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Uncertainty aggregation is an important reasoning for making decisions in the real world, which is full of uncertainty. The paper proposes an information source model for aggregating epistemic uncertainties about truth and discusses uncertainty aggregation in the form of possibility distributions. A new combination rule of possibilities for truth is proposed. Then, this paper proceeds to discussion about a traditional but seemingly forgotten representation of uncertainty (i.e., certainty factors (CFs)) and proposes a new interpretation based on possibility theory. CFs have been criticized because of their lack of sound mathematical interpretation from the viewpoint of probability. Thus, this paper first establishes a theory for a sound interpretation using possibility theory. Then it examines aggregation of CFs based on the interpretation and some combination rules of possibility distributions. The paper proposes several combination rules for CFs having sound theoretical basis, one of which is exactly the same as the oft-criticized combination. [ABSTRACT FROM AUTHOR]

Robust optimization is a significant tool to deal with the uncertainty of parameters. However, the robust versions of the mean - variance (MV) model have serious shortcomings. Thus, we propose new robust versions of the MV model and its possibilistic counterpart, based on the Principal Component Analysis. We also derive their analytical solutions when the risk-free asset and short positioning are allowed. In addition, we suggest an eigenvalue approach to manage their conservativeness. After laying down the theoretical points, we illustrate them by using a real data set of six holding stocks trading on the Borsa Istanbul (BIST). We also compare the profitability and performance results of the existing models and the proposed robust models. [ABSTRACT FROM AUTHOR]

• Presents a novel fuzzy DEA based on a general fuzzy measure. • Develops an adjustable and flexible fuzzy DEA model to consider DMUs' preferences. • Applying the adjustable FDEA model for measuring efficiency of hospitals in USA. Possibilistic Data Envelopment Analysis (PDEA) is one of the most applicable and popular approaches in the literature to deal with imprecise and ambiguous data in DEA models. In this approach, with respect to tendency of decision maker (DM) in taking optimistic, pessimistic and compromise attitude, three measures including possibility, necessity and credibility measures are used to form the Fuzzy DEA (FDEA) models, respectively. However, decision makers may have different preference and so it is necessary to customize fuzzy DEA models according to properties of DMUs. This paper proposes a novel fuzzy DEA model based on general fuzzy measure in which the attitude of DMUs could be determined by the optimistic-pessimistic parameters. As a result, the proposed FDEA model is general, applicable, flexible, and adjustable based on each DMUs. A numerical example is used to explain the proposed approach while usefulness and applicability of this approach have been illustrated using a real data set to measure efficiency of 38 hospital in United States. [ABSTRACT FROM AUTHOR]

Possibility theory is applied to introduce and reason about the fundamental notion of a key for uncertain data. Uncertainty is modeled qualitatively by assigning to tuples of data a degree of possibility with which they occur in a relation, and assigning to keys a degree of certainty which says to which tuples the key applies. The associated implication problem is characterized axiomatically and algorithmically. Using extremal combinatorics, we then characterize the families of non-redundant possibilistic keys that attain maximum cardinality. In addition, we show how to compute for any given set of possibilistic keys a possibilistic Armstrong relation, that is, a possibilistic relation that satisfies every key in the given set and violates every possibilistic key not implied by the given set. We also establish an algorithm for the discovery of all possibilistic keys that are satisfied by a given possibilistic relation. It is shown that the computational complexity of computing possibilistic Armstrong relations is precisely exponential in the input, and the decision variant of the discovery problem is NP-complete as well as W[2]-complete in the size of the possibilistic key. Further applications of possibilistic keys in constraint maintenance, data cleaning, and query processing are illustrated by examples. The computation of possibilistic Armstrong relations and discovery of possibilistic keys from possibilistic relations have been implemented as prototypes. Extensive experiments with these prototypes provide insight into the size of possibilistic Armstrong relations and the time to compute them, as well as the time it takes to compute a cover of the possibilistic keys that hold on a possibilistic relation, and the time it takes to remove any redundant possibilistic keys from this cover. [ABSTRACT FROM AUTHOR]

