191 research outputs found
On the geometric mean method for incomplete pairwise comparisons
When creating the ranking based on the pairwise comparisons very often, we
face difficulties in completing all the results of direct comparisons. In this
case, the solution is to use the ranking method based on the incomplete PC
matrix. The article presents the extension of the well known geometric mean
method for incomplete PC matrices. The description of the methods is
accompanied by theoretical considerations showing the existence of the solution
and the optimality of the proposed approach.Comment: 15 page
A general unified framework for pairwise comparison matrices in multicriterial methods
In a Multicriteria Decision Making context, a pairwise comparison matrix is a helpful tool to determine the weighted
ranking on a set of alternatives or criteria. The entry of the matrix can assume different meanings: can be a preference ratio (multiplicative case) or a preference difference (additive case) or belongs to and measures the
distance from the indifference that is expressed by 0.5 (fuzzy
case). For the multiplicative case, a consistency index for the
matrix has been provided by T.L. Saaty in terms of maximum eigenvalue.
We consider pairwise comparison matrices over an abelian linearly
ordered group and, in this way, we provide a general framework
including the mentioned cases. By introducing a more general notion
of metric, we provide a consistency index that has a natural
meaning and it is easy to compute in the additive and multiplicative cases; in the other cases, it can be computed easily starting from a suitable additive or multiplicative matrix
Comparing inconsistency of pairwise comparison matrices depending on entries
Pairwise comparisons have been a long-standing technique for comparing alternatives/criteria and their role has been pivotal in the development of modern decision-making methods. Since several types of pairwise comparison matrices (e.g., multiplicative, additive, fuzzy) are proposed in literature, in this paper, we investigate, for which type of matrix, decision-makers are more coherent when they express their subjective preferences. By performing an experiment, we found that the additive approach provides the worst level of coherence
The development of a project typology and selection tool to improve decision-making in sustainable projects
Decision-making in sustainable projects is a complex and challenging process, especially during the initiating and planning phases of project development, due to influence from several external factors, as well as the uncertain environments surrounding their creation. It is essential to improve the decision-making process in sustainable projects during these two phases by relying on strong decision-making tools. The first contribution in this work identifies gaps in the literature of how institutionalization can impact sustainable projects through the effects of institutional isomorphisms from institutional theory. This helps decision makers better understand the relationship between institutionalization and sustainable projects. The second contribution is a sustainable project typology based on the affects that the coercive, normative, and mimetic institutional pressures have on common key sustainable project characteristics. The typology can improve decision-making by providing realistic predictions about the project early in the planning phase. The third contribution further develops this typology into a project selection tool that can be used in the initiating phase. It applies the Fuzzy Analytic Hierarchy Process (FAHP) to rank the key project characteristics based on importance as selection criteria by utilizing the literature as the voice of expert opinion. Because using the literature as a source of expert opinion can present its own set of challenges, the fourth contribution considers how the choice of selection tool inputs can impact project selection. Accordingly, Subject Matter Experts (SMEs) are utilized as an alternative source of expert opinion in an effort to validate the previously generated results and compare how these selection criteria are prioritized in literature and practice --Abstract, page iv
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Introduction to the Analytic Hierarchy Process
The Analytic Hierarchy Process (AHP) has been one of the foremost mathematical methods for decision making with multiple criteria and has been widely studied in the operations research literature as well as applied to solve countless real-world problems. This book is meant to introduce and strengthen the readers’ knowledge of the AHP, no matter how familiar they may be with the topic.
This book provides a concise, yet self-contained, introduction to the AHP that uses a novel and more pedagogical approach. It begins with an introduction to the principles of the AHP, covering the critical points of the method, as well as some of its applications. Next, the book explores further aspects of the method, including the derivation of the priority vector, the estimation of inconsistency, and the use of AHP for group decisions. Each of these is introduced by relaxing initial assumptions. Furthermore, this booklet covers extensions of AHP, which are typically neglected in elementary expositions of the methods. Such extensions concern different numerical representations of preferences and the interval and fuzzy representations of preferences to account for uncertainty. During the whole exposition, an eye is kept on the most recent developments of the method.Peer reviewe
A graph theoretic approach to scene matching
The ability to match two scenes is a fundamental requirement in a variety of computer vision tasks. A graph theoretic approach to inexact scene matching is presented which is useful in dealing with problems due to imperfect image segmentation. A scene is described by a set of graphs, with nodes representing objects and arcs representing relationships between objects. Each node has a set of values representing the relations between pairs of objects, such as angle, adjacency, or distance. With this method of scene representation, the task in scene matching is to match two sets of graphs. Because of segmentation errors, variations in camera angle, illumination, and other conditions, an exact match between the sets of observed and stored graphs is usually not possible. In the developed approach, the problem is represented as an association graph, in which each node represents a possible mapping of an observed region to a stored object, and each arc represents the compatibility of two mappings. Nodes and arcs have weights indicating the merit or a region-object mapping and the degree of compatibility between two mappings. A match between the two graphs corresponds to a clique, or fully connected subgraph, in the association graph. The task is to find the clique that represents the best match. Fuzzy relaxation is used to update the node weights using the contextual information contained in the arcs and neighboring nodes. This simplifies the evaluation of cliques. A method of handling oversegmentation and undersegmentation problems is also presented. The approach is tested with a set of realistic images which exhibit many types of sementation errors
The moduli space of matroids
In the first part of the paper, we clarify the connections between several
algebraic objects appearing in matroid theory: both partial fields and
hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are
compatible with the respective matroid theories. Moreover, fuzzy rings are
ordered blueprints and lie in the intersection of tracts with ordered
blueprints; we call the objects of this intersection pastures.
In the second part, we construct moduli spaces for matroids over pastures. We
show that, for any non-empty finite set , the functor taking a pasture
to the set of isomorphism classes of rank- -matroids on is
representable by an ordered blue scheme , the moduli space of
rank- matroids on .
In the third part, we draw conclusions on matroid theory. A classical
rank- matroid on corresponds to a -valued point of
where is the Krasner hyperfield. Such a point defines a
residue pasture , which we call the universal pasture of . We show that
for every pasture , morphisms are canonically in bijection with
-matroid structures on .
An analogous weak universal pasture classifies weak -matroid
structures on . The unit group of can be canonically identified with
the Tutte group of . We call the sub-pasture of generated by
``cross-ratios' the foundation of ,. It parametrizes rescaling classes of
weak -matroid structures on , and its unit group is coincides with the
inner Tutte group of . We show that a matroid is regular if and only if
its foundation is the regular partial field, and a non-regular matroid is
binary if and only if its foundation is the field with two elements. This
yields a new proof of the fact that a matroid is regular if and only if it is
both binary and orientable.Comment: 83 page
Fuzzy Toric Geometries
We describe a construction of fuzzy spaces which approximate projective toric
varieties. The construction uses the canonical embedding of such varieties into
a complex projective space: The algebra of fuzzy functions on a toric variety
is obtained by a restriction of the fuzzy algebra of functions on the complex
projective space appearing in the embedding. We give several explicit examples
for this construction; in particular, we present fuzzy weighted projective
spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction
is actually suited for arbitrary subvarieties of complex projective spaces, one
can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on
fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number
of available fuzzy spaces significantly, we show that the fuzzification of a
projective toric variety amounts to a quantization of its toric base.Comment: 1+25 pages, extended version, to appear in JHE
- …