Fuzzy Characteristic Functions Research Articles

Mathematical encouragement of companies to cooperate by using cooperative games with fuzzy approach

In this paper, we will analyze the phenomenon of strategic cooperation between companies competing with each other so far. The analysis will concern the real situation of establishing a strategic alliance between companies. The analysis will consist in assessing the effects of cooperation for each of the companies, depending on whether and in what composition the companies would decide to cooperate. For the analysis, we will use a mathematical model of a cooperative game with fuzzy characteristic functions. Finally, we propose fuzzy interval solutions are a timely extension to classical game theoretic solutions.

A real Shapley value for evidential games with fuzzy characteristic function

Game theory is a famous issue of expert decision making. The real Shapley value for cooperative games with fuzzy characteristic function has high performance in deal with cooperative games, which is an effective tool in deal with issues of game theory. The real Shapley value for cooperative games with fuzzy characteristic function is based on the level sets, which is the extent of fuzzy sets. However, the real Shapley value for cooperative games with fuzzy characteristic function cannot solve the evidential games problems. What is the real Shapley value for evidential games with fuzzy characteristic function is still an open problem. This paper proposes the real Shapley value for evidential games with characteristic function, which consists of level sets, the real evidential Shapley value, basic probability assignment function. The real Shapley value for evidential games with fuzzy characteristic function can solve the expert decision making issues under evidential environment, with the aid of basic probability assignment function. Meanwhile, the theorem of the proposed model has been discussed. Numerical examples has been applied to illustrate the effectiveness of the proposed model. The experimental results show that proposed model can obtain the real evidential Shapley value of a given evidential games and address the issues of expert decision making.

Banzhaf values for cooperative games with fuzzy characteristic function

ABSTRACT The Banzhaf value, which determines the power of each agent in a cooperative game, has been used in the literature to analyze fuzzy cooperative situations. In this paper, we propose two Banzhaf values for games with fuzzy characteristic function. In one of them, the players' payoffs are fuzzy quantities. In the other one, the payoffs are real numbers. In each case, we provide an axiomatization with reasonable properties.

A real Shapley value for cooperative games with fuzzy characteristic function

There are cooperative situations in which the players have only imprecise expectations about the profit that can be obtained by each coalition. In order to model these situations, several families of games have been introduced in the literature. Games with fuzzy characteristic function are among them. The main problem that arises when dealing with one of these games is how to allocate among the players the total profit derived from the cooperation. In this regard it seems reasonable that the vagueness in the payments of the coalitions will cause vagueness in the payoffs of the players. In fact, the values introduced for these games assign a fuzzy payoff to each player in the game. However, in some of these situations it might be necessary to assign a precise payoff to each player. With this purpose, in this paper we use a well known ranking for fuzzy numbers to introduce a real Shapley value for games with fuzzy characteristic function.

A characterization of the Shapley value for cooperative games with fuzzy characteristic function

The characteristic function of a cooperative game determines the payment that each coalition can obtain when the players in the coalition cooperate. But there are cooperative situations in which the players have only imprecise expectations about the profit that can be achieved by each coalition. These situations are modeled through cooperative games with fuzzy characteristic function, in which the payment of each coalition is a fuzzy quantity. A value for these games assigns to each player in a game a fuzzy quantity that indicates the vaguely expected payoff for the player. There is a Shapley value for games with fuzzy characteristic function, but no characterization of this value has been given in the literature. In this paper a characterization of the Shapley value for games with fuzzy characteristic function is presented.

The graph games with fuzzy characteristic functions

The graph games with fuzzy characteristic functions

A TRANSITION FROM TWO-PERSON ZERO-SUM GAMES TO COOPERATIVE GAMES WITH FUZZY PAYOFFS

In this paper, we deal with games with fuzzy payoffs. We proved that players who are playing a zero-sum game with fuzzy payoffs against Nature are able to increase their joint payoff, and hence their individual payoffs by cooperating. It is shown that, a cooperative game with the fuzzy characteristic function can be constructed via the optimal game values of the zero-sum games with fuzzy payoffs against Nature at which players' combine their strategies and act like a single player. It is also proven that, the fuzzy characteristic function that is constructed in this way satisfies the superadditivity condition. Thus we considered a transition from two-person zero-sum games with fuzzy payoffs to cooperative games with fuzzy payoffs. The fair allocation of the maximum payoff (game value) of this cooperative game among players is done using the Shapley vector.

Fuzzy Multichoice Games with Fuzzy Characteristic Functions

In this paper, a generalized form of fuzzy multichoice games with fuzzy characteristic functions is proposed, which can be seen as an extension of traditional fuzzy games. Based on the extension Hukuhara difference, fuzzy multichoice games with fuzzy characteristic functions are studied, and a Shapley function is discussed. The notion of fuzzy multichoice population monotonic allocation scheme (FMPMAS) is defined. When the given fuzzy multichoice game with fuzzy characteristic functions is convex, we show that the proposed Shapley function is a FMPMAS. Furthermore, two special kinds of fuzzy multichoice games with fuzzy characteristic functions called fuzzy multichoice games with multilinear extension form and fuzzy characteristic functions and fuzzy multichoice games with Choquet integral form and fuzzy characteristic functions are researched.