In decision problems involving two dimensions (like several agents in uncertainty) the properties of expected utility ensure that the result of a two-stepped procedure evaluation does not depend on the order with which the aggregations of local evaluations are performed (e.g., agents first, uncertainty next, or the converse). We say that the aggregations on each dimension commute. In a previous conference paper, Ben Amor, Essghaier and Fargier have shown that this property holds when using pessimistic possibilistic integrals on each dimension, or optimistic ones, while it fails when using a pessimistic possibilistic integral on one dimension and an optimistic one on the other. This paper studies and completely solves this problem when more general Sugeno integrals are used in place of possibilistic integrals, leading to double Sugeno integrals. The results show that there are capacities other than possibility and necessity measures that ensure commutation of Sugeno integrals. Moreover, the relationship between two-dimensional capacities and the commutation property for their projections is investigated. [ABSTRACT FROM AUTHOR]

A novel machine-learning-based framework to evaluate the effect of design parameters affected by epistemic uncertainty on the performance of textile antennas is presented in this letter. In particular, epistemic variations are characterized in the framework of possibility theory, which is combined with Bayesian optimization to accurately and efficiently perform uncertainty quantification. A suitable application example validates the proposed method. [ABSTRACT FROM AUTHOR]

As a potential tool for handling fuzziness, possibility theory has been proposed for decades but is still far beyond mature. The fundamental reason is that we lack a clear understanding of the nature of randomness/fuzziness and then the connotation of probability/possibility. This work presented a clear definition of randomness/fuzziness and an intuitive definition of possibility as reasonable physical interpretation for existing axiomatic definition of possibility. The concepts of random/fuzzy sample spaces were introduced, upon which the axiomatic definition of possibility was properly reformulated in a structure parallel to the axiomatic definition of probability. Possibility update equation as well as possibility operators of disjunction/conjunction is discussed and justified. Though a simple rule of probability/possibility transformation is available, the whole systems of probability/possibility are found to be not coherent yet still comparable. In situations where both kinds of uncertainties are involved, fusion of one kind of uncertainty into another uncertain inference system is workable, which was further illustrated by an application example of recognizing noncooperative target using feature observations of radar cross section. Parallel computing of probability/possibility is also discussed to cope with the intensive computation challenge of practical problems of high dimension and/or with big data. We conclude that probability/possibility systems are complementary methods for handling of random/fuzzy information. [ABSTRACT FROM AUTHOR]

Formal Concept Analysis (FCA) is a field of applied mathematics with its roots in order theory, in particular the theory of complete lattices. It is not only a method for data analysis and knowledge representation, but also a formal formulation for concept formation and learning. Over the past 20 years, FCA has been widely studied. In this paper, the current research progresses and the existing problems of similarity measures in FCA are analyzed. To address the drawbacks of the existing methods, we propose a kind of novel semantic similarity measure for FCA by using Linked Data and WordNet. We aim to develop a method that is fully automatic without requiring predefined domain ontologies and can be used independently of the domain in applications requiring semantic similarity measures in FCA. To realize the semantic similarity estimation for FCA, we firstly extend the similarity assessment methods for resources (or entities) in Linked Data into semantic cases by using WordNet. Furthermore, we propose two kinds of semantic similarity measures (i.e., context-free method and context-aware method) for FCA concepts and concept lattices, respectively. Compared with the existing similarity measure methods in FCA, the proposed approach uses concept of possibility theory to determine lower and upper bounds of similarity intervals. Finally, we evaluate the proposed similarity assessment approaches by applying them to real-worlds datasets. [ABSTRACT FROM AUTHOR]

Abstract The paper investigates the problem of defining a specificity relation in Pyt'ev possibility theory. That branch of possibility theory developed by prof. Yu.P. Pyt'ev in the late 1990s defines possibility as a monotone measure Π which numerical values except 0 and 1 are meaningless. The following information and its consequent implications are meaningful: Π (A) = 0 (A is impossible), or Π (A) > Π (B) (A is more possible than B), or Π (A) = Π (B) (A and B are equally possible). All results obtained with Pyt'ev possibility theory are invariant to any strictly increasing lower semi-continuous transformation of all possibility values with the fixed points: 0 and 1. This fact allows anyone to assign numerical values to possibilities using an individual subjective scale of possibility values. Unlike Zadeh possibility theory based on fuzzy sets, where specificity of possibility distributions is evaluated by means of fuzzy set inclusion and distribution π 1 is more specific than π 2 iff π 1 (⋅) ⩽ π 2 (⋅) pointwise, in Pyt'ev possibility theory such an approach can not be applied if numerical values of π 1 and π 2 are assigned through different subjective scales. A thesis we put forward in this paper states that the specificity relation definition has to be consistent with the possibilistic decision-making approaches. The following is intended to be true: the more specific an underlying possibility and a corresponding possibility distribution are, the narrower (more specific) a set of the optimal decisions is (a decision is optimal if it minimizes the possibility of error). A specificity relation satisfying such a requirement is defined and studied. [ABSTRACT FROM AUTHOR]

