Model of Cooperation in Multi Agent Systems with Fuzzy Coalitions José C Romero Cortés 1,2 , Leonid B. Sheremetov1,3 1
Center for Computing Research of the National Technical University, (CIC-IPN), Mexico 2 Universidad Autónoma Metropolitana, México 3 St. Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences (SPIIRAS), Russia
[email protected],
[email protected] Abstract. Agent-based computing is a new paradigm to build complex distributed computer systems. The article e xplores one of the key issues of agent-based computing - the problem of interactions in multi-agent systems (MAS) in dynamic organizational context. Particularly, the article describes an approach to the problem of coalition forming based on fuzzy coalitio n games with associated core, as well as fuzzy linear programming and genetic algorithms for the game solution search. T he proposed approach enables coalition forming based on the fuzzy game theory and permits to change the way of MAS programming from the predefined ad-hoc architectures to dynamic flexible and agile systems with dynamic configurations developed on-line by the MAS itself. The proposed model is applied for the coalition forming for management of supply chain networks.
1. Introduction Agents, high-level interactions and organizational relationships are considered as essential concepts of agent-based computing [6, 9, 10]. To cope with the variety of interactions in dynamic organizational context, agent researchers have devised protocols that enable organizational groupings to be formed; specified mechanisms to ensure groupings act together in a coherent fashion; and developed structures to characterize the macro behaviour of collectives. Nevertheless, when it comes to developing MAS systems with currently available techniques a problem that interactions between the various agents are too rigidly defined still exist. For real applications, such as management processes in supply chain networks (SCN) [4, 8], agents operate within the dynamic organizational context and have to make decisions about the nature and scope of interactions at run time, so it is imperative that this key shaping factor is taken into account while developing models of interaction supporting cooperative activity of agents. This article describes an approach that enables coalition forming based on the fuzzy game theory. The theory of games in deterministic coalitions has been treated by numerous authors [14, 20], nevertheless, it is more natural to form coalitions as a product of negotiation that occur in a fuzzy environment [1, 3, 12]. The main problem
in application of this approach in MAS, even in deterministic case [16], is a difficulty of analytical solution of a game. The problem of formation of fuzzy coalitions in the context of the theory of games is formulated in this article under two approaches: (i) based in the fuzzy core; (ii) using fuzzy linear programming. In both cases, for simplicity linear membership functions are assumed. In addition, methods for solution are given based on genetic algorithms and conventional algorithms of linear programming permitting simplify significantly a problem of the game solution search. Finally, the proposed model is applied for the coalition forming for SCN management.
2. Fuzzy coalition games model The common idea in all distributed artificial intelligence contributions is that agents cooperate during problem solving and hence the basic goal behind cooperation is reaching a consensus. The cooperation like function of utility, is motivated by the endeavour of agents to increase their individual benefits by means of coordinated activities [11, 18]. The theory of games with coalitions offers results showing the general form of possible cooperation and conditions under which this is obtained. In many cases, a broad class of attainable patterns of cooperation and of distributions of the final payments exists, and it is important to exactly suggest one of them like the best one or the most unbiassed. Coalition formation generally takes part before the application of coordinated strategies when agents have only a vague idea about the expected gains of the coalitions, so distribution of the gains can be vague, imprecise, and uncertain. Using the theory of the fuzzy sets and the theory of games in coalitions it is possible to study the vagueness and to follow the way from the input of the vague idea about the gains via the bargaining process to the output corresponding to the possibilities of the vagueness of the distributions of the individual gains. Considering the nature of the application domain, the approach developed in this article is debt to Milan Mares [12, 13] and is based on fuzzy coalition game of integer players with full involvement. Let I be the players set with: I = {1, 2,...,n}, where n can be finite or infinite. Then the set of possible coalitions is the power set:
