Fuzzy sets (membership functions) are numeric constructs. In spite of the underlying semantics of fuzzy sets (which is inherently linked with the higher level of abstraction), the membership grades and processing of fuzzy sets themselves emphasize the numeric facets of all pursuits stressing the numeric nature of membership grades and in this way reducing the interpretability and transparency of results. In this study, we advocate an idea of a granular description of membership functions where instead of numeric membership grades, introduced are more interpretable granular descriptors (say, low, high membership, etc.). Granular descriptors are formalized with the aid of various formal schemes available in Granular Computing, especially sets (intervals), fuzzy sets, and shadowed sets. We formulate a problem of a design of granular descriptors as a certain optimization task, elaborate on the solutions and highlight some areas of applications.
Fuzzy cognitive maps (FCMs) form a class of graph-oriented fuzzy models describing causal relationships among concepts. In this study, we augment these models by introducing their generalization coming in the form of granular FCMs. In contrast with FCMs, in the granular FCMs, the connections between the nodes (states) are described in the form of information granules, especially intervals and fuzzy sets. Key scenarios in which granular models (and granular FCMs) arise are presented in order to offer a compelling rationale behind the formation of such models. In the context of system modeling, we show that information granularity emerges as an important design asset. We discuss detailed schemes of allocation of information granularity and quantify a performance of the resulting granular FCM in terms of a coverage criterion. For illustrative purposes, the detailed studies are completed for granular FCMs with interval-valued connections.
This article is concerned with the development of fuzzy models realized with the aid of genetic programming (GP). The proposed architecture employs GP to form fuzzy logic expressions involving logic operators and information granules (fuzzy sets) located in the input space, used to predict information granules located in the output space. We propose an architecture realizing logic processing, with the structural optimization of the model accomplished by a multitree genetic programming and the parametric optimization completed by gradient-based learning. The granulation of information used in this architecture is developed using the Fuzzy C-means clustering algorithm. The novelty of this study is two-fold: 1) it comes with the flexibility of the logic-oriented structure of fuzzy models, and 2) our architecture is designed to handle high-dimensional data by alleviating the detrimental effect of distance concentration hampering the effectiveness of standard Takagi-Sugeno-Kang fuzzy rule-based models. The article is illustrated through some experiments that provide a detailed insight into the performance of the fuzzy models. A comprehensive comparative analysis is also covered.
In routing problems, the contribution of a variable to fitness often depends on the states of other variables. This phenomenon is referred to as linkage. High linkage level typically makes a routing problem more challenging for an evolutionary algorithm (EA). An entire linkage measure, named entire linkage index (ELI), has been proposed in this paper for such routing problems. Aiming at solving high linkage-based routing problems, we presented a real adjacency matrix-coded evolution algorithm (RAMEA) that is capable of learning and evolving correlation matrix of decision variables. The efficiency of RAMEA was tested on two familiar routing problems: travelling salesman problem (TSP) and generalised travelling salesman problem (GTSP). The experimental results show that the RAMEA is promising for those highly linkage-based routing problems, especially for those of large-scale.
In this paper, we propose a consistency and consensus-based method to solve the group decision-making (GDM) problems within the framework of interval-valued intuitionistic multiplicative preference relations (IVIMPRs). First, we introduce a similarity measure that expresses the similarity between two decision makers (DMs). Then, a similarity-based consensus index is offered to evaluate closeness between individuals' judgments. Based on the consensus index, the concept of acceptable consensus for IVIMPRs is presented. In the sequel, a consistency and consensus improvement model that aims at retaining original opinions of the DMs is introduced to make consistency and consensus of IVIMPRs acceptable. Moreover, the DMs' comprehensive weights are obtained by combining their subjective weights and objective weights. Afterward, to derive priority weights of alternatives, a programming model is established and solved by three approaches considering DMs' different risk attitudes. Finally, a GDM method with IVIMPRs is offered, and its application to select the partners is offered.
Proposed is an idea of partial supervision realized in the form of a neural-network front end to the schemes of unsupervised learning (clustering). This neural network leads to an anisotropic nature of the induced feature space. The anisotropic property of the space provides us with some of its local deformation necessary to properly represent labeled data and enhance efficiency of the mechanisms of clustering to be exploited afterwards. The training of the network is completed based upon available labeled patterns-a referential form of the labeling gives rise to reinforcement learning. It is shown that the discussed approach is universal and can be utilized in conjunction with any clustering method. Experimental studies are concentrated on three main categories of unsupervised learning including FUZZY ISODATA, Kohonen self-organizing maps, and hierarchical clustering.
The objective of this paper is to study facility-location problems in the presence of a hybrid uncertain environment involving both randomness and fuzziness. A two-stage fuzzy-random facility-location model with recourse (FR-FLMR) is developed in which both the demands and costs are assumed to be fuzzy-random variables. The bounds of the optimal objective value of the two-stage FR-FLMR are derived. As, in general, the fuzzy-random parameters of the FR-FLMR can be regarded as continuous fuzzy-random variables with an infinite number of realizations, the computation of the recourse requires solving infinite second-stage programming problems. Owing to this requirement, the recourse function cannot be determined analytically, and, hence, the model cannot benefit from the use of techniques of classical mathematical programming. In order to solve the location problems of this nature, we first develop a technique of fuzzy-random simulation to compute the recourse function. The convergence of such simulation scenarios is discussed. In the sequel, we propose a hybrid mutation-based binary ant-colony optimization (MBACO) approach to the two-stage FR-FLMR, which comprises the fuzzy-random simulation and the simplex algorithm. A numerical experiment illustrates the application of the hybrid MBACO algorithm. The comparison shows that the hybrid MBACO finds better solutions than the one using other discrete metaheuristic algorithms, such as binary particle-swarm optimization, genetic algorithm, and tabu search.
