A fuzzy cooperative game model for configuration management of open supply networks Текст научной статьи по специальности «Математика»

A Fuzzy Cooperative Game Model for Configuration Management of Open Supply Networks*

Leonid B. Sheremetov1 and Alexander V. Smirnov2

1 Mexican Petroleum Institute, Av. Eje Central Lazaro Cardenas Norte, 152, México, DF, CP 07730, E-mail: sher@imp.mx

2 St. Petersburg Institute for Informatics and Automation, Russian Academy of Sciences, Chetyrnadtsataya Liniya, 39, St. Petersburg, 199179, Russia E-mail: smir@iias.spb.su

Abstract. The paper considers the problem of open supply networks (OSNs) configuring in a highly dynamic economic environment. A novel coalition formation mechanism is proposed, which helps to resolve conflic-tions between the objectives of the OSN participants and to agree upon effective solutions. This mechanism is based on a generalized model of a fuzzy cooperative game with core. The model was applied for configuring of an automotive OSN. Simulation results are considered.

Keywords: cooperative game, core, supply network, configuring. 1. Introduction

In a todays highly competitive market manufacturers face the challenge of reducing manufacturing cycle time, delivery lead-time and inventory. As a consequence, new organizational forms of enterprise integration emerge to address these challenges resulting in more agile structures of federated enterprises known as adaptive, agile and open supply chains and networks. These organizations are based on the principles of partnership between the enterprises, agile network structures instead of linear chains and are driven by novel business strategies based on the product demand (Fig. 1.). In contrast to conventional chains and supply networks, such organizations can be called open supply networks (OSNs), which are characterized by:

— Availability of alternative providers.

— Availability of alternative configurations meeting orders specifications.

— Expediency of dynamic configuration and reconfiguration of the network depending on the order stream and economic benefit of every enterprise.

— Conflicting objectives of each organization and non-integrated decision making processes.

An OSN belongs to the class of systems with dynamically changing structures, which means that once a new order comes, a new configuration emerges. Thus OSN configuring can be considered one of the main supply chain management

* This work was supported by the European Commission under the Project FP6-IST-NMP 507592-2 "ILIPT", Russian Foundation for Basic Research under the grant No.08-07-00264 and by the Presidium of the Russian Academy of Sciences under the project No.213.

Fig. 1. Organizational forms of enterprise integration (adopted from (McBeath et al., 2010))

tasks (Garavelli, 2003; Chandra and Grabis, 2007). Traditionally, configuring has been solved in a two-stage fashion: i) a structure of a network is formed at a strategic level and ii) its behavior is optimized at tactical and operational levels based on demand forecast. Being suitable for vertical and even virtual enterprises (Fig. 1.), such practices, unfortunately, do not meet the requirements of a highly dynamic environment (Smirnov, 1999). One of the consequences is a so-called bullwhip effect, when even small demand fluctuations in a loosely balanced forecast-driven distribution systems lead to increased inventories, and, as a consequence, spatial constraints, unused capital, obsolete inventories and so on (Suckya, 2009).

Federated enterprises composed of self-interested entities with probably conflicting goals require flexible dynamic configurations. Unfortunately, adoption of more flexible and dynamic practices, like constraint satisfaction, auctions and knowledge-based approaches, which offer the prospect of better matches between suppliers and customers as market conditions change, has faced difficulties, due to the complexity of many supply chain relationships and the difficulty of effectively supporting dynamic trading practices (Hosam and Khaldoun, 2006; Sandkuhl et al., 2007; Smirnov et al., 2006). Due to the conflictions among the objectives of each organization and non-integrated decision making processes, there has been a need for new mechanisms, which help to resolve those conflictions and to agree upon effective solutions.

For advanced business strategies of OSN like demand-driven or build-to-order SN, a task of configuring of virtual production channels can be defined as a coalition formation task. The benefit distribution among the OSN members has proved to be fuzzy, uncertain, and ambiguous (Roth, 1995). Using the theory of fuzzy cooperative games (FCGs), we can process the uncertainty and pass from the introduction of a fuzzy benefit concept through the bargaining process to the conclusion about the corresponding fuzzy distribution of individual benefits among the coalition members.

