Nitrate is the primary form of nitrogen in natural waters and it can easily pass through soil to groundwater. Some levels of nitrate concentration in groundwater can cause some health problems such as methemoglobinemia in infants and several cancers. Since geological structures are not homogeneous, investigation of spatial variability of nitrate concentrations in groundwater is characterized by particularly high uncertainties. In this paper, a novel methodology for measure of uncertainty in groundwater nitrate variability is presented. To appraise the fuzziness, which is a type of uncertainty in spatial models, point cumulative semimadogram (PCSM) measure and a metric distance were employed. Measures of fuzziness have been carried out for each location using the experimental and model PCSMs. Finally an uncertainty map, which defines the regional variation of the uncertainty in different categories, has been composed.
Uncertainty measure can supply a new viewpoint for analyzing knowledge conveyed by an Atanassov's intuitionistic fuzzy set (AIFS). So uncertainty measurement is a key topic in AIFS theory, analogous to the role of entropy in probability theory. After reviewing the existing measures of uncertainty (entropy) for AIFSs, we argue that the existing measures of uncertainty cannot capture all facets of uncertainty associated with an AIFS. Then we point out and justify that there are at least three kinds of uncertainty for an AIFS, namely non-specificity, fuzziness, and intuitionism. We provide formal measures of non-specificity, fuzziness, and intuitionism, together with their properties and proofs. Properties of the proposed non-specificity measure are especially investigated. Finally, a general uncertainty measure consisting of these three uncertainties is presented. Illustrative examples show that the proposed uncertainty measure is consistent with intuitive cognize, and it is more sensitive to changes of AIFSs. Moreover, the proposed uncertainty measure can also discriminate uncertainty hiding in classical sets. Thus, it provides an alternative way to construct unified uncertainty measures.
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.
Wireless sensor networks are deployed in complex and uncertain environments, and multiple objectives of routing algorithms are expected to be optimal. However, routing algorithms based on deterministic single objective optimization may not flexibly meet the above needs of applications. This paper adopts fuzzy random optimization and multi-objective optimization, introduces fuzzy random variables to describe both fuzziness and randomness of link delay, link reliability and nodes' residual energy, and proposes a routing model based on fuzzy random expected value and standard deviation model. A hybrid routing algorithm based on fuzzy random multi-objective optimization is designed, which embeds fuzzy random simulation into genetic algorithm with Pareto optimal solution. Simulation results show that the presented algorithm, by adjusting the parameters of fuzzy random variables for depicting both fuzziness and randomness, achieves a longer lifetime and wider performances of delay, latency jitter, reliability, communication interference, energy and balanced energy distribution. Therefore, the presented algorithm can meet different application needs of the cluster head network in the two-tiered wireless sensor networks.
Owing to product proliferation and expanding high customer service provision, firms nowadays face severe competitive challenges in globalisation. The revised process or product design often involves delaying products differentiation until later production stages. Form postponement applying fuzzy theory and mathematical programming is implemented to formulate a multi-period, multistage assembly network model with multi-product under uncertain leadtimes to find a set of optimal alternative for searching optimal differentiation plans. The inherent fuzziness involved in determination of corresponding leadtimes is handled by expected values of fuzzy numbers, and a set of crisp values of leadtimes had executed by centre of gravity defuzzification.
The purpose of this paper is to develop a linear programming methodology for solving multiattribute group decision making problems using intuitionistic fuzzy (IF) sets. In this methodology, IF sets are constructed to capture fuzziness in decision information and decision making process. The group consistency and inconsistency indices are defined on the basis of pairwise comparison preference relations on alternatives given by the decision makers. An IF positive ideal solution (IFPIS) and weights which are unknown a priori are estimated using a new auxiliary linear programming model, which minimizes the group inconsistency index under some constraints. The distances of alternatives from the IFPIS are calculated to determine their ranking order. Moreover, some properties of the auxiliary linear programming model and other generalizations or specializations are discussed in detail. Validity and applicability of the proposed methodology are illustrated with the extended air-fighter selection problem and the doctoral student selection problem.
The Kolmogorov-Sinai (K-S) entropy is used to quantify the average amount of uncertainty of a dynamical system through a sequence of observations. Sequence probabilities therefore play a central role for the computation of the entropy rate to determine if the dynamical system under study is deterministically non-chaotic, deterministically chaotic, or random. This paper extends the notion of the K-S entropy to measure the entropy rate of imprecise systems using sequence membership grades, in which the underlying deterministic paradigm is replaced with the degree of fuzziness. While constructing sequential probabilities for the calculation of the K-S entropy is difficult in practice, the estimate of the K-S entropy in the setting of fuzzy sets in an image is feasible and can be useful for modeling uncertainty of pixel distributions in images. The fuzzy K-S entropy is illustrated as an effective feature for image analysis and texture classification. (C) 2015 Elsevier Ltd. All rights reserved.
In this study, a hybrid land-water-environment (LWE) model is developed for identifying ecological effect and risk under uncertain precipitation in an agroforestry ecosystem. A simulation-based fuzzy-stochastic programming with risk analysis (SFSR) method is used into LWE model to reflect the meteorological impacts; meanwhile, it also can quantify artificial fuzziness (e.g., risk attitude of policymaker) and natural vagueness (e.g., ecological function) in decision-making. The developed LWE model with SFSR method is applied to a practical agroforestry ecosystem in China. Results of optimized planting scale, irrigative water schedule, pollution mitigation scheme, and system benefit under changed rainfall, precise risk-adoption and vague ecological function are obtained; meanwhile their corresponding ecological effects and risks are analyzed. It found that current LWE plans could generate massive water deficits (e.g., 23.22 x 10(6) m(3) in crop irrigation and 26.32 x 10(6) m(3) in forest protection at highest) due to over-cultivation and excessive pollution discharges (e.g., the highest excessive TP and TN discharges would reach 460.64 and 15.30 x 10(3) ton) due to irrational fertilization, which would increase regional ecological risks. In addition, fifteen scenarios associated with withdrawing cultivation and recovering forest based on regional environment heterogeneity (such as soil types) have been discussed to adjust current agriculture-environment policies. It found that, the excessive pollution discharges (TN and TP) could be reduced 12.95% and 18.32% at highest through ecological expansions, which would generate higher system benefits than that without withdrawing farmland and recovering forest. All above can facilitate local policymakers to modulate a comprehensive LWE with more sustainable and robust manners, achieving regional harmony between socio-economy and eco-environment. (c) 2018 Elsevier B.V. All rights reserved.