Solution to fuzzy system of linear equations with crisp coefficients

This paper presents a new and simple method to solve fuzzy real system of linear equations by solving two n × n crisp systems of linear equations. In an original system, the coefficient matrix is considered as real crisp, whereas an unknown variable vector and right hand side vector are considered as fuzzy. The general system is initially solved by adding and subtracting the left and right bounds of the vectors respectively. Then obtained solutions are used to get a final solution of the original system. The proposed method is used to solve five example problems. The results obtained are also compared with the known solutions and found to be in good agreement with them.     

An overlapping cluster algorithm to provide non-exhaustive clustering

The partitioning clustering is a technique to classify n objects into k disjoint clusters, and has been developed for years and widely used in many applications. In this paper, a new overlapping cluster algorithm is defined. It differs from traditional clustering algorithms in three respects. First, the new clustering is overlapping, because clusters are allowed to overlap with one another. Second, the clustering is non-exhaustive, because an object is permitted to belong to no cluster. Third, the goals considered in this research are the maximization of the average number of objects contained in a cluster and the maximization of the distances among cluster centers, while the goals in previous research are the maximization of the similarities of objects in the same clusters and the minimization of the similarities of objects in different clusters. Furthermore, the new clustering is also different from the traditional fuzzy clustering, because the object–cluster relationship in the new clustering is represented by a crisp value rather than that represented by using a fuzzy membership degree. Accordingly, a new overlapping partitioning cluster (OPC) algorithm is proposed to provide overlapping and non-exhaustive clustering of objects. Finally, several simulation and real world data sets are used to evaluate the effectiveness and the efficiency of the OPC algorithm, and the outcomes indicate that the algorithm can generate satisfactory clustering results.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

An alternative formulation for the fuzzy assignment problem

The existing assignment problems for assigning n jobs to n individuals are limited to the considerations of cost or profit measured as crisp. However, in many real applications, costs are not deterministic numbers. This paper develops a procedure based on Data Envelopment Analysis method to solve the assignment problems with fuzzy costs or fuzzy profits for each possible assignment. It aims to obtain the points with maximum membership values for the fuzzy parameters while maximizing the profit or minimizing the assignment cost. In this method, a discrete approach is presented to rank the fuzzy numbers first. Then, corresponding to each fuzzy number, we introduce a crisp number using the efficiency concept. A numerical example is used to illustrate the usefulness of this new method.     

Maximum likelihood estimation from fuzzy data using the EM algorithm

A method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample. This method is based on maximizing the observed-data likelihood defined as the probability of the fuzzy data. It is shown that the EM algorithm may be used for that purpose, which makes it possible to solve a wide range of statistical problems involving fuzzy data. This approach, called the fuzzy EM (FEM) method, is illustrated using three classical problems: normal mean and variance estimation from a fuzzy sample, multiple linear regression with crisp inputs and fuzzy outputs, and univariate finite normal mixture estimation from fuzzy data.     

Overlap functions

In this paper we address a key issue in scenario classification, where classifying concepts show a natural overlapping. In fact, overlapping needs to be evaluated whenever classes are not crisp, in order to be able to check if a certain classification structure fits reality and still can be useful for our declared decision making purposes. In this paper we address an object recognition problem, where the best classification with respect to background is the one with less overlapping between the class object and the class background. In particular, in this paper we present the basic properties that must be fulfilled by overlap functions, associated to the degree of overlapping between two classes. In order to define these overlap functions we take as reference properties like migrativity, homogeneity of order 1 and homogeneity of order 2. Hence we define overlap functions, proposing a construction method and analyzing the conditions ensuring that t-norms are overlap functions. In addition, we present a characterization of migrative and strict overlap functions by means of automorphisms.     

Time-cost trade-off analysis of project networks in fuzzy environments

This paper proposes a novel approach for time-cost trade-off analysis of a project network in fuzzy environments. Different from the results of previous studies, in this paper the membership function of the fuzzy minimum total crash cost is constructed based on Zadeh’s extension principle and fuzzy solutions are provided. A pair of two-level mathematical programs parameterized by possibility level α is formulated to calculate the lower and upper bounds of the fuzzy minimum total crash cost at α. By enumerating different values of α, the membership function of the fuzzy minimum total crash cost is constructed, and the corresponding optimal activity time for each activity is also obtained at the same time. An example of time-cost trade-off problem with several fuzzy parameters is solved successfully to demonstrate the validity of the proposed approach. Since the minimum total crash cost is expressed by a membership function rather than by a crisp value, the fuzziness of parameters is conserved completely, and more information is provided for time-cost trade-off analysis in project management. The proposed approach also can be applied to time-cost trade-off problems with other characteristics.     