Encoding symbolic features in simple decision systems over ontological graphs for PSO and neural network based classifiers

In the paper, we present the idea of encoding symbolic features appearing in simple decision systems over ontological graphs for building classifiers based on Particle Swarm Optimization (PSO) as well as Neural Networks . Simple decision systems over ontological graphs refer to a general trend in computations proposed by Zadeh and called “computing with words”. In case of such decision systems, we deal with attribute values, describing objects of interest, which are concepts placed in semantic spaces expressed by means of ontological graphs. Ontological graphs deliver us some additional knowledge which can be useful in classification processes. Symbolic data, in our approach in the form of concepts from ontologies, require special treatment to be used in classifiers based on searching for the numerical mapping functions between the known inputs and the corresponding known outputs. • Encoding symbolic features in decision systems over ontological graphs is shown. • Classifiers based on numerical mapping functions can be used for encoded data. • The idea is based on characteristic functions defined according to data semantics. • Encoding based on fuzzy characteristic function becomes a promising method.

Generalized Fuzzy Shapley Function for Fuzzy Games

A generalized fuzzy Shapley function for fuzzy games is proposed. First, a game with fuzzy characteristic function is introduced. Based on Hukuhara difference, the fuzzy Hukuhara-Shapley function is proposed as a solution concept to this class of fuzzy games. Some of its properties are shown. An equivalent axiomatic characterization of the fuzzy Hukuhara-Shapley function is given. Furthermore, a generalized fuzzy Shapley function for games with fuzzy coalition and fuzzy characteristic function is developed. It is shown that the simplified expression of the generalized fuzzy Shapley function can be regarded as the generalization of the fuzzy Shapley function defined for some particular games with fuzzy coalition and fuzzy characteristic function.

Fuzzy games on augmenting systems with fuzzy characteristic functions

Fuzzy games on augmenting systems with fuzzy characteristic functions

The Shapley function for fuzzy games with fuzzy characteristic functions

In this paper, we research fuzzy games with fuzzy characteristic functions, where the player participation levels are real numbers in [0, 1], and their payoffs are fuzzy numbers. Based on the Hukuhara difference between fuzzy numbers and the calculating formula of the Shapley function, we discuss a general form of fuzzy games with fuzzy characteristic functions, and study the Shapley function for this kind of fuzzy games. When the given fuzzy game is convex, we research some properties of fuzzy games with fuzzy characteristic functions. Furthermore, we research a special kind of fuzzy games with fuzzy characteristic functions, which is can be seen as an extension of fuzzy games with multilinear extension form. Finally, a numerical example is given.

Water Resources Allocation Using a Cooperative Game with Fuzzy Payoffs and Fuzzy Coalitions

In this paper, two fuzzy cooperative games are utilized for modeling equitable and efficient water allocation among water users in both inter-basin and intra-basin water allocation problems. The proposed all-inclusive water allocation approach consists of three main steps, following Sadegh et al. (Water Resour Manage 24(12):2991–2310, 2010). In the first step, an initial water allocation is carried out using an optimization model taking into account an equity criterion. In the second step, the water users form crisp coalitions with fuzzy characteristic functions to increase the total net benefit of the system and also their own benefits. In the methodology used in this step, the water users do not need to have exact information about their payoffs and they can evaluate their payoffs as fuzzy numbers. In the second step, based on the Hukuhara difference of fuzzy numbers, optimum water allocation strategies are determined using a game with fuzzy characteristic function. In the third step, we applied another methodology which considers a class of fuzzy games with fuzzy characteristic functions and also fuzzy coalitions for water allocation. The methodology of this step is on the basis of the Hukuhara difference and the Choquet integral. The usefulness of the mentioned methodologies is studied by applying them to three defined real life scenarios in a case study of water allocation in Iran. The results showed that the proposed methodologies are professionally appropriate to real-world uncertain problems of equitable and economic inter-basin and intra-basin water resources allocations.

An extension of cooperative fuzzy games

In this paper, a generalized form of fuzzy game is proposed that can be seen as an extension of the cooperative fuzzy game. Players in the proposed fuzzy game do not need to know precise information about the payoff value or to form a crisp coalition. Based on the Hukuhara difference, a game with a fuzzy characteristic function is studied, and the explicit form of a Shapley function is shown. Furthermore, we introduce and investigate a new class of fuzzy game with fuzzy coalitions and a fuzzy characteristic function. Based on the Hukuhara difference and the Choquet integral, we give the explicit form of the Shapley value for the new class of fuzzy game. It has been seen that most of the properties of a crisp game hold well in the proposed fuzzy game. A practical application of the proposed model in joint production problem is also provided.

