A Possibilistic Approach of the Max-Product Bernstein Kind Operators

By analogy with the probabilistic approach of the classical Bernstein polynomials, in this paper firstly we give the proof for the uniform convergence of the nonlinear max-product Bernstein operator by using the theory of possibility. This new approach, which interprets the max-product Bernstein operator as a possibilistic expectation of a particular fuzzy variable having a possibilistic Bernoulli distribution, does not only offer a good justification for the max-product Bernstein operator, but also allows to extend the method to other discrete max-product Bernstein type operators.     

Sum-based weighted belief base merging: From commensurable to incommensurable framework

Different methods have been proposed for merging multiple and potentially conflicting information. The merging process based on the so-called “Sum” operation offers a natural method for merging commensurable prioritized belief bases. Their popularity is due to the fact that they satisfy the majority property and they adopt a non-cautious attitude in deriving plausible conclusions.This paper analyzes the sum-based merging operator when sources to merge are incommensurable, namely when they do not share the same meaning of uncertainty scales. We first show that the obtained merging operator can be equivalently characterized either in terms of an infinite set of compatible scales, or by a well-known Pareto ordering on a set of propositional logic interpretations. We also study some restrictions on compatible scales based on different commensurability hypothesis.Moreover, this paper provides a postulate-based analysis of our merging operators. We show that when prioritized bases to merge are not commensurable, the majority property is no longer satisfied. We provide conditions to recovering it. We also analyze the fairness postulate, which represents the unique postulate unsatisfied when belief bases to merge are commensurable and we propose a new postulate of consensuality. This postulate states that the result of the merging process must be consensual. It obtains the consent of all parties by integrating a piece of belief of each base.Finally, in the incommensurable case, we show that the fairness and consensuality postulates are satisfied when all compatible scales are considered. However, we provide an impossibility theorem stating that there is no way to satisfy fairness and consensuality postulates if only one canonical compatible scale is considered.     

Qualitative possibilistic influence diagrams based on qualitative possibilistic utilities

This paper proposes a new approach for decision making under uncertainty based on influence diagrams and possibility theory. The so-called qualitative possibilistic influence diagrams extend standard influence diagrams in order to avoid difficulties attached to the specification of both probability distributions relative to chance nodes and utilities relative to value nodes. In fact, generally, it is easier for experts to quantify dependencies between chance nodes qualitatively via possibility distributions and to provide a preferential relation between different consequences. In such a case, the possibility theory offers a suitable modeling framework. Different combinations of the quantification between chance and utility nodes offer several kinds of possibilistic influence diagrams. This paper focuses on qualitative ones and proposes an indirect evaluation method based on their transformation into possibilistic networks. The proposed approach is implemented via a possibilistic influence diagram toolbox (PIDT).     

Linear programming with a possibilistic objective function

This paper proposes some ways for dealing with a linear program when the coefficients of the objective function are subject to possibilistic imprecision, i.e. they are characterized by fuzzy restrictions. Emphasis is placed upon a passive approach that yields a satisfying solution via an appropriate semi-infinite program, and an active one that allows to reach a solution with a high possibility level of optimality. Extensions to the possibilistic constraints case and to the case of multiple-objective programming problems with possibilistic coefficients are also hinted. We end up with some concluding remarks and indicate axes for further developments.     

An axiomatic derivation of the coding-theoretic possibilistic entropy

We re-take the possibilistic (strictly non-probabilistic) model for information sources and information coding put forward in (Fuzzy Sets and Systems 132–1 (2002) 11–32); the coding-theoretic possibilistic entropy is defined there as the asymptotic rate of compression codes, which are optimal with respect to a possibilistic (not probabilistic) criterion. By proving a uniqueness theorem, in this paper we provide also an axiomatic derivation for such a possibilistic entropy, and so are able to support its use as an adequate measure of non-specificity, or rather of “possibilistic ignorance”, as we shall prefer to say. We compare our possibilistic entropy with two well-known measures of non-specificity: Hartley measure as found in set theory and U-uncertainty as found in possibility theory. The comparison allows us to show that the latter possesses also a coding-theoretic meaning.     

