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1

Applying Data Envelopment Analysis Principle in Ordinal Multi Criteria Decision Analysis

Kress Moshe

Foundations of Computing and Decision Sciences

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2021

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Vol. 46, No. 2

147--157

EN

We consider a multicriteria decision analysis (MCDA) problem where importance of criteria, and evaluations of alternatives with respect to the criteria, are expressed on a qualitative ordinal scale. Using the extreme-point principle of Data Envelopment Analysis (DEA), we develop a two-parameter method for obtaining overall ratings of the alternatives when preferences and evaluations are made on an ordinal scale. We assume no parametric setup other than the two parameters that reflect minimum intensities of discriminating among rank positions: one parameter for the alternatives’ ranking and one for the criteria ranking. These parameters are bounded by the ordinal input data, and they imply a universal tie among the alternatives when both parameters are selected to be zero. We describe the model, discuss its theoretical underpinning, and demonstrate its application.

2

Faces for two-qubit separable states and the convex hulls of trigonometric moment curves

Kye S.-H.

Probability and Mathematical Statistics

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2013

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Vol. 33, Fasc. 2

385--400

EN

We analyze the facial structures of the convex set consisting of all two-qubit separable states. One of the faces is a four-dimensional convex body generated by the trigonometric moment curve arising from polyhedral combinatorics. Another one is an eight-dimensional convex body, which is the convex hull of a homeomorphic image of the two-dimensional sphere. Extreme points consist of points on the surface, and any two of tchem determine an edge. We also reconstruct the trigonometric moment curve in any even-dimensional affine space using the qubit-qudit systems, and characterize the facial structures of the convex hull.

3

An application of the Choquet theorem to the study of randomly-superinvariant measures

Rajba T.

Opuscula Mathematica

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2012

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Vol. 32, no. 2

317-326

EN

Given a real valued random variable Θ we consider Borel measures μ on Β (R), which satisfy the inequality μ(B) ≥ Eμ (B-Θ) (B ∈ Β (R)) or the integral inequality [formula].We apply the Choquet theorem to obtain an integral representation of measures μ satisfying this inequality. We give integral representations of these measures in the particular cases of the random variable Θ.

4

Uniqueness of Cartesian Products of Compact Convex Sets

Lipecki Z., Losert V., Spurný J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 2

175--183

EN

Let Xi, i∈I, and Yj, j∈J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of ∏i∈IXi onto ∏j∈JYj. We show that there exists a bijection b:I→J such that φ is the product of affine homeomorphisms of Xi onto Yb(i), i∈I.

5

I-sets and cominimal projections

Lipieta A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2006

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Vol. 46, [Z] 1

37-62

EN

Let Y [...] be a subspace of codimension two. For n = 4 the formula for cominimal projections will be presented. For n = 5 the complete characterization of I-sets which determined cominimal projections and consequently the manner of finding cominimal projections will be given. This completes the results from [15].

6

Three boundary value problems for second order differential inclusions in Banach spaces

Azzam D., Castaing C., Thibault L.

Control and Cybernetics

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2002

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Vol. 31, no 3

659-693

EN

The paper studies, in the context of Banach spaces, the problem of three boundary conditions for both second order differential inclusions and second order ordinary differential equations. The results are obtained in several new settings of Sobolev-type spaces involving Bochner and Pettis integrals. Some classes of second order multivalued evolution equations associated with m-accretive operators are also considered. Applications to some control problems are provided with the help of narrow convergence for Young measures.

7

Strongly extreme points in Orlicz sequence spaces

Wisła M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

241-255

EN

Let l^[fi] be an Orlicz space endowed with the Luxemburg norm. The main result reads as flooows: the set of all strongly extreme points of the unit ball B(l^[fi]) is nonempty iff it coincides with the set of all extreme points of B(l^[fi]) iff the function [fi] either satisfies the condition [ro]_2 or vanishes at some point outside zero.

8

Closedness of the set of extreme points in Orlicz spaces with Orlicz norm

Wisła M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

221-240

EN

In this paper the problem of closedness of the set of exreme points of the unit ball of Orlicz spaces equipped with Orlicz norm over a non-negative, atomless, [ro]-finite and complete measure space in discussed.

9

H-points in Köthe-Bochner spaces

Krassowska D.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

97-108

EN

In the paper we prove that if a measure space is atom-free then under some conditions every H-point of the unit ball of the Köthe-Bochner space must be an extreme point of this ball. This gives a local version of a result of Casting and Płuciennik and generalizes a result of Hu and Lin.