In this paper, a game-theoretic approach is used to form coalitions: a class of FCG with core is considered. The basic definition of the fuzzy core was proposed by (Mares, 2001) and extended to the area of multiagent systems by (Sheremetov and Romero-Cortes, 2003). In this paper, an extended definition of the core is considered. The definition introduces fuzzy individual payments and binary values fij added to the fuzzy core to form the structure of effective coalitions.

The rest of the paper is structured as follows. In the next section, a OSN configuring task is defined as a problem of coalition formation. In section 3, different approaches to coalition formation in cooperative games theory are analyzed. In section 4, the mathematical structure of the model is described; it is shown that the model represents an extension of the model proposed in (Mares, 2001). A case study applying the proposed approach is discussed in the context of OSNs in section 5. A prototype consisting of seven enterprises and generating a structure of three coalitions is considered. Finally, the results of the paper are discussed in Conclusions section.

2. Supply network configuring as a coalition formation task

The task of configuring can be defined as a selection of those agents (enterprises), which have available competencies to complete the demand/order, and joining them together in the most efficient structure according to the selected criteria. The main

components of the configuring task are: order, resource and configuration. Let us consider that to fulfill the order T, I tasks should be executed: T={ T1, T2,..., Tj}. Each task Ti(i = 1, 2,...,I), is defined by a tuple {{BTi}, {PrefT*}}, where {BTi} -is a vector of numerical values of the dimension r: {bt* } = (bTi, bT*,..., bbp. 0, characterizing a capacity on each competency k = 1, 2,...,r, required to perform a task Ti. If the tasks are ordered, then T1 < T2 < ... < Tl < Tm ^ ... ^ Tj, where Tl ^ Tm means that Ti precedes the task Tm. The preferences vector PrefT* may include additional parameters like the preferred lot size, penalties for backorders, etc. Fulfillment of each order and each task Ti implies a payoff: Payoff (Ti).

Example 1. Suppose that the order is to produce 100 products (cars of a specific model) per week. A car consists of four basic components: 1) the body Ti (14 external tubes bT1 and 5 exterior sheets bT2); 2) the interior T2 (a dash board bT2, 3 seats bT2: two front and one rear); 3) the chassis T3 (4 wheels bT3, 2 axles bT3: a front and a rear, 4 dampers bT3: two front and two rear); 4) the power train T4 (a motor blTi and a transmission bT4). In other words: T = {Tu T2, T3, T4}, BTl = (1400, 500), bt2 = (100, 300), BT3 = (400, 200,400), BTi = (100,100).

The enterprises of the supply network represent resources. Depending upon their role in the OSN, these resources can be suppliers of raw materials and components, assembly plants or warehouses. They are modeled as active autonomous entities with purposeful actions and, thus, may be called agents. Let us consider a finite set of agents Agent = {A\,A2,. ..,AN}. Then each agent Aj G Agent(j = 1, 2,..., N) is defined as a tuple {{ba, }, {PrefAj}}. For simplicity lets designate Aj as j. Then Bj - is a vector of numerical values of the dimension r: Bj = (bl,b'2,..., br), bkj > 0, characterizing agents available capacity on each competency k' = 1, 2,...,r. The preferences vector Prefk denote agents preferences on the lot size, orders time lag, etc.

Finally, a configuration is such a set of agents (resources) CT C Agent that their joint capacity of competencies satisfies the requirements of an order T. To solve the configuring task for the case when the agents and tasks competencies coincide (bT and bj mean the capacities on the same competency) means to assign resources to the tasks in such a way that the order T is fulfilled. Each agent Aj G Agent may be assigned to a task Ti iff it has available capacities 3k G{1, 2,... ,r}, bj > 0. Being self-interested, each agent will try to optimize this assignment according to one of the following criteria:

r

— Maximize the use of his capacities Y1 (bj — bT) ^ min;

fc=i *

— To get the most profitable task (to increase the payoff)

rr

2 g(bj.) — J2 fTi(bj) ^ max, where g(bj ) - is a reward function associated

j=i * k=i * *

with the payoff Payoff (Ti), fTi (bj) - a cost of agents j G Agent capacity bj

required to fulfill the task Ti;

r

— Reduce the task Ti fulfillment time: ^ tTi (bj) ^ min, where tTi (bj) - time

k=1 * *

of fulfillment of the task Ti by agent k G Agent using his capacity bj.

