We expand the classical model of a two-player game to select the best strategies, whose action is expected to maintain the values of a certain variable on the neutral level. By inserting fuzzy sets as payoff values in the game matrix we facilitate the procedure of formulations of payoff expectations by players. Instead of making difficult decisions about the choice of accurate numerical entries of the matrix the players are able to use words, which should simplify a communication between them when designing the preliminaries of the game. The players also have the possibility of making a ranking of their favorite strategies.
We expand the classical model of a two-player game by inserting of fuzzy sets as payoff values in the game matrix. Players can thus formulate their payoff expectations with words instead of deciding on numerical entries of the matrix. In this way we count on the better verbal communication between players when designing the preliminaries of the game. As a final result we expect to obtain samples of the players optimal strategies, which will preserve the profit of the game on the neutral level.
The work is particularly addressed to some beginners in performing operations on continuous fuzzy numbers. We discuss three different approaches to operations on fuzzy numbers to make comparisons of results in the aspect of their advantages and disadvantages. By demonstrating different possibilities of making calculations on fuzzy numbers we intend to help users in the proper selection of such operators that are adapted to their tasks in the most adequate way.
We discuss two computational techniques in the current paper. In the first part, we aim at employing FCM (fuzzy c-means) clustering to compute membership degrees of two clusters providing decisions to perform surgery or not for a testing set of 25 gastric cancer patients. The second part handles mathematical modelling of a common function approximating the information obtained from the c-means procedure. After constructing the equation of the function, we can make the decision about the surgery in the form of the surgery degree for an arbitrary gastric cancer patient. A centre, dealing with mathematical techniques concerning surgery prognoses, can quickly decide about surgery for the patient who lives in a remote place. A transmission of information among the centre and some hospitals, interested in adopting the centre services, can facilitate surgery decision-making. This trial can be treated as a contribution in the telemedicine domain.
We explore the classical model of a two-player game to select the best strategies, where action is expected to maintain the values of a certain variable on the neutral level. By inserting fuzzy sets as payoff values in the game matrix, we facilitate the procedure of formulations of payoff expectations by players. Instead of making inconvenient decisions about the choice of accurate numerical entries of the matrix, the players are able to use words, which should simplify communication between them when designing the preliminaries of the game. The players also have the possibility of making a ranking of their favourite strategies. At the next stage of the play, we involve group decision-making in order to aggregate results coming from several paired games, when more than two players contradict each other.
Strict analytic formulas are the tools usually derived for determining the formal relationships between a sample of independent variables and a variable which they affect. If we cannot formalize the function tying the independent and dependent variables then we will utilize some control actions. Apart from crisp versions of control we often adopt their fuzzy variants developed by Mamdani and Assilian or Sugeno. Fuzzy control algorithms are furnished with softer mechanisms, when comparing them to classical control. The algorithms are particularly adaptable to support medical systems, often handling uncertain premises and conclusions. From the medical point of view it would be desirable to prognosticate the survival length for patients suffering from gastric cancer. We thus formulate the objective of the current paper as the utilization of fuzzy control actions for the purpose of making the survival prognoses.
Fuzzy set theory was presented for the first time by Professor Lotfi A. Zadeh from Berkeley University in 1965. In conventional binary logic a statement can be true or false, and there is no place for even a little uncertainty in this judgment. An element either belongs to a set or does not. We call these kinds of sets crisp sets. In practice we often experience those real situations that are represented by crisp sets as impossible to describe accurately. A two-valued logic assumes that precise symbols must be employed, and it is therefore not applicable to the real existence. If the information demanded by a system is lacking, the future state of such a system may not be known completely. One of the instruments used to handle the vagueness in the real-world situations is fuzzy set theory, which has been frequently applied in a wide range of areas like, e.g., dynamic systems, militaries, medicine and other domains. Another theory, which copes with the problem of imprecision, is known as rough set theory. It was proposed by Professor Zdzisław Pawlak in Warsaw in the 1980ties. Whereas imprecision is expressed in the category of a membership degree in fuzzy set theory, this is a matter of the set approximation in rough set theory. Due to the definition of a rough set formulated by means of the decision attribute value, two approximate sets of the rough set are determined. These contain sure and possible members of the universe considered, in which the rough set has been defined. One of the objectives of this study is to apply some classical methods of fuzzy set theory to medicine in order to estimate the survival length of gastric cancer patients. We have decided to test the action of fuzzy controllers of the Mamdani and Sugeno type. Two clinical markers, playing roles of the independent variables, have been included in the algorithm as the base information assisting the survival prognosis. Since the model results have been convergent to the expected experimental values then we will intend to make some extensions of the model concerning the larger number of independent variables. We have also utilized rough set classification, to verify the types of operations. These items are discussed in the thesis in conformity with the physicians’ wishes to support results of statistical investigations. The current research is funded by the scientific grant obtained from Blekinge Research Board.