A class of multiobjective linear programming model with fuzzy random coefficients

The aim of this paper is to deal with a multiobjective linear programming problem with fuzzy random coefficients. Some crisp equivalent models are presented and a traditional algorithm based on an interactive fuzzy satisfying method is proposed to obtain the decision maker’s satisfying solution. In addition, the technique of fuzzy random simulation is adopted to handle general fuzzy random objective functions and fuzzy random constraints which are usually hard to be converted into their crisp equivalents. Furthermore, combined with the techniques of fuzzy random simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy random multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.     

Numerical solution of algebraic fuzzy equations with crisp variable by Gauss–Newton method

In this paper we introduce an algebraic fuzzy equation of degree n with fuzzy coefficients and crisp variable, and we present an iterative method to find the real roots of such equations, numerically. We present an algorithm to generate a sequence that can be converged to the root of an algebraic fuzzy equation.     

Optimal compromise solution of multi-objective minimal cost flow problems in fuzzy environment

Ghatee and Hashemi [M. Ghatee, S.M. Hashemi, Ranking function-based solutions of fully fuzzified minimal cost flow problem, Inform. Sci. 177 (2007) 4271–4294] transformed the fuzzy linear programming formulation of fully fuzzy minimal cost flow (FFMCF) problems into crisp linear programming formulation and used it to find the fuzzy optimal solution of balanced FFMCF problems. In this paper, it is pointed out that the method for transforming the fuzzy linear programming formulation into crisp linear programming formulation, used by Ghatee and Hashemi, is not appropriate and a new method is proposed to find the fuzzy optimal solution of multi-objective FFMCF problems. The proposed method can also be used to find the fuzzy optimal solution of single-objective FFMCF problems. To show the application of proposed method in real life problems an existing real life FFMCF problem is solved.     

Some duality results on linear programming problems with symmetric fuzzy numbers

Recently, linear programming problems with symmetric fuzzy numbers (LPSFN) have considered by some authors and have proposed a new method for solving these problems without converting to the classical linear programming problem, where the cost coefficients are symmetric fuzzy numbers (see in [4]). Here we extend their results and first prove the optimality theorem and then define the dual problem of LPSFN problem. Furthermore, we give some duality results as a natural extensions of duality results for linear programming problems with crisp data.     

Information dynamics algorithm for detecting communities in networks

The problem of community detection is relevant in many scientific disciplines, from social science to statistical physics. Given the impact of community detection in many areas, such as psychology and social sciences, we have addressed the issue of modifying existing well performing algorithms by incorporating elements of the domain application fields, i.e. domain-inspired. We have focused on a psychology and social network-inspired approach which may be useful for further strengthening the link between social network studies and mathematics of community detection. Here we introduce a community-detection algorithm derived from the van Dongen’s Markov Cluster algorithm (MCL) method [4] by considering networks’ nodes as agents capable to take decisions. In this framework we have introduced a memory factor to mimic a typical human behavior such as the oblivion effect. The method is based on information diffusion and it includes a non-linear processing phase. We test our method on two classical community benchmark and on computer generated networks with known community structure. Our approach has three important features: the capacity of detecting overlapping communities, the capability of identifying communities from an individual point of view and the fine tuning the community detectability with respect to prior knowledge of the data. Finally we discuss how to use a Shannon entropy measure for parameter estimation in complex networks.     

Measuring matrix-based cell formation with alternative routings

A good measure is necessary to evaluate the goodness of grouping in cellular manufacturing. For simple group technology (GT) problems, there are some grouping measures available, the most cited ones are grouping efficiency and grouping efficacy. A grouping measure, which considers alternative routings, is not available for the generalised GT problems. In this paper a new measure called alternative routing grouping (ARG) efficiency is proposed and its desirable properties are derived. Not only can this alternative routing grouping efficiency be used in the generalised GT problems, but it can also be used in the simple GT problems. The computation of the alternative routing grouping efficiency is illustrated with examples, and results of empirical tests on a set of known problems are also reported in this paper. The alternative routing grouping efficiency is extended to the simple GT measures. Grouping efficiency suffers from a weak discriminating capability and grouping efficacy has relatively a better discriminating capability. It is proved that the ARG efficiency has better discriminating capability than the existing measures, and this measure can evaluate the effectiveness of grouping in both simple and generalised cell formation problems with alternative process plans.     