2I = {φ ,{1},{2},...{n}, {1, 2},...,{n −1, n}, {1, 2,3},...,{n − 2, n −1, n},...,{1,2,...n}} A structure of n coalitions K
1
,...,
K
m
is: κ
=
{K
1
,...,
(1) K
m
}
( I , v ) , where I is nonempty and finite set of players, subsets of I are called coalitions, and v is called a characteristic function of the game, which is a mapping: v : 2 I → R , then v ( K ) ∈ R , ∀K ⊂ I , with v (φ) = 0 and v ( K ) ≥ 0 for any K ⊂ I . The game ( I , v ) is super-additive, for any K, L⊂I with K∩L =φ, if the inequality v( K ∪ L) ≥ v( K ) + v(L ) holds. The real vector x = ( xi ) i∈ I ∈ R I is called an The crisp coalition game is defined as a pair
imputation, and it is accessible for a coalition, K ⊂ I ⇔
X i ≤ v( K ) . It should be ∑ i∈K
mentioned that our game is of non-zero sum, so the standard form of the imputation definition is not applied in our case. Definition.- The core of the game (I,v) is defined as a set of imputations:
C = {x = ( xi ) i∈I :∑ xi ≤ v ( I ), ∀K ⊂ I : ∑ xi ≥ v ( K )} i ∈I
(2)
i∈K
where C (nonempty) is the set of possible distributions of the total payment achievable by the coalition of all players, this is accessible for the coalition of all players and none of coalitions can offer to its members uniformly more than what they obtain accepting some imputation from the core. The first argument of C , unlike the conventional definition of the core in the case of cooperative game with transferable utilities, or TUgame [18], indicates that the payments for the great coalition are less than the characteristic of I . This restriction is added by the nature of the problem at hand. The second argument are the imputations relative to each coalition. The fuzzy core for this game is defined as an extension of crisp core in [13]. Let
( I , v ) be a crisp coalition game and let w : 2 I → R + be a mapping connecting every + coalition K ⊂ I with a fuzzy quantity w( K ) ∈ R , called characteristic function of the game, with membership function λk : R → [0,1] , where v (K ) is the modal value of w(K ) with w(φ) = 0 . Then ( I , w) is called a fuzzy coalition game and is a fuzzy extension of ( I , v ) . In this case, the fuzzy core C F has the following membership function:
γ c F ( x ) = min [ν f = ( w ( I ), K⊂I
where: ν f = ( w ( I ),
∑x i∈ I
i
∑x
i
∑
),ν f = (
i∈ I
x i , w ( K )))]
(3)
i∈ K
) , with the preference
f= as a weak order fuzzy
[ ]
relation with membership function ν f = : R × R → 0,1 . This definition differs significantly from the approach of cooperative fuzzy games with scalable players proposed in [1, 19], where coalitions involve fractions of players. In that case, the fuzzy core of a fuzzy game ( I , v, Λ) with positively homogeneous v is equal to the subdifferential ∂v (Λ ) of v at Λ , which is not the case for our problem with integer full-involved players. Moreover, the fuzzy nature of the coalitions as they are defined in [1], is based mainly on the partial participation of the players in the game. In our case, a fuzzy coalition generates the membership function of the players' payments, which are fuzzy. The class of games with nonempty core is also a fuzzy subset of the class of coalition games and in some games the core is achievable, however in others the
agreement occurs with some degree of possibility. In the latter case, the possibility of such agreement is full excluded. The possibility that C F ≠ φ for the game ( I , w) is given by:
γ C F ( I , w ) = sup( γ C F ( x )) : x ∈ R I ) To assure that C F ≠ φ , we have that if the characteristics of the game fuzzy quantities then: 1. The possibility that the game ( I , w) is superadditive is given by:
(4)
( I , w) are
σ (I , w ) = min(ν f ≈ (ω (K ∪ L),ω (K ) ⊕ ω ( K ) : with K ∩ L = φ , K , L ⊂ I ) 2. The possibility that the game
( I , w) is convex is given by:
δ (I , w) = min(ν f ≈ (ω ( K ∪ L) ⊕ ω (K ∩ L),ω ( K ) ⊕ ω (K ) : K , L ⊂ I ) It is evident from 1) and 2) that δ ( I , w) ≤ σ (I , w) , and if (I ,ν ) is convex then
γ C ( I , w) = 1 . If (I , w) is an extension of ( I ,ν ) with µ K (x) increasing for F
x < ν (K ) and decreasing for x > ν (K ) for K ⊂ I y x ∋ µK ( x) ≠ 0 then: γ C ( I , w) ≥ δ (I ,ω ) F
(5)
From (5), if ( I , w) is convex then the game has a solution, that is C F ≠ φ . The fuzziness is a fundamental component of realistic models of cooperation, which can be considered from the following two points: (i) when a simultaneous participation of players in several coalitions, similar to TU-games, takes place, and (ii) when fuzziness is in a preliminary shape or is a priori expressed by players and coalitions about their prospective earnings. Considered approach is more oriented on the second case. This focus of coalitions formation can be supplemented by knowledge administration, individual one as well as of a group.