Possibilistic nested logic programs and strong equivalence

In this paper, the class of possibilistic nested logic programs is introduced. These possibilistic logic programs allow us to use nested expressions in the bodies and heads of their rules. By considering a possibilistic nested logic program as a possibilistic theory, a construction of a possibilistic logic programing semantics based on answer sets for nested logic programs and the proof theory of possibilistic logic is defined. In order to define a general method for computing the possibilistic answer sets of a possibilistic nested program, the idea of equivalence between possibilistic nested programs is explored. By considering properties of equivalence between possibilistic programs, a process of transforming a possibilistic nested logic program into a possibilistic disjunctive logic program is defined. Given that our approach is an extension of answer set programming, we also explore the concept of strong equivalence between possibilistic nested logic programs. To this end, we introduce the concept of poss SE-models. Therefore, we show that two possibilistic nested logic programs are strong equivalents whenever they have the same poss SE-models.The expressiveness of the possibilistic nested logic programs is illustrated by a scenario from the medical domain. In particular, we exemplify how possibilistic nested logic programs are expressive enough for capturing medical guidelines which are pervaded by vagueness and qualitative information.     

An informational distance for estimating the faithfulness of a possibility distribution,viewed as a family of probability distributions,with respect to data

An acknowledged interpretation of possibility distributions in quantitative possibility theory is in terms of families of probabilities that are upper and lower bounded by the associated possibility and necessity measures. This paper proposes an informational distance function for possibility distributions that agrees with the above-mentioned view of possibility theory in the continuous and in the discrete cases. Especially, we show that, given a set of data following a probability distribution, the optimal possibility distribution with respect to our informational distance is the distribution obtained as the result of the probability-possibility transformation that agrees with the maximal specificity principle. It is also shown that when the optimal distribution is not available due to representation bias, maximizing this possibilistic informational distance provides more faithful results than approximating the probability distribution and then applying the probability-possibility transformation. We show that maximizing the possibilistic informational distance is equivalent to minimizing the squared distance to the unknown optimal possibility distribution. Two advantages of the proposed informational distance function is that (i) it does not require the knowledge of the shape of the probability distribution that underlies the data, and (ii) it amounts to sum up the elementary terms corresponding to the informational distance between the considered possibility distribution and each piece of data. We detail the particular case of triangular and trapezoidal possibility distributions and we show that any unimodal unknown probability distribution can be faithfully upper approximated by a triangular distribution obtained by optimizing the possibilistic informational distance.     

Using possibilistic logic for modeling qualitative decision: Answer Set Programming algorithms

A qualitative approach to decision making under uncertainty has been proposed in the setting of possibility theory, which is based on the assumption that levels of certainty and levels of priority (for expressing preferences) are commensurate. In this setting, pessimistic and optimistic decision criteria have been formally justified. This approach has been transposed into possibilistic logic in which the available knowledge is described by formulas which are more or less certainly true and the goals are described in a separate prioritized base. This paper adapts the possibilistic logic handling of qualitative decision making under uncertainty in the Answer Set Programming (ASP) setting. We show how weighted beliefs and prioritized preferences belonging to two separate knowledge bases can be handled in ASP by modeling qualitative decision making in terms of abductive logic programming where (uncertain) knowledge about the world and prioritized preferences are encoded as possibilistic definite logic programs and possibilistic literals respectively. We provide ASP-based and possibilistic ASP-based algorithms for calculating optimal decisions and utility values according to the possibilistic decision criteria. We describe a prototype implementing the algorithms proposed on top of different ASP solvers and we discuss the complexity of the different implementations.     