As we all know the classical set theory has a deep-rooted influence in the traditional mathematics. According to the two-valued logic, an element can belong to a set or cannot. In the former case, the element’s membership degree will be assigned to one, whereas in the latter case it takes the zero value. With other words, a feeling of imprecision or fuzziness in the two-valued logic does not exist. With the rapid development of science and technology, more and more scientists have gradually come to realize the vital importance of the multi-valued logic. Thus, in 1965, Professor Lotfi A. Zadeh from Berkeley University put forward the concept of a fuzzy set. In less than 60 years, people became more and more familiar with fuzzy set theory. The theory of fuzzy sets has been turned to be a favor applied to many fields. The study aims to apply some classical and extensional methods of fuzzy set theory in life expectancy and treatment prognoses for cancer patients. The research is based on real-life problems encountered in clinical works by physicians. From the introductory items of the fuzzy set theory to the medical applications, a collection of detailed analysis of fuzzy set theory and its extensions are presented in the thesis. Concretely speaking, the Mamdani fuzzy control systems and the Sugeno controller have been applied to predict the survival length of gastric cancer patients. In order to keep the gastric cancer patients, already examined, away from the unnecessary suffering from surgical operation, the fuzzy c-means clustering analysis has been adopted to investigate the possibilities for operation contra to nonoperation. Furthermore, the approach of point set approximation has been adopted to estimate the operation possibilities against to nonoperation for an arbitrary gastric cancer patient. In addition, in the domain of multi-expert decision-making, the probabilistic model, the model of 2-tuple linguistic representations and the hesitant fuzzy linguistic term sets (HFLTS) have been utilized to select the most consensual treatment scheme(s) for two separate prostate cancer patients. The obtained results have supplied the physicians with reliable and helpful information. Therefore, the research work can be seen as the mathematical complements to the physicians’ queries.
Like data analysis, pattern recognition and data mining, fuzzy clustering also has been applied widely, and successful applications have been reported. In this paper we aim to employ the technique of fuzzy c-means (FCM) cluster to prognosticate the operation possibility on gastric cancer patients. Our purpose is to partition some clinical data in two fuzzy clusters. One of them considers patients who have a chance for successful surgery whereas the other cluster contains the patients without a view for surgery. Each patient is given by characteristic biological markers. The initial values of membership degrees taking place in the partition matrix are usually determined randomly. In this work we will use particularly designed membership functions to calculate the degrees of membership.
The chapter is composed of two parts. In the first part we aim at employing fuzzy c-means (FCM) clustering to prognosticate membership degrees pointing out possibilities for operation and none operation for a set of 25 gastric cancer patients characterized by values of decisive biological markers. The second part handles the technique of mathematical modelling of a common membership function approximating the information collected from the given set of patients. When constructing the equation of the function we are able to determine the operation and none operation diagnosis for an arbitrary gastric cancer patient.
Strict analytic formulas are the tools derived for determining the formal relationships between a sample of independent variables and a variable which they affect. If we cannot formalize the function tying the independent and dependent variables then we will utilize fuzzy control actions. The algorithm is particularly adaptable to support the problem of prognosticating the survival length for gastric cancer patients. We thus formulate the objective of the current paper as the utilization of fuzzy control action for the purpose of making the survival prognoses.
Abstract—In this paper, two models, one is called the probabilistic model and the other is known as the model of 2-tuple fuzzy linguistic representations, are applied to solve multi-expert decision making issues (MEDM). A MEDM problem is considered, in which a group of physicians are independently asked about assessing the effectiveness of a set of treatment therapies for a prostate cancer patient. The objective of this paper is to find the most common judgment by means of these two models. Moreover, fuzzy linguistic terms are used to express the experts’ opinions and s-parametric membership functions are designed to depict the fuzzy linguistic terms.
Apart from the probabilistic model and the model of 2-tuple linguistic representations, a new extension of the fuzzy set, known as the hesitant fuzzy linguistic term set can be seen as the third representative of linguistic approaches. In this paper, we focus on multi-expert decision-making problems, in which a group of physicians are independently asked for assessing the effectiveness of a set of treatment therapies. Our goal is to rank the effectiveness of treatment modalities from the most recommended to the contraindicated. Two individual prostate cancer patients have been taken into account in the practical studies. For the first patient, the probabilistic model and the model of 2-tuple linguistic representations have been adopted to accomplish the medical application. Whereas, for the second patient, the approach of hesitant fuzzy linguistic term set has been used to make the medication prognoses. Moreover, the continuous fuzzy numbers in the Left-Right representations are used to mathematically express the experts’ judgments and s-parametric membership functions are designed to represent the fuzzy linguistic terms.