Agents can form coalitions to execute tasks. The notion of coalition is widely used in organizational systems. A coalition can be defined as a group of self-interested

agents that by means of negotiation protocols decide to cooperate in order to solve a problem or to achieve a goal (Gasser, 1991). Within the context of this paper, a coalition is defined as a group of agents joining their capacities for task Ti fulfillment. A coalition is described by a tuple: (KTi, alloc Ti,uTi), where KTi Q Agent and KTi = 0; allocTi - allocation function assigning each task i a group of m agents such that alloc Ti = KTi, if ^ bm > bT.. If for each competency k, bj > bT.,

m

KTi may consist of a single agent j G KTi, then alloc Ti = j. The coalition of

all agents involved in the orders T execution is called grand coalition KT. The

utility of a coalition is defined by a characteristic function: v(KTi) = Payoff (Ti) —

J2 J2 fTi (bj) ' V>(Ti, k, j), where p - a binary variable, defining the fact that an agent k j

participates in the task execution with his capacity bj:

(T k j / —1, if an agent executes bj , j) \ 0 , in the opposite case

The coalitions utility v(KTi) is distributed between the coalition members according to the vector of payoff distribution uTi = j uT., uT ,■■■, ' j, where ujT. -

payoff of agent j G Agent, and u' - coalition payoff. If within a coalition KTi,

an agent j executes different competencies then ujT, =^2,9j (bT-),

Ti k Ti

&•(&*,) = -^v{KTi) ■ <p{Ti, k,j) is fulfilled.

i E blT.

1=1 i

The grand coalition KT , joining together all the agents participating in the orders fulfillment corresponds to the configuration of the supply network CT. Thus to form a coalition means to find the appropriate coalition structure which permits to maximize the payoff for all agents belonging to this structure.

3. Coalition formation in cooperative game theory

Taxonomy of coalition formation algorithms includes both distributed and centralized algorithms. In this paper we restrict this analysis by those represented by the theory of games (Aubin, 1981). Until recently, in the domain of supply chains and networks management, non-cooperative game theory was usually used for modeling of the competing enterprises as a game with zero-sum. In that context, all the players are considered being self-interested trying to optimize their own profits without taking into account their effects on the other players. The main purpose of such a game is to find the optimal strategy for each player and determine if the obtained strategy coordinates the supply chain, i.e. maximizes the global profit. Nevertheless, the cooperative nature of federated enterprises causes necessity of considering OSN within the context of cooperative game theory in order to model and understand the behavior of cooperating network partners. The principal difference between both approaches lies in different assumptions about the nature of the game and of the rational behavior of the players. In other words, cooperative games are considered in those cases when the players can form coalitions.

In the context of supply networks configuring, the theory of cooperative games offers results that show the structure of possible interaction between partners and the conditions required for it. The main questions to be answered are: what coalitions will be formed, how the common wealth will be distributed and if the obtained

coalition structure is stable. Cooperative game theory represents a variety of models and the selection of the appropriate approach for OSN configuring is a challenging task. The models of coalition formation are usually classified based upon the type of the environment and the principles of the payoff distribution (Fig. 3.). The environment can de superadditive and subadditive. Usually, coalitions joining together can increase the wealth of their players. If they form a single coalition (grand coalition), the only question is to find acceptable distributions of the payoff of the grand coalition. But in the latter case, at least one coalition does not meet this condition. The payoff distribution should guarantee the stability of the coalition structure when no one player has an intention to leave a coalition because of the expectation to increase its payoff. Moreover, profit distribution can be fuzzy, uncertain, and ambiguous (Mares, 2001). Using the theory of fuzzy cooperative games (FCGs), we can process the uncertainty and pass from the introduction of a fuzzy profit concept through the bargaining process to the conclusion about the corresponding fuzzy distribution of individual payoffs.

