Basic uncertain information (BUI) in the form <inline-formula><tex-math notation="LaTeX">\langle x;c \rangle </tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">x \in ...0,1</tex-math></inline-formula> is an observed datum and <inline-formula><tex-math notation="LaTeX">c \in 0,1</tex-math></inline-formula> is its reliability, is discussed and studied. The concept of BUI's aggregation is introduced and some construction methods are proposed and exemplified. The connection with interval-valued data and their aggregation is also discussed.
Ordered weighted averaging (OWA) operators, a family of aggregation functions, are widely used in human decision-making schemes to aggregate data inputs of a decision maker's choosing through a ...process known as OWA aggregation. The weight allocation mechanism of OWA aggregation employs the principle of linear ordering to order data inputs after the input variables have been rearranged. Thus, OWA operators generally cannot be used to aggregate a collection of <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> inputs obtained from any given convex partially ordered set (poset). This poses a problem since data inputs are often obtained from various convex posets in the real world. To address this problem, this paper proposes methods that practitioners can use in real-world applications to aggregate a collection of <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> inputs from any given convex poset. The paper also analyzes properties related to the proposed methods, such as monotonicity and weighted OWA aggregation on convex posets.
In multicriteria group decision-making problems, we need to determine the relative importances among criteria as well as among experts. When doing so, however, we often face the situations where ...consensuses within experts over different criteria need to be considered, and where uncertainties arise when experts do evaluations. Therefore, we need some special and reasonable methods to generate weights in such situations. In this study, three elaborately devised methods suggest ways to generate relative importance among experts, criteria, and the combination of them, respectively. The first one elicits the consensus extents within experts over different criteria, by which it can generate suitable weights among different criteria. The second one fully considers the uncertain nature when experts do evaluations, and proposes a fuzzy model which can generate weighting vector among experts according to the certainty degrees of valuations given by all the experts. In the last method, when relative importances among both experts and criteria are predetermined in the form of two capacities with dimension n and m , respectively, we find an interesting mechanism to successfully melt them into one nm -dimensional capacity which is based on given cognitive strength and on the proposed concept of compromised active/passive consensus.
Given probability information, i.e., a probability measure m with a random variable x on the outcome space N , the expected value of that random variable is commonly used as some valuable evaluation ...result for further decision making. However, there is no guarantee that the given probability information will be convincing to every decision maker. This is possible because decision makers may question the reliability of that provided probability information and can also be because decision makers often have their own different optimistic/pessimistic preferences. Often, such optimistic/pessimistic preferences can be easily embodied and expressed by some ordered weighted average (OWA) weight functions w . This study first compares and analyzes some simpler methods to melt the given OWA weight functions w with the given probability measure m to generate a new probability measure, pointing out their respective advantages and shortcomings. Then, this study proposes the melting axioms, which will both conform to our intuition and have mathematical reasonability. As the main finding of this study, we then propose the Crescent Method, which will effectively melt the given OWA weight function w with the given probability measure m to generate a final resulted fuzzy measure. Based on that melted fuzzy measure, we perform the Choquet integral of x as the more convincing evaluation result to decision makers with preference w . The study also proposes several interesting mathematical results such as the orness of resulted fuzzy measure will always be equal to the orness of the given OWA weight function w .
The famous Hirsch index has been introduced just ca. ten years ago. Despite that, it is already widely used in many decision-making tasks, like in evaluation of individual scientists, research grant ...allocation, or even production planning. It is known that the h-index is related to the discrete Sugeno integral and the Ky Fan metric introduced in the 1940s. The aim of this paper is to propose a few modifications of this index as well as other fuzzy integrals-also on bounded chains-that lead to better discrimination of some types of data that are to be aggregated. All of the suggested compensation methods try to retain the simplicity of the original measure.
Motivated by some cognition styles, this study first proposes a new type of preaggregation functions called cognitive integrals with its generalized forms. Rather than being affected only by selected ...capacity, generalized cognitive integrals consider other two parameters, the cognitive strength and the involved semicopula. Our main focus is on cognitive strength, by which the integrals values will be monotonically decreasing with respect to our cognitive strength. After some appropriated adaptations, we later propose the concept of adapted cognitive integrals with two equivalent forms of itself. Not only being a type of the preaggregation functions, we prove that adapted cognitive integrals are indeed one new type of aggregation functions, still equipped with the two new parameters like in cognitive integrals.
Choquet Integral is a powerful aggregation function especially in merging finite real inputs. However in real life, many inputs exist in continuum, e.g., the Riemann Integrable functions. The ...standard Choquet Integral formulas can not accommodate such inputs. This study proposes a new expression which enables merging Riemann Integrable inputs using a discrete Choquet integral. Relevant properties arising therein are discussed. A few application domains are identified which include time-dependent multicriteria decision aid and dynamic fuzzy cooperative games, etc.
The ordinal sum of triangular norms on the real unit interval
has been used to construct other triangular norms. But, in general, the standard approach to the ordinal sum construction of triangular ...norms and triangular conorms may not work on an arbitrary bounded lattice. In this study, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an arbitrary bounded lattice. Also, we give some illustrative examples for the clarity.