Strong guidance on mitigating the effects of uncertainties in the brass casting blending problem: a hybrid optimization approach

The scrap charge optimization problem in the brass casting process is a critical management concern that aims to reduce the charge while preventing specification violations. Uncertainties in scrap material compositions often cause violations in product standards. In this study, we have discussed the aleatory and epistemic uncertainties and modelled them by using probability and possibility distributions, respectively. Mathematical models including probabilistic and possibilistic parameters are generally solved by transforming one type of parameter into the other. However, the transformation processes have some handicaps such as knowledge losses or virtual information production. In this paper, we have proposed a new solution approach that needs no transformation process and so eliminates these handicaps. The proposed approach combines both chance-constrained stochastic programming and possibilistic programming. The solution of the numerical example has shown that the blending problem including probabilistic and possibilistic uncertainties can be successfully handled and solved by the proposed approach.     

On the uniqueness of possibilistic measure of uncertainty and information

It is demonstrated, through a series of theorems, that the U-uncertainty (introduced by Higashi and Klir in 1982) is the only possibilistic measure of uncertainty and information that satisfies possibilistic counterparts of axioms of the well established Shannon and hartley measures of uncertainty and information. Two complementary forms of the possibilistic counterparts of the probabilistic branching (or grouping) axiom, which is usually used in proofs of the uniqueness of the Shannon measure, are introduced in this paper for the first time. A one-to-one correspondence between possibility distributions and basic probabilistic assignments (introduced by Shafer in his mathematical theory of evidence) is instrumental in most proofs in this paper. The uniqueness proof is based on possibilistic formulations of axioms of symmetry, expansibility, additivity, branching, monotonicity, and normalization.     

Fuzzy data analysis by possibilistic linear models

Since fuzzy data can be regarded as distribution of possibility, fuzzy data analysis by possibilistic linear models is proposed in this paper. Possibilistic linear systems are defined by the extension principle. Fuzzy parameter estimations are discussed in possibilistic linear systems and possibilistic linear models are employed for fuzzy data analysis with non-fuzzy inputs and fuzzy outputs defined by fuzzy numbers. The estimated possibilistic linear system can be obtained by solving a linear programming problem. This approach can be regarded as fuzzy interval analysis.     

On the transformation between possibilistic logic bases and possibilistic causal networks

Possibilistic logic bases and possibilistic graphs are two different frameworks of interest for representing knowledge. The former ranks the pieces of knowledge (expressed by logical formulas) according to their level of certainty, while the latter exhibits relationships between variables. The two types of representation are semantically equivalent when they lead to the same possibility distribution (which rank-orders the possible interpretations). A possibility distribution can be decomposed using a chain rule which may be based on two different kinds of conditioning that exist in possibility theory (one based on the product in a numerical setting, one based on the minimum operation in a qualitative setting). These two types of conditioning induce two kinds of possibilistic graphs. This article deals with the links between the logical and the graphical frameworks in both numerical and quantitative settings. In both cases, a translation of these graphs into possibilistic bases is provided. The converse translation from a possibilistic knowledge base into a min-based graph is also described.     

Ample fields as a basis for possibilistic processes

Ample fields play an important role in possibility theory. These fields of subsets of a universe, which are additionally closed under arbitrary unions, act as the natural domains for possibility measures. A set provided with an ample field is then called an ample space. In this paper we generalise Wang's notions of product ample field and product ample space. We make a topological study of ample spaces and their products, and introduce ample subspaces, extensions and one-point extensions of ample spaces. In this way, a first step towards a mathematical theory of possibilistic processes is made.     

Group decision making with expertons and uncertain generalized probabilistic weighted aggregation operators

Expertons and uncertain aggregation operators are tools for dealing with imprecise information that can be assessed with interval numbers. This paper introduces the uncertain generalized probabilistic weighted averaging (UGPWA) operator. It is an aggregation operator that unifies the probability and the weighted average in the same formulation considering the degree of importance that each concept has in the aggregation. Moreover, it is able to assess uncertain environments that cannot be assessed with exact numbers but it is possible to use interval numbers. Thus, we can analyze imprecise information considering the minimum and the maximum result that may occur. Further extensions to this approach are presented including the quasi-arithmetic uncertain probabilistic weighted averaging operator and the uncertain generalized probabilistic weighted moving average. We analyze the applicability of this new approach in a group decision making problem by using the theory of expertons in strategic management.     