Fig. 2. Cooperative games' taxonomy

Due to the model complexity, most of the models of cooperative games have been developed for superadditive environments and for fuzzy settings allow us to consider only linear membership functions. Nevertheless, for realistic applications additive environments and the absence of the restrictions on the type of membership functions is a time challenge.

According to (Kahan and Rapoport, 1984), cooperative games can be divided into two classes based on the way a solution of the game is obtained: games with a solution set and games with a single solution. To the former class belong the approaches of the stable sets (Von Neumann and Morgenstern, 1944), the core (Gillies, 1953), the kernel (Davis and Maschler, 1965) and bargaining set (Aumann and Maschler, 1964). To the latter - Shapley value (Shapley, 1953),

t value in the TU-games (Tijs, 1981) and the nucleolus (Schmeidler, 1969). Core and Stable sets are two widely used mechanisms for analyzing the possible set of stable outcomes of cooperative games with transferable utilities. The concept of a core is attractive since it tends to maximize the so called social wealth, i.e. the sum of coalition utilities in the particular coalition structure. Such imputations are called C-stable. The core of a game with respect to a given coalition structure is defined as a set of such imputations that prevent the players from forming small coalitions by paying off all the subsets an amount which is at least as much they would get if they form a coalition. Thus the core of a game is a set of imputations which are stable. The problem of the core is that, on the one hand, the computational complexity of finding the optimal structure is high since for the game with n players at least 2ln—1 of the total |n|ln'/2 coalition structures should be tested. On the other hand, for particular classes of the game a core can be empty. Because of these problems, using the C-stable coalition structures has been quite unpopular so far (Klusch and Gerber, 2002). In this paper we show that these problems can be solved in a proposed generalized model of a FCG.

4. Generalized model of a fuzzy cooperative game with core

A FCG is defined as a pair (Agent, w), where Agent is nonempty and finite set of players, subsets of Agent joining together to fulfil some task Ti are called coalitions K, and w is called a characteristic function of the game, being w : 2n ^ a mapping connecting every coalition K C Agent with a fuzzy quantity w(K) G , with a membership function jK : R ^ [0,1]. A modal value of w(K) corresponds to the characteristic function of the crisp game v(K): max (w (K)) = (v (K)). For an empty coalition w(0) = 0. A fuzzy core for the game (Agent, w) with the imputation X = (xij)iei,jeAgent G as a fuzzy subset CF of :

Cp = Ixij € v ^ (w(Agent), Xj¡j), min (v ^ ( Xj¡j,w(Ki)))

i e I, j e Agent

KiEk

i e I, jEAgent jEKi

h (!)

where xij is the fuzzy payment of an agent j participating in a coalition i, i = 1, 2,...,I, j = 1, 2,...,N, k = [K1, K2, ..., Ki] is the ordered structure of effective coalitions; ^ - is a fuzzy partial order relation with a membership function v ^ : R x R ^ [0,1], and pij is a binary variable such that:

1, if an agent j participates in a coalition i;

[ 0, otherwise.

This variable can be considered as a result of some agents strategy on joining a coalition. A fuzzy partial order relation is defined as follows.

Definition 1. Let a,b be fuzzy numbers with membership functions ja and jb respectively, then the possibility of partial order a ^ b is defined as v ^ (a, b) G [0,1] as follows: v ^ (a, b) = sup (min(ja (x) ,jb (y) )). x,y e R x > y

The core CF is the set of possible distributions of the total payment achievable by the coalitions, and none of coalitions can offer to its members more than they can obtain accepting some imputation from the core. The first argument of the core CF indicates that the payments for the grand coalition are less than the characteristic function of the game. The second argument reflects the property of group rationality of the players, that there is no other payoff vector, which yields more to each player. The membership function fCF : R ^ [0,1], is defined as:

fcF =min v > (w(Agent), Xjpj), min (v y ( Xjpj,w(Ki))) (2)

KiEk K

i e I, jE Agent jEKi

j e Agent

With the possibility that a non-empty core CF of the game (Agent, w) exists:

Ycf (Agent, w) = sup (f cF (x) : x 6S") (3)

The solution of a cooperative game is a coalition configuration (S, x) which consists of (i) a partition S of Agent, the so-called coalition structure, and (ii) an efficient payoff distribution x which assigns each agent in Agent its payoff out of the utility of the coalition it is member of in a given coalition structure S. A coalition configuration (S, x) is called stable if no agent has an incentive to leave its coalition in S due to its assigned payoff xi.