Underground engineering has complicated environment and unforeseen factors, so there are generally high risks. The traditional risk assessment method was mainly based on the loss theory of probability theory, which had limitations in practical projects. The essence of risk assessment was quantitative analysis and evaluation of uncertainty. The risk assessment of water inrush in the construction of karst tunnels in underground engineering as an example, and a risk margin model was conducted based on the possibility theory. It was found that under the condition of lack of information and people's cognitive differences, the margin model under the possibility theory was more in line with the actual situation of the project, which could provide quantitative basis for the decision-making and management of the project. [ABSTRACT FROM AUTHOR]

Abstract In this paper we present FTCLogic, a formal first-order logic that can manage fuzzy temporal constraints between variables efficiently. In this logic, the use of explicit temporal axioms is unnecessary, and therefore the deduction mechanism doesn't slow down for this reason. FTCLogic has an immediate precedent: the Extended Fuzzy Temporal Constraint Logic or EFTCL. However, while EFTCL is based on Timed Possibilistic Logic, FTCLogic uses the Possibilistic Logic to formulate an original semantics according to its syntax. In fact, FTCLogic defines both syntax and semantics from a powerful combination of two formalisms: the Possibilistic Logic and the Fuzzy Temporal Constraints Networks. FTCLogic has provided the basis for the creation of a fuzzy temporal PROLOG: PROLogic, which is implemented through Haskell, and which is currently undergoing evaluation. [ABSTRACT FROM AUTHOR]

Possibility Theory is a great tool to deal with the imprecise probability. However, the possibilistic counterpart of the mean -- variance (MV) model has serious shortcomings. Thus, we propose a new possibilistic MV model, which depends on the Principal Components Analysis. The proposed model enables to incorporate subjective judgments into the portfolio selection. In addition, it captures the asymmetry in the return data unlike the MV model. The proposed model is also tractable as the MV model since it can be expressed as a concave quadratic maximization problem. After laying down the theoretical points, we illustrate it by using a real data set of six holding stocks trading on the Borsa Istanbul (BIST). We also compare the profitability and performance results of the proposed model and the MV model. [ABSTRACT FROM AUTHOR]

In this paper, we suggest NPCm, a new Naïve Bayesian-like Possibilistic Classifier for mixed categorical and numerical data. The proposed classifier is based on a bi-module belief estimation as well as the Generalized Minimum-based (G-Min) algorithm which has been recently proposed for the classification of categorical data. Distinctively, in the design of both categorical and numerical belief estimation modules, we make use of a probability-to-possibility transform-based possibilistic approach as a strong alternative to the probabilistic one when dealing with decision-making under uncertainty. Thereafter, we use the G-Min algorithm as an improvement of the minimum algorithm to make decision from possibilistic beliefs. Experimental evaluations on 12 datasets taken from University of California Irvine (UCI) and containing all mixed data, confirm the effectiveness of the proposed new G-Min-based NPCm. Indeed, with the used datasets, the proposed classifier outperforms all the classical Bayesian-like classification methods. Consequently, we prove the efficient use of the bi-module possibilistic estimation approach together with the G-Min algorithm for the classification of mixed categorical and numerical data. [ABSTRACT FROM AUTHOR]

Abstract Safety assessment of oil and gas pipeline containing corrosion defects is one of the significant techniques for ensuring safe operation of the pipeline. In practice, the parameters involved in conventional assessment procedures have the nature of uncertainty (i.e. randomness and fuzziness). The traditional methods deal with the uncertainty through probability method or fuzzy set theory. However, each of them cannot effectively handle the coexistence of both random parameters and fuzzy parameters. In addition, corrosion is time-dependent, which means only if the aggressive environment exists, corrosion defects could gradually propagate until the pipeline failure occurs. With respect to these problems, this paper proposed a new safety assessment method for oil and gas pipeline containing internal corrosion defects. The method can be used to estimate the failure probability bounds of a pipeline with time by introducing corrosion rate into the limit state function of the pipeline, meanwhile, combining probability theory with possibility theory to deal with random and fuzzy uncertainties. To demonstrate the feasibility of the model, an application example was presented. In the case, the failure probability bounds at different service times were calculated. Furthermore, the effects of uncertainty of parameters (i.e. pipe geometry, corrosion defect size, material mechanical properties, and operating pressure) and their mean values on failure probability bounds were discussed. The results show that the pipe wall thickness has the most important impact on corrosion failure probability of the pipe, following by the yield stress of pipe material, the outer diameter of the pipe, corrosion defect depth, ultimate tensile stress of the material, and the defect length. Highlights • A new dynamic safety assessment method is proposed. • Parameter uncertainty and corrosion defect growth are considered in the method. • Probability theory and possibility theory are combined to handle hybrid uncertainty. • Effect of parameter uncertainties on pipe failure was investigated. [ABSTRACT FROM AUTHOR]