Possibilistic networks for information retrieval

This paper proposes an information retrieval (IR) model based on possibilistic directed networks. The relevance of a document w.r.t a query is interpreted by two degrees: the necessity and the possibility. The necessity degree evaluates the extent to which a given document is relevant to a query, whereas the possibility degree evaluates the reasons of eliminating irrelevant documents. This new interpretation of relevance led us to revisit the term weighting scheme by explicitly distinguishing between informative and non-informative terms in a document. Experiments carried out on three standard TREC collections show the effectiveness of the model.     

Possibilistic sequential decision making

When the information about uncertainty cannot be quantified in a simple, probabilistic way, the topic of possibilistic decision theory is often a natural one to consider. The development of possibilistic decision theory has lead to the proposition a series of possibilistic criteria, namely: optimistic and pessimistic possibilistic qualitative criteria [7], possibilistic likely dominance [2], [9], binary possibilistic utility [11] and possibilistic Choquet integrals [24]. This paper focuses on sequential decision making in possibilistic decision trees. It proposes a theoretical study on the complexity of the problem of finding an optimal strategy depending on the monotonicity property of the optimization criteria – when the criterion is transitive, this property indeed allows a polytime solving of the problem by Dynamic Programming. We show that most possibilistic decision criteria, but possibilistic Choquet integrals, satisfy monotonicity and that the corresponding optimization problems can be solved in polynomial time by Dynamic Programming. Concerning the possibilistic likely dominance criteria which is quasi-transitive but not fully transitive, we propose an extended version of Dynamic Programming which remains polynomial in the size of the decision tree. We also show that for the particular case of possibilistic Choquet integrals, the problem of finding an optimal strategy is NP-hard. It can be solved by a Branch and Bound algorithm. Experiments show that even not necessarily optimal, the strategies built by Dynamic Programming are generally very good.     

Hybrid possibilistic networks

Possibilistic networks and possibilistic logic are two standard frameworks of interest for representing uncertain pieces of knowledge. Possibilistic networks exhibit relationships between variables while possibilistic logic ranks logical formulas according to their level of certainty. For multiply connected networks, it is well-known that the inference process is a hard problem. This paper studies a new representation of possibilistic networks called hybrid possibilistic networks. It results from combining the two semantically equivalent types of standard representation. We first present a propagation algorithm through hybrid possibilistic networks. This inference algorithm on hybrid networks is strictly more efficient (and confirmed by experimental studies) than the one of standard propagation algorithm.     

Measuring the Student Group Capacity for Obtaining Geometric Information in the van Hiele Developmental Thought Process：A Fuzzy Approach

1　 IntroductionThe van Hiele level theory of geometric reasoning[1,5,9] describes the ways of studentreasoning inEuclidean geometry.The ontogeny of thiscognitive activity of mental developmentis characterised byfive hierarchical and qualitatively differe…     

A simplified kinematic wave model at a merge bottleneck

A merge is a point of a highway where two or more streams of traffic flow into one. It is always easy to solve the demand–supply problem at a merge when the merge is operating under uncongested condition. However, when congestion backs up exceeding the merging point where multiple streams of traffic meet, one is typically faced with splitting downstream supply among the merging branches. Solutions of this situation are multiple and several merge queuing models have been proposed in the past. To address the drawbacks of the past models, this paper proposes a capacity-based weighted fair queuing (CBWFQ) model that is characterized by its fidelity (approximation to real situation), simplicity, and extensibility. Based on the CBWFQ merge queuing model, a simplified kinematic wave model is formulated to model traffic operation at a merge bottleneck.