This article investigates group decision-making (GDM) problems, where the decision makers' (DMs) preference information is represented by incomplete interval-valued intuitionistic fuzzy preference relations (IVIFPRs). First, a multiplicative consistency property and an acceptably multiplicative consistency property for IVIFPRs are offered. Then, an optimization model to estimate the missing values in an incomplete IVIFPR is constructed. Subsequently, two optimization models are, respectively, established to derive a perfectly consistent IVIFPR and an acceptably consistent IVIFPR from a given inconsistent IVIFPR. Furthermore, a model is offered to gain the DMs' weights. Afterward, the consensus index is defined. When the consensus for IVIFPRs is unacceptable, a model is presented to reach the consensus requirement. Moreover, a novel GDM method for incomplete IVIFPRs is presented. Finally, the presented method is applied to an illustrative example that shows the feasibility of the offered method.
Software cost estimation techniques predict the amount of effort required to develop a software system. Cost estimates are needed throughout the software lifecycle to determine feasibility of software projects and to provide for appropriate allocation or reallocation of available resources. To assess the effect of imprecise evaluations, a comprehensive sensitivity analysis was performed on a major cost estimation model, COCOMO II. Results of this analysis are described and explicated in this paper. To reduce risk of drawing biased conclusions, three different methods for sensitivity analysis were employed: the mathematical analysis of the estimating equation, Monte Carlo simulation, and error propagation. The results of the first two methods are very consistent and confirm expected highest sensitivity of the model to the imprecision of the size estimate. Error propagation allows determination of the combined impact of imprecision in multiple inputs and it is therefore most valuable from the practical point of view. The results obtained by this technique also indicate very strong sensitivity to the imprecision in size estimates. A possible way to cope with imprecise information in software cost estimation is also indicated.
The learning abilities and high transparency are the two important and highly desirable features of any model of software quality. The transparency and user-centricity of quantitative models of software engineering are of paramount relevancy as they help us gain a better and more comprehensive insight into the revealed relationships characteristic to software quality and software processes. In this study, we are concerned with logic-driven architectures of logic models based on fuzzy multiplexers (fMUXs). Those constructs exhibit a clear and modular topology whose interpretation gives rise to a collection of straightforward logic expressions. The design of the logic models is based on the genetic optimization and genetic algorithms, in particular. Through the prudent usage of this optimization framework, we address the issues of structural and parametric optimization of the logic models. Experimental studies exploit software data that relates software metrics (measures) to the number of modifications made to software modules.
The results of different fuzzy clustering algorithms are dealt with collectively in a formal framework of probabilistic set theory in order to interpret the structure of data. Special attention is paid to calculation of entropy of the fuzzy clusters detected by various grouping methods. Two numerical examples illustrate applicability of the proposed way of cluster evaluation.
The paper introduces an idea of logical filtering viewed as a new tool for solving fuzzy relational equations. Considering the panoply of the existing methods, the proposed approach can be classified as a semi-analytic method in the sense it departures from the individual analytical solutions to the individual equations in the system and combines them through an optimization process of logical filtering (masking). Several types of filtering are studied and provided with the detailed learning schemes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
In this paper, we develop a comprehensive conceptual and algorithmic framework to cope with a problem of clustering homogeneous information granules. While there have been several approaches to coping with granular (viz. non-numeric) data, the origin of granular data themselves considered there is somewhat unclear and, as a consequence, the results of clustering start lacking some full-fledged interpretation. In this paper, we offer a holistic view at clustering information granules and an evaluation of the results of clustering. We start with a process of forming information granules with the use of the principle of justifiable granularity (PJG). With this regard, we discuss a number of parameters used in this development of information granules as well as quantify the quality of the granules produced in this manner. In the sequel, Fuzzy C -Means is applied to cluster the derived information granules, which are represented in a parametric manner and associated with weights resulting from the usage of the PJG. The quality of clustering results is evaluated through the use of the reconstruction criterion (quantifying the concept of information granulation and degranulation). A suite of experiments using synthetic and publicly available datasets is reported to quantify the performance of the proposed approach and highlight its key features.
