensions of this approach could be considered as well. The number of agents, N , can vary during the MAS evolution. One can estimate the maximum number of agents that the system could include and set N by this value. The agents that are not actually present in a state can be replaced by virtual agents within singleton clusters. The real action of an agent is not concerned in this model, so these virtual agents can be considered as doing anything (eventually, as waiting for start their actions). It is important to preserve a constant number of agents among all source-plans, in order to have the same dimension of the matrices H k ,m (otherwise, the fuzzy relations cannot be constructed). In this case, the virtual agents produce only one non null value in H k ,m , which is the unit on their corresponding diagonal position. This does not affect the final fuzzy relation and provides only a dimensionality increase. The model is flexible about how the degrees of occurrence can be initially set. The use of a System Identification technique is not the only possibility. The operator of the model can construct these numbers by his own means. The elementary crisp relation defined by H k ,m could be defined in a different manner. We defined it here by the statement: “two agents are in the relation if they belong to the same cluster”. If it is too general, this definition could be replaced by another ones, more specific, e.g. by taking into consideration some possible agent actions such as: one agent constructs another agent, one agent coordinates another agents, two agents collaborate on a task or negotiate a deal, etc. As long as more precise information about agents is available, the
Note This research was developed with the financial support of NORTEL Chair affiliated to Department of Mechanical and Manufacturing Engineering within the University of Calgary (Canada).
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