A game (Agent, w) is defined as superadditive, subadditive, or simply additive for any two coalitions K, L C Agent, K n L = 0 as follows:

w(K U L) y w(K) ® w(L) —superadditive,

w* (K U L) < w* (K) © w* (L) —subadditive, (4)

w* (K U L) = w* (K) © w* (l) —additive,

where © - is a sum of fuzzy numbers with a membership function defined as: l^a®b(x) = sup (min(^a(y), ¡j>b(x — y))), * defines superoptimal values of the corre-

x,yER

sponding coalitions (Mares, 2001).

The properties of the game are defined in three lemmas and two theorems (Sheremetov and Romero-Cortes, 2003). One of them proves that the fuzzy set of coalition structures forming the game core represents a subset of the fuzzy set formed by the structure of effective coalitions. In turn, this inference allows us to specify the upper possibility bound for the core, which is a very important condition for the process of solution searching, because in this case, the presence of a solution that meets the efficiency condition may serve as the signal to terminate the search algorithm.

Definition 2. A coalition K is called effective if it can't be eliminated from the coalition structure by a subcoalition L C K. A set of effective coalitions is called a coalition structure. A possibility that a coalition K is effective is defined as follows: supxER„ (min(^fc(x), fi,* (x) : L C K)).

Theorem 1. Let (Agent, w) be a fuzzy coalition game. Then for some structure of effective coalitions k, its possibility is at least equal to the possibility of forming the

core.

Proof. From formula 2, if all pij are equal to 1, then we obtain the structure of coalitions that belong to the core; otherwise, the coalition structure corresponds to the generalized model. In addition, the inequality v ^ ( £ xij, £ xijpij, i G I)

jeKi jeKi

holds with positive possibility and, consequently, the possibility of the structure is higher for the generalized model than for the basic one. □

It should be noted that the above statements take into account only the characteristics of the game (Agent, w); therefore, any real argument can be introduced into the fuzzy core. For example, such restrictions as a number of agents in each coalition and those defining coalitions to be overlapping or not or regulating the tasks order are admissible. This feature is very important for the application of the model for OSN configuration management.

To find the analytical (exact) solution of the FCG, it is necessary to determine the fuzzy super-optimum and the fuzzy relation of domination (Mares, 2001), which is extremely difficult in real applications. Therefore, it is proposed to use a heuristic technique of finding solutions that are close to the optimal one. In the considered case, the techniques of soft computing using genetic algorithms (GA) in the context of fuzzy logic are applied. It is equivalent to binary encoding of the fuzzy core with the fitness function equal to the supremum of all minimums of the membership function. Application of GA allows one to obtain an approximate solution for the games with a large number of players and a membership function of any type. Being an anytime algorithm that steadily improves the solution, the GA can find the best solution under the time constraints.

5. Case study: a cooperative game for 3-echelons automotive OSN

The developed model of a cooperative game was used for configuring of an OSN's production channel for a specific car's model. The demand is represented by a uniform distribution around the linear trend:

dt = a + b ■ t + a ■ j, (5)

where t - time, d - demand (dt corresponds to time interval [t — 1,t]), a - basis value, b - trend (equals 0 for a demand without trend), j - random noise uniformly distributed within [0,1], and a - distribution amplitude. For the demand forecasting Simple Moving Average (SMA) is used:

_£ di

ft+2 = ft+1 = i~t~"+1 , (6) n

where f - forecast, n - forecast base.

Suppose that the OSN contains several enterprises capable of satisfying the demand both in components' production and vehicle's assembly. The configuring task can be defined as follows: to select an effective configuration of a production channel (both the enterprises and the demand's distribution between them) such that an ordered quantity of vehicles (a = 100) can be produced on five consecutive week intervals (n = 5) with a low noise (a = 5) and without fluctuations associated with storing and delivery of the final and intermediate products. The enterprises pursuit a goal of maximizing their payoffs. The following parameters are considered:

production capacity (units per week), production cost (per unit), stocking costs (per unit per week) and penalties for backorders (per unit per week). Stocks are unlimited. Payoffs for each component are fuzzy variables defined, for simplicity, by a uniform positive ramp membership function. The forecasting model for the demand is the following:

100 + 5t + 5^, for t = 1,...,5, (7)

Component production can be performed by 6 enterprises, each with different competencies (Table 1). For simplicity, the competencies are restricted to the task level. The payoff for the assembled car is $20000.