An exact algorithm for the fuzzy p-median problem

In this paper we propose a fuzzy version of the classical p-median problem. We consider a fuzzy set of constraints so that the decision-maker will be able to take into account solutions which provide significantly lower costs by leaving a part of the demand uncovered. We propose an algorithm for solving the problem which is based on Hakimi's works and we compare the crisp and the fuzzy approach by means of an example.     

Measuring performances of multiple-channel queueing systems with imprecise data: a membership function approach

This paper proposes a mixed integer nonlinear programming (MINLP) approach to measure the system performances of multiple-channel queueing models with imprecise data. The main idea is to transform a multiple-channel queue with imprecise data to a family of conventional crisp multiple-channel queues by applying the α-cut approach in fuzzy theory. On the basis of α-cut representation and the extension principle, two pairs of parametric MINLP are formulated to describe the family of crisp multiple-channel queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure, a real-world case of multiple-channel fuzzy queue is investigated successfully. Since the performance measures are expressed by membership functions rather than by crisp values, the fuzziness of input information is completed conserved. Thus, the results obtained from the proposed approach can represent the system more accurately, and more information is provided for system design in practice.     

The fuzzy core in games with fuzzy coalitions

In this paper, the fuzzy core of games with fuzzy coalition is proposed, which can be regarded as the generalization of crisp core. The fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero. The nonempty condition of the fuzzy core is given based on the fuzzy convexity. Three kinds of special fuzzy cores in games with fuzzy coalition are studied, and the explicit fuzzy core represented by the crisp core is also given. Because the fuzzy Shapley value had been proposed as a kind of solution for the fuzzy games, the relationship between fuzzy core and the fuzzy Shapley function is also shown. Surprisingly, the relationship between fuzzy core and the fuzzy Shapley value does coincide, as in the classical case.     

A genetic algorithm with local search strategy for improved detection of community structure

On the basis of modularity optimization, a genetic algorithm is proposed to detect community structure in networks by defining a local search operator. The local search operator emphasizes two features: one is that the connected nodes in a network should be located in the same community, while the other is “local selection” inspired by the mechanisms of efficient message delivery underlying the small‐world phenomenon. The results of community detection for some classic networks, such as Ucinet and Pajek networks, indicate that our algorithm achieves better community structure than other methodologies based on modularity optimization, such as the algorithms based on betweenness analysis, simulated annealing, or Tasgin and Bingol's genetic algorithm. © 2009 Wiley Periodicals, Inc. Complexity, 2010     

The fuzzy p-median problem: A global analysis of the solutions

We apply fuzzy techniques to incorporate external data into p-median problems. So we can detect certain solutions that would be discarded by usual crisp and fuzzy algorithms but that contrasted with this additional information can be advantageous. This usually reveals a pathology of the model and hence our methods provide some fuzzy validation criteria for p-median models.     

Parametric regression analysis of imprecise and uncertain data in the fuzzy belief function framework

In this paper, parametric regression analyses including both linear and nonlinear regressions are investigated in the case of imprecise and uncertain data, represented by a fuzzy belief function. The parameters in both the linear and nonlinear regression models are estimated using the fuzzy evidential EM algorithm, a straightforward fuzzy version of the evidential EM algorithm. The nonlinear regression model is derived by introducing a kernel function into the proposed linear regression model. An unreliable sensor experiment is designed to evaluate the performance of the proposed linear and nonlinear parametric regression methods, called parametric evidential regression (PEVREG) models. The experimental results demonstrate the high prediction accuracy of the PEVREG models in regressions with crisp inputs and a fuzzy belief function as output.     

Frame potential minimization for clustering short time series

The short time series expression miner by Ernst et al. (Bioinformatics 21:i159?Ci168, 2005) assigns time series data to the closest of suitably selected prototypes followed by the selection of significant clusters and eventual grouping. We prove that the proposed dissimilarity measure 1 ? ??, with correlation coefficient ??, can be interpreted as the distance of projected data onto the (d ? 1)-dimensional unit sphere ${\mathcal{S}^{d-1}}$ . The choice of prototypes is closely related to classical problems in optimization theory. Moreover, we propose a new functional, which has a data-driven component and connects the choice of prototypes to the theory of finite unit norm tight frames by Benedetto and Fickus (Adv Comput Math 18:357?C385, 2003).