We review the existing measures of uncertainty (entropy) for Atanassov's intuitionistic fuzzy sets (AIFSs). We demonstrate that the existing measures of uncertainty for AIFS cannot capture all facets of uncertainty associated with an AIFS. We point out and justify that there are at least two facets of uncertainty of an AIFS, one of which is related to fuzziness while the other is related to lack of knowledge or non-specificity. For each facet of uncertainty, we propose a separate set of axioms. Then for each of fuzziness and non-specificity we propose a generating family (class) of measures. Each family is illustrated with several examples. In this context we prove several interesting results about the measures of uncertainty. We prove some results that help us to construct new measures of uncertainty of both kinds.
In practice, many practical problems occur in uncertain environments, especially in situations that involve human subjective evaluation such as that in the analytic hierarchy process (AHP). This paper presents a practical multi-criteria group decision-making method for decision making under uncertainty. To handle the randomness and fuzziness of individual judgments, the normal Cloud model, group decision-making technique, and the Delphi feedback method are adopted. In the proposed Cloud Delphi hierarchical analysis (CDHA), experts are asked to express their judgments using interval numbers. Individual fuzziness and randomness are then mined from the interval-value comparison matrices. Subsequently, the interval-value pairwise comparison matrices are converted into the corresponding Cloud matrices, and the one-iteration Delphi process is executed to diminish individual judgment mistakes. The individual Cloud weight vectors are calculated using the geometric mean technique and are finally weighted to form the group Cloud weight vector. A simple case study that involved reproducing the relative area sizes of six provinces in China shows that the CDHA method can effectively reduce mistakes and improve decision makers' judgments in situations that require subjective expertise and judgmental inputs. In addition, a practical decision-making problem in which houses are ranked by home buyers shows that the proposed method is effective when applied to complex, large, multidisciplinary problems with considerable uncertainties.
This paper investigates uncertainties in complex supply chain situations and proposes a fuzzy-based decision support model for determining the chance of meeting on-time delivery in a complex supply chain environment. It integrates fuzzy logic principles and unitary structure-based supply chain model and enables addressing uncertainties associated with key inputs of on-time delivery performance for effective decision making process. The proposed pragmatic model deals with the fuzziness of the key inputs including, variations in demand forecasting, materials shortages and distribution lead time, and combines a fuzzy reasoning approach for monitoring on-time delivery of finished products. In systematically dealing with the uncertainties of complex supply chains, this model supports the minimizing of business losses that result from penalties and customer dissatisfaction, and the consequent reduced market share. Application of the proposed model is illustrated using a textile industry case study.
The selection of the optimal ensembles of classifiers inmultiple-classifier selection technique is un-decidable in many cases and it is potentially subjected to a trialand- error search. This paper introduces a quantitative metalearning approach based on neural network and rough set theory in the selection of the best predictive model. This approach depends directly on the characteristic, metafeatures of the input data sets. The employed meta-features are the degree of discreteness and the distribution of the features in the input data set, the fuzziness of these features related to the target class labels and finally the correlation and covariance between the different features. The experimental work that consider these criteria are applied on twenty nine data sets using different classification techniques including support vector machine, decision tables and Bayesian believe model. The measures of these criteria and the best result classification technique are used to build ameta data set. The role of the neural network is to perform a black-box prediction of the optimal, best fitting, classification technique. The role of the rough set theory is the generation of the decision rules that controls this prediction approach. Finally, formal concept analysis is applied for the visualization of the generated rules.
We focus on the Gibbs free energy ΔG for nucleating a droplet of the stable phase (e.g., solid) inside the metastable parent phase (e.g., liquid), close to the first-order transition temperature. This quantity is central to the theory of homogeneous nucleation, since it superintends the nucleation rate. We recently introduced a field theory describing the dependence of ΔG on the droplet volume V, taking into account besides the microscopic fuzziness of the droplet-parent interface, also small fluctuations around the spherical shape whose effect, assuming isotropy, was found to be a characteristic logarithmic term. Here we extend this theory, introducing the effect of anisotropy in the surface tension, and show that in the limit of strong anisotropy ΔG(V) once more develops a term logarithmic on V, now with a prefactor of opposite sign with respect to the isotropic case. Based on this result, we argue that the geometrical shape that large solid nuclei mostly prefer could be inferred from the prefactor of the logarithmic term in the droplet free energy, as determined from the optimization of its near-coexistence profile.
The occurrence of imprecision in the real world is inevitable due to some unexpected situations. The imprecision is often involved in any engineering design process. The imprecision and uncertainty are often interpreted as fuzziness. Fuzzy systems have an essential role in the uncertainty modelling, which can formulate the uncertainty in the actual environment. In this paper, a new approach is proposed to solve a system of fuzzy polynomial equations based on the Gr?bner basis. In this approach, first, the h-cut of a system of fuzzy polynomial equations is computed, and a parametric form for the fuzzy system with respect to the parameter of h is obtained. Then, a Gr?bner basis is computed for the ideal generated by the h-cuts of the system with respect to the lexicographical order using Faugère's algorithm, i.e., F _4 algorithm. The Gr?bner basis of the system has an upper triangular structure. Therefore, the system can be solved using the forward substitution. Hence, all the solutions of the system of fuzzy polynomial equations can easily be obtained. Finally, the proposed approach is compared with the current numerical methods. Some theorems together with some numerical examples and applications are presented to show the efficiency of our method with respect to the other methods.
This paper discusses a new approach to segment different types of skin cancers using fuzzy logic approach. The traditional skin cancer segmentation involves the analysis of image features to delineate the cancerous region from the normal skin. Using low level features such as colour and intensity, segmentation can be done by obtaining a threshold level to separate the two regions. Methods like Otsu optimisation provide a quick and simple process to optimise such threshold level; however this process is prone to the lighting and skin tone variations. Fuzzy clustering algorithm has also been widely used in image processing due to its ability to model the fuzziness of human visual perception. Classical fuzzy C means (FCM) clustering algorithm has been applied to image segmentation with good results; however, the classical FCM is based on type-1 fuzzy sets and is unable to handle uncertainties in the images. In this paper, we proposed an optimum threshold segmentation algorithm based on type-2 fuzzy sets algorithms to delineate the cancerous area from the skin images. By using the 3D colour constancy algorithm, the effect of colour changes and shadows due to skin tone variation in the image can be significantly reduced in the preprocessing stage.We applied the optimum thresholding technique to the preprocessed image over the RGB channels, and combined individual results to achieve the overall skin cancer segmentation. Compared to the Otsu algorithm, the proposed method is less affected by the shadows and skin tone variations. The results also showed more tolerance at the boundary of the cancerous area. Compared with the type-1 FCM algorithm, the proposed method significantly reduced the segmentation error at the normal skin regions.