Table 1. Input data for the fuzzy cooperative game for automotive OSN configuring

Enter- capa- Competency Membership Compo- Pro- Sto- Penal- Asso -

prise city function nent's price duc- cking ties for ciated

(units (MF) (parameters tion cost back- variable

per of the MF) cost (per orders

week) unit per week) (per unit per week)

1 100 Body Positive ramp (+) $(65007000) $4500 $250 $400 X11, P11

2 100 Motor Positive ramp (+) $(45005000) $3500 $150 $300 X22, P22

3 100 Transmission Positive ramp (+) $(38004000) $2500 $50 $250 X33, P33

4 300 Body Positive ramp (+) $(65007000) $4900 $300 $400 X14, <P14

Motor Positive ramp (+) $(45005000) $3800 $200 $300 X24, P24

Transmission Positive ramp (+) $(38004000) $2700 $80 $250 X34, P34

5 100 Motor Positive ramp (+) $(45005000) $3600 $170 $300 X15, P15

Transmission Positive ramp (+) $(38004000) $2600 $60 $250 X25, P25

6 200 Body Positive ramp (+) $(65007000) $4700 $270 $400 X16, P16

Motor Positive ramp (+) $(45005000) $3600 $170 $300 X26, P26

7 150 Assembly Positive ramp (+) $(20004000) $1500 $750 $1500 X47, P47

The order is decomposed into tasks which correspond to each car component's assembly. As a result, an effective structure of three coalitions (according to the number of the components) is to be formed considering capacity constraints. The structure of the core of the cooperative game is shown in Table 2. Additional constraints define the viability of the obtained solution.

The following notation is used: Xj - the quantity of the i component to be produced by agent j in time t, w(Agent) - fuzzy payoff per unit for car production,

Table 2. The structure of the core of the cooperative game

Core's component

Definition

C = (2500x1« + 2100xi4t + 2300xi6t + 1500x22t + 1200x24t + 1400x25t + 1400x26t + 1500x33t + 25 00x47t + 1300xs4t + 1400x35t > (100 + 5t + 5p) w (Agent),

2500xnt + 2100xi4t + 2300xi6t < (100 + 5t + 5^) w (ki)

1500x22t + 1200x24t + 1400x25t + 1400x26t < (100 + 5t + 5^) w k)

1500x33t + 1300x34t + 1400x35t < (100 + 5t + 5^) w k)

25 00x47t < (100 + 5t + 5p) w (k4)

xiit + xi4t + xi6t < 100 + 5t + 5^ x22t + x24t + x25t + x26t < 100 + 5t +5^ x33t + x34t + x35t < 100 + 5t + 5^ x47t < 100 + 5t + 5^

xiit < 100 xi4t < 300 xi6t < 200 x22t < 100 x24t < 300 x25t < 100 x26t < 200 x33t < 100 x34t < 300 x35t < 100 x47t < 150

Xijt eR+, i= 1,..., 4; j = 1,..., 7 t = 1,..., 5_

Constraint on the grand coalition

Constraints on the components' coalitions

Constraints on the forecasted demand for each component

Capacity constraints on the payoffs

w(k\) - fuzzy payoff per unit for Body Production, w(k2) - fuzzy payoff per unit for Motor Production, w(k3) - fuzzy payoff per unit for Transmission Production, w(k4) - fuzzy payoff per unit for car assembly, and ^ - uniform random variable in [0,1]. The solution of the game obtained using Evolver package and genetic algorithms is shown in Table 3.