Highlights • This study focuses on design and implementation of risk mitigation strategies. • Proposes an integrated approach in selection of risk mitigation strategy. • Interrelationships between risks and cost of mitigation have been studied. • We investigated the optimal budget allocation in strategy selection process. • We offers an effective risk management decision making approach considering ROI. Abstract Trucking industry plays a major role in the transportation of goods across different geographical locations. The operational complexities of the trucking industry lead to various risks. This study focuses on effective design and implementation of risk-mitigation strategies for the trucking industry with consideration for budget restrictions. In this paper, both subjective and objective attributes are considered in the mathematical modeling and thereby tries to capture realism in the strategic decision-making process. The results of the study provide novel insights that relate the impact of risk on the cost of mitigation. Further, the effect of three characteristics, targeted risk level (TRL), implementation cost (IC) of strategy, and the probability of risk occurrence (PO) is shown in designing and developing risk-mitigation strategies. The experimental analysis not only augments theoretical knowledge related to risk management decision-making processes but also contributes to designing and developing a risk-mitigation strategy under economic constraints. From the managerial perspective, the study demonstrates how decision-makers can benefit from an integrated approach to develop a more holistic understanding of risk-management processes. This study also provides guidelines in policy selection considering higher return on investment (ROI). The paper concludes by highlighting the key findings and discussing opportunities for future research. [ABSTRACT FROM AUTHOR]

In intelligent technical multi-sensor systems, information is often at least partly redundant—either by design or inherently due to the dynamic processes of the observed system. If sensors are known to be redundant, (i) information processing can be engineered to be more robust against sensor failures, (ii) failures themselves can be detected more easily, and (iii) computational costs can be reduced. This contribution proposes a metric which quantifies the degree of redundancy between sensors. It is set within the possibility theory. Information coming from sensors in technical and cyber–physical systems are often imprecise, incomplete, biased, or affected by noise. Relations between information of sensors are often only spurious. In short, sensors are not fully reliable. The proposed metric adopts the ability of possibility theory to model incompleteness and imprecision exceptionally well. The focus is on avoiding the detection of spurious redundancy. This article defines redundancy in the context of possibilistic information, specifies requirements towards a redundancy metric, details the information processing, and evaluates the metric qualitatively on information coming from three technical datasets.

Reliability analyses based on probability theory are widely applied in geotechnical engineering, and several analytical or numerical methods have been built upon the concept of failure occurrence. Nevertheless, common geotechnical engineering real-world problems deal with scarce or sparse information where experimental data are not always available to a sufficient extent and quality to infer a reliable probability distribution function. This paper rigorously combines Fuzzy Clustering and Possibility Theory for deriving a data-driven, quantitative, reliability approach, in addition to fully probability-oriented assessments, when useful but heterogeneous sources of information are available. The proposed non-probabilistic approach is mathematically consistent with the failure probability, when ideal random data are considered. Additionally, it provides a robust tool to account for epistemic uncertainties when data are uncertain, scarce, and sparse. The Average Cumulative Function transformation is used to obtain possibility distributions inferred from the fuzzy clustering of an indirect database. Target Reliability Index Values, consistent with the prescribed values provided by Eurocode 0, are established. Moreover, a Degree of Understanding tier system based on the practitioner's local experience is also proposed. The proposed methodology is detailed and discussed for two numerical examples using national-scale databases, highlighting the potential benefits compared to traditional probabilistic approaches. • Novel non-probabilistic and data-driven reliability method. • Average Cumulative Function used as Probability to Possibility transformation. • Possibilistic Reliability Targets consistent with the probabilistic values provided by the Eurocode 0. • Three-tier Degree of Understanding system for considering Local Experience and Subjective Information. • Numerical applications on shallow and pile group foundations performed by using indirect databases. [ABSTRACT FROM AUTHOR]

This paper proposes two mixed models to study a consumer’s optimal saving in the presence of two types of risk: income risk and background risk. In the first model, income risk is represented by a fuzzy number and background risk by a random variable. In the second model, income risk is represented by a random variable and background risk by a fuzzy number. For each model, three notions of precautionary savings are defined as indicators of the extra saving induced by income and background risk on the consumer’s optimal choice. In conclusion, we can characterize the conditions that allow for extra saving relative to optimal saving under certainty, even when a certain component of risk is modelled using fuzzy numbers.