In this paper, we elaborate on a new design approach to the development and analysis of granular input spaces and ensuing granular modeling. Given a numeric model (no matter what specific design methodology has been used to construct it and what architecture has been adopted), we form a granular input space through allocating a certain level of information granularity across the input variables. The formation of granular input space helps us gain a better insight into the ranking of input variables with respect to their precision (the variables with a lower level of information granularity need to be specified in a precise way when estimating the inputs). As a consequence, for granular inputs, the outputs of the granular model are also information granules (say, intervals, fuzzy sets, rough sets, etc.). It is shown that the process of forming granular input space can be sought as an optimization of allocation of information granularity across the input variables so that the specificity of the corresponding granular outputs of the granular model becomes the highest while coverage of data becomes maximized. The construction of granular input space dwells upon two fundamental principles of granular computing-the principle of justifiable granularity and the optimal allocation of information granularity. The quality of the granular input space is quantified in terms of the two conflicting criteria, that is, the specificity of the results produced by the granular model and the coverage of experimental data delivered by this model. In the ensuing optimization problem, one maximizes a product of specificity and coverage. Differential evolution is engaged in this optimization task. The experimental studies involve both synthetic dataset and data coming from the machine learning repository.
The identification of fuzzy rule-based systems is considered. By their nature, these fuzzy models are geared toward capturing relationships between information granules — fuzzy sets. The level of granularity of fuzzy sets helps establish a required level of detail that is of interest in the given modelling environment. The form of the information granules themselves (in particular their distribution and type of membership functions) becomes an important design feature of the fuzzy model, contributing to its structural as well as parametric optimisation. This, in turn, calls for a comprehensive and efficient framework of information (data) granulation, and the one introduced in the study involves a hard C-means (HCM) clustering method and genetic algorithms (GAs). HCM produces an initial collection of information granules (clusters) that are afterwards refined in a parametric way with the aid of a genetic algorithm. The rules of the fuzzy model assume the form `if x1 is A and x2 is B and · · · and xn is W then y=phis(x1 , x2 ,…, xn , param) and come in two forms: a simplified one that involves conclusions that are fixed numeric values (that is, phis is a constant function), and a linear one where the conclusion part (phis) is viewed as a linear function of inputs. The parameters of the rules are optimised through a standard method of linear regression (least square error method). An aggregate objective function with weighting factor used in this study helps maintain a balance between the performance of the model for training and testing data. The proposed identification framework is illustrated with the use of two representative numerical examples.
Fuzzy cognitive maps (FCMs) are a convenient tool for modeling of dynamic systems by means of concepts connected by cause-effect relationships. The FCM models can be developed either manually (by the experts) or using an automated learning method (from data). Some of the methods from the latter group, including recently proposed Nonlinear Hebbian Learning (NHL) algorithm, use Hebbian law and a set of conditions imposed on output concepts. In this paper, we propose a novel approach named data-driven NHL (DD-NHL) that extends NHL method by using historical data of the input concepts to provide improved quality of the learned FCMs. DD-NHL is tested on both synthetic and real-life data, and the experiments show that if historical data are available, then the proposed method produces better FCM models when compared with those formed by the generic NHL method.
The study is devoted to linguistic data mining, an endeavor that exploits the concepts, constructs, and mechanisms of fuzzy set theory. The roles of information granules, information granulation, and the techniques therein are discussed in detail. Particular attention is given to the manner in which these information granules are represented as fuzzy sets and manipulated according to the main mechanisms of fuzzy sets. We introduce unsupervised learning (clustering) where optimization is supported by the linguistic granules of context, thereby giving rise to so-called context-sensitive fuzzy clustering. The combination of neuro, evolutionary, and granular computing in the context of data mining is explored. Detailed numerical experiments using well-known datasets are also included and analyzed.