3. Fuzzy coalition game example and solution techniques Let us consider one example of a game defined above. In order to show how the solution methods proposed below work, we shall consider first the same example discussed in [13], for which the analytical solution exists. Consider a fuzzy game 2 ( I , w) with I = {1,2} , where for some ( x1 , x 2 ) ∈ R we have:
for x2 ≤ 6 1 for x1 ≤ 4 for x2 ≥ 10 1 0 µ{1} ( x1 , x2 ) = 0 for x1 ≥ 6 , µ{2} ( x1 , x 2 ) = . 5 for x2 ∈ [ 7,9 ] 3 − x / 2 for x ∈ (4 ,6 ) 4 − x / 2 for x ∈ (6 ,7 ) 1 1 2 2 5 − x2 / 2 for x2 ∈ (9 ,10 ) for x1 + x 2 ≤ 8 1 µI ( x1 , x 2 ) = 0 for x1 + x 2 ≥ 16 2 − ( x + x ) / 8 for x + x ∈ ( 8,16 ) 1 2 1 2 Although the membership functions in our case are linear (triangular and pyramidal forms) [3], it is a complicated task to find the analytical solution of the fuzzy game ( I , w ) . For example, the Pareto optimal and the superoptimal solutions are fuzzy sets in the plane with imputation ( x1 , x 2 ) = (4, 6) which occurs with a possibility of 0.75:
γ
C
F
( I , w ) = sup( µ C F ( x 1 = 4 , x 2 = 6 )) = . 75
3.1 Solution technique using genetic algorithms Due to the complexity of the problem of analytical (exact) solution, it is proposed in this article to apply heuristic techniques and to find an approximate one. In our case, we apply the soft computing techniques using genetic algorithms in a context of fuzzy logic; this is equivalent to a binary codified fuzzy core with adjustment fusion of the supreme of the minimums of membership functions. We fixed the rates of mutation and crossover in the typical values of 6 and 50% respectively, with a size of population of 50 organisms, time convergence parameters and number of iterations are also fixed [15]. In order to obtain the solution, we use the legacy software Evolver from Microsoft, which requires of the formulation of the problem in an Excel spreadsheet [2]. For this, it is necessary to store a supreme of the minimums as a formulae in one cell, and to store variables and the arguments of fuzzy core in the others. In addition, initial values of the parameters of the genetic algorithm are provided. It is also required to give any initial solution that can or not be a feasible one, this is an arbitrary imputation. As it is shown in figure 1, the solution very close to the analytical one is obtained with this algorithm. The curve of the evolution of fuzzy core can also be observed, as well as the variable and the supreme that have already arrived at convergence levels. As it can be observed in figure 1, the cell A1 shows the possibility of the imputation (4.0098, 6.0412) in iteration 2659 which is of 0.743625. The criterion of termination of the algorithm was 2000 minutes, nevertheless the shown solution required only 3 minutes. The mathematical effort is reduced enormously, although for real cases where a number of players and therefore of coalitions is large, it is necessary to simulate a
great number of games of fuzzy coalitions and to see what happens to the percentage of successes. We shall consider another more real example in the following section.
Fig. 1. Solution of the game, evolution of objective function, rates of mutation and crossover and levels of the varia bles in convergence iteration.
3.2 Solution technique using fuzzy linear programming There are other approaches to this problem, although many of them do not have such a mathematical support as the case of the core. For instance, the problem also can be formulated in terms of the linear programming according to: Min Z = CX