The article describes various methods of assessment of load-bearing capacity (reliability) of a pile as a main element of pile foundation. It is shown that deterministic, probabilistic and possibilistic methods have a number of advantages and limitations. An actual task is to develop new approaches to assessment of foundations load-bearing capacity. The combined method providing the optimal assessment according to the given examples is developed in the article. Some features of the proposed methods are in the probability and possibility theories application to account uncertainty or incompleteness of initial data in quantifying the reliability of a pile. Presented in the article methods for a quantitative assessment of single piles reliability can be used for more complex computational models, including multielement pile foundations and more complex models of soil foundations. These methods have not been applied to piles reliability research so far and the article pre-sented is a pioneering one and has no analogues known to the authors.

The paper presents new and improved numerical methods designed for analysis of mechanical systems with uncertain parameters. Uncertain parameters in mechanical systems are described by means of possibility theory. The main representatives of this theory are fuzzy and interval numbers. Algorithms using interval and fuzzy numbers are incorporated into the analysis. The possibilities and efficiency of suggested methods and algorithms are tested by programs created in the environment of the software package MATLAB.

Uncertainty is at the heart of decision-making processes in most real-world applications. Uncertainty can be broadly categorized into two types: aleatory and epistemic. Aleatory uncertainty describes the variability in the physical system where sensors provide information (hard) of a probabilistic type. Epistemic uncertainty appears when the information is incomplete or vague such as judgments or human expert appreciations in linguistic form. Linguistic information (soft) typically introduces a possibilistic type of uncertainty. This paper is concerned with the problem of classification where the available information, concerning the observed features, may be of a probabilistic nature for some features, and of a possibilistic nature for some others. In this configuration, most encountered studies transform one of the two information types into the other form, and then apply either classical Bayesian-based or possibilistic-based decision-making criteria. In this paper, a new hybrid decision-making scheme is proposed for classification when hard and soft information sources are present. A new Possibilistic Maximum Likelihood (PML) criterion is introduced to improve classification rates compared to a classical approach using only information from hard sources. The proposed PML allows to jointly exploit both probabilistic and possibilistic sources within the same probabilistic decision-making framework, without imposing to convert the possibilistic sources into probabilistic ones, and vice versa.

In this paper, we propose a family of indistinguishability operators, that we have called Yager Possibilitic Response Functions (YPRFs for short), as an appropriate tool for allocating tasks to a collective of agents. In order to select the best agent to carry out each task, we have used the so-called response threshold method, where each agent decides the next task to perform following a probabilistic Markov process and, in addition, involves a response function which models how appropriate the task is for the agent. In previous works, we developed a new response threshold method which incorporates the use of indistinguishability operators as response functions and possibility theory instead of probability, for task allocation from a very general perspective without taking into account the specific characteristics of the agents except their limitations to carry out a task. Such an allocation is modelled by means of possibilistic, instead of probabilisitic, Markov chains. We show that possibilistic Markov chains outperform its probabilistic counterparts for the aforementioned propose. All the indistinguishability operators considered in previous papers were not able to take into account the agents’ restrictions for moving from a task to another one, or equivalently to carry out a task instead of another one. In order to avoid this handicap, we introduce a new kind of response functions, YPRFs, which are modelled by means of indistinguishability operators obtained via Yager t-norms. This new type of response functions drops to zero when an agent, due to its limitations, is not able to execute a task and, therefore, is able to model a generic multi-agent system with restrictions. The performed simulation, under Matlab, allows us to compare the results obtained using the new YPRFs with those obtained applying celebrated response functions also generated via indistinguishability operators (that we call Original Possibilitic Response Functions, OPRFs for short). Moreover, the results confirm that the YPRFs are able to take into account agent’s restrictions while the OPRFs are not able. Finally, in the light of the experimental results, we can confirm that those systems modelled.