A Note on Prototypes, Convexity and Fuzzy Sets

The work on prototypes in ontologies pioneered by Rosch [10] and elaborated by Lakoff [8] and Freund [3] is related to vagueness in the sense that the more remote an instance is from a prototype the fewer people agree that it is an example of that prototype. An intuitive example is the prototypical “mother”, and it is observed that more specific instances like ”single mother”, “adoptive mother”, “surrogate mother”, etc., are less and less likely to be classified as “mothers” by experimental subjects. From a different direction Gardenfors [4] provided a persuasive account of natural predicates to resolve paradoxes of induction like Goodman’s “Grue” predicate [5]. Gardenfors proposed that “quality dimensions” arising from human cognition and perception impose topologies on concepts such that the ones that appear “natural” to us are convex in these topologies. We show that these two cognitive principles — prototypes and predicate convexity — are equivalent to unimodal (convex) fuzzy characteristic functions for sets. Then we examine the case when the fuzzy set characteristic function is not convex, in particular when it is multi-modal. We argue that this is an indication that the fuzzy concept should really be regarded as a super concept in which the decomposed components are subconcepts in an ontological taxonomy.

The measure of interaction among players in games with fuzzy coalitions

In this paper, the derivative among participation levels of players in games with fuzzy coalitions, which measures the simultaneous interaction among players with certain participation levels, is defined. Some properties of the derivative of fuzzy game with k-monotonicity are studied. Furthermore, the interaction index among participation levels is established. Based on the assumption that fuzzy characteristic functions are continuous, the integral index of (absolute) interaction among levels is proposed and the integral index of (absolute) interaction among players is defined as well. At last, the Shapley total integral index of interaction among players is constructed, and a series of properties relative to the index are proved.

UNCERTAIN SPATIO-TEMPORAL REASONING FOR DISTRIBUTED TRANSPORTATION SCHEDULING PROBLEM

Distributed artificial intelligence (DAI) is suitable for applications where there is no central control. One of these applications with which we are concerned is transportation scheduling. We noticed that all the approaches dedicated to this application use a weak representation of time and a simple reasoning. Furthermore, these approaches ignore the uncertain behavior of agents. What we propose is an approach based on fuzzy temporal characteristic functions (FTCF), which allows a powerful representation of agent company behaviors informing us at each time about the degree that the agent is available. Thanks to this representation, we develop a spatio-temporal reasoning allowing a cooperation inter-and intra-companies to allocate trucks and delegate orders. First, we use an offline scheduling algorithm that takes charge of new transportation tasks only when trucks are at a destination. To increase the performance of the system, we need to introduce the ability to take charge of the new transportation task immediately and to delegate it to a truck while this latter is moving toward a destination. For this purpose, the system needs to address two issues: (1) determining the current location of the truck and its proximity to the departure and arrival of the new transportation task and (2) respecting the temporal constraints. For this purpose, we use a spatio-temporal reasoning allowing us to deal with the first issue using a spatial reasoning and then addressing the second issue using a temporal reasoning. This approach is a first step towards a design of a temporally situated multi-agent system that allows us to take location and the time at which agents are at this location to determine the suitable actions.

Cooperative games with fuzzy coalitions and fuzzy characteristic functions

In this paper, an extension of a cooperative fuzzy game is proposed, which admits the representation of the rate of participation of every player in a coalition and also associates fuzziness with the value of the game. Games, subject to fuzzy coalitions as well as those pertaining to fuzzy characteristic functions or vague expectations are separately studied in the literature. We propose an extension of a fuzzy game with fuzzy coalitions and vague expectations together and obtain some interesting properties. It has been observed that most of the properties satisfied by a crisp game hold good in the fuzzy sense in this extension. A practical application of the proposed model in Investment Theory is being provided. Further, a Shapley function in the fuzzy sense has been proposed as a solution concept to this class of games. The notion of a fuzzy population monotonic allocation function (FPMAF) is defined and established that the proposed Shapley function is an FPMAF also.

Knowledge engineering for a document retrieval system

A document retrieval system based on fuzzy set theory is described in this paper. Weights, as fuzzy characteristic functions, can be assigned to descriptors in the query expression and to index terms in the document description. Fuzzy set theory allows one to calculate a relevance value for each document from weights assigned to documents and queries. The relevance value for each document is calculated following different models. An elicitation mechanism is used to generate and to enhance the thesaurus structure. The thesaurus guides the retrieval operations.

Applications of fuzzy sets in curve fitting

This paper presents several methods for functional approximation with variable-knot variable-degree splines, with variable-knot first order splines, which are relatively easy to find, as the intermediate input. By means of a fuzzy characteristic function determining how “sharp” the angle at each knot is (the higher the polynomial degree, the “sharper” its first order spline approximation bends), we can decide whether we can group certain adjacent segments together to be approximated by a single higher order polynomial segment. Some simulation experiments have also been done.