Table 3. The coalition structure and the number of produced components for five time intervals

t xiit x22t x33t xi4t x24t x34t x25t x35t xi6t x26t x47t

1 100 100 100 0 0 0 5,299 5,3 5,3 0,001 105,3

2 100 100 100 0 0 0 8,399 11,5 11,5 3,101 111,5

3 100 100 100 0 0 0 10,25 15,2 15,2 4,95 115,2

4 100 100 100 0 0 0 14,50 23,7 23,7 9,20 123,7

5 100 100 100 0 0 0 16,75 28,2 28,2 11,45 128,2

The common network payoffs per car obtained for each time interval are equal to 7,578.46, 7,578.22, 7,578.22, 7,577.84, 7,577.84 respectively. The payoffs (p) of the participating enterprises per car/component are as follows: p1 = 2500; p2 = 1500; p3 = 1500; p4 = 0; p5 = 1400 (motor and transmission); p6 = 2300 (body); p6 = 1400 (motor); p7 = 2500. The same gross payoffs per enterprise were obtained for each time interval for each component: w(Agent) = 20,000; w(k1 )= 7,000; w(k2)=5,000; w(k3)=4,000 w(k4)=4,000. The possibility of the fuzzy game Yc(Agent,w) = 1.00 (because of the simplicity of the case study), though the imputation obtained took into account the subjective estimations of the players defined by their fuzzy payments.

The analysis of the obtained solution shows the following. The constraint capacity of the first 3 units though having minimal production costs, does not permit them to satisfy all the demand. That is why, while demand is increasing, other enterprises are involved in the production. In the case of Motor Production (k2), the incrementing production of this component is assigned to both enterprises 5 and 6 (Table 3). If we compare the parameters of these enterprises (Table 2), it can be seen that they are the same both for the production cost ($3600 per motor) and for stocking ($170 per motor /week). That means that the solution strategy looks for a balanced final solution.

In the conducted experiments on model complexity the number of iterations needed to approach the optimal solution served as the investigated variable with the following factors: the number of agents and coalitions, the accuracy, and the order of fuzzy payments. Results show that the number of iterations (computation time) decreases or remains constant when the number of agents increases. In other words, it takes less time to form coalitions. On the other hand, the results demonstrate almost linear relation between the numbers of coalitions and agents. On the whole, the experiments justified that all factors are highly significant; the only surprise was that the order of fuzzy payments substantially influence the number of iterations (the convergence time).

6. Conclusions

In this paper, the approach to OSN configuration based on formation of enterprise coalitions as a result of a fuzzy cooperative game was considered. Uncertainty in realistic cooperation models occurs in two cases: when players participate in several coalitions, and when there exist fuzzy expectations of player and coalition benefits. The presented approach is mostly aimed at the latter case. This uncertainty of the agent payments may be caused by such dynamic events as production failures, changes in confidence estimations and reputations of potential coalition partners, and receiving unclear or even incomplete information and data during the task performance and negotiation.

The proposed model considers the coalitions' efficiency by introducing binary variables pj into the fuzzy core. This permits not only to increase individual benefits for players but also the possibility to find an effective and stable agreement. Using the constraints of the application domain the number of viable coalitions can be significantly decreased, thus reducing the algorithmic complexity of the problem. Though in the case study a positive ramp membership function was used (to be able to use also conventional Excel solver), the general solution method (applying genetic algorithms) permits the use of function of any type (linear or nonlinear, universal or not). Obviously, there is no guaranty that the obtained solution corresponds to a global optimum, but for a game with side payments, there is no algorithm to obtain the optimal solution.

The fields of FCGs and dynamic coalition formation are still in their infancy and require further research efforts. For example, the notions of a superadditive FCG and a "stable" distribution of fuzzy payments in the games using fuzzy extension of the core and Shapley values were examined in (Mares, 2001). Some aspects of application of the coalition game models to the development of dynamic coalition formation schemes were considered in (Klusch and Gerber, 2002). Nevertheless, sub-additive fuzzy games and the notions of "uncertain" stability and effective algo-

rithms for FCGs represent the subjects for current research. In the future work, the development of algorithms for dynamic formation of fuzzy coalitions seems to be the promising and challenging problem in the field of self-organizing system research.

Acknowlegments. The authors are very grateful to Dr. Andrew Krizhanovsky for his help and assistance in the preparation of the final version of the paper.

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