Publication: Pointed order polytopes: Studying geometrical aspects of the polytope of bi-capacities
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2021-11-08
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Elsevier Science Bv
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In this paper we study some geometrical questions about the polytope of bi-capacities. For this, we introduce the concept of pointed order polytope, a natural generalization of order polytopes. Basically, a pointed order polytope is a polytope that takes advantage of the order relation of a partially ordered set and such that there is a relevant element in the structure.
We study which are the set of vertices of pointed order polytopes and sort out a simple way to determine whether two vertices are adjacent. We also study the general form of its faces. Next, we show that the set of bi-capacities is a special case of pointed order polytope. Then, we apply the results obtained for general pointed order polytopes for bi-capacities, allowing to characterize vertices and adjacency, and obtaining a bound for the diameter of this important polytope arising in Multicriteria Decision Making.
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[1] J. Abbas. The bipolar Choquet integral based on ternary-element sets. Journal of Arti�cial
Intelligence and Soft Computing Research, 6(1):13{21, 2016.
[2] G. Beliakov and J.Z. Wu. Learning fuzzy measures from data: simpli�cations and optimisation
strategies. Information Sciences, 494:100{113, 2019.
[3] O.V. Borodin and A.O. Ivanova. Describing faces in 3-polytopes with no vertices of degree from
5 to 7. Discrete Mathematics, 342(11):3208{3215, 2019.
[4] D. Candeloro, R. Mesiar, and A.R. Sambucini. A special class of fuzzy measures. Choquet integral
and applications. Fuzzy Sets and Systems, (355):83{99, 2019.
[5] D. Catanzaro and R. Pesenti. Enumerating vertices of the balanced minimum evolution polytope.
Computers and Operations Research, (109):209{217, 2019.
[6] M. R. Chithra and A. Vijayakumar. The diameter variability of the cartesian product of graphs.
Discrete Mathematics, Algorithms and Applications, 6(1), 2014.
[7] G. Choquet. Theory of capacities. Annales de l'Institut Fourier, (5):131{295, 1953.
[8] E. F. Combarro and P. Miranda. Identi�cation of fuzzy measures from sample data with genetic
algorithms. Computers and Operations Research, 33(10):3046{3066, 2006.
[9] E. F. Combarro and P. Miranda. Adjacency on the order polytope with applications to the theory
of fuzzy measures. Fuzzy Sets and Systems, 180:384{398, 2010.
[10] R. Dedekind. �Uber Zerlegungen von Zahlen durch ihre gr�ossten gemeinsamen Teiler. Festschrift
Hoch Braunschweig Ges. Werke, II:103{148, 1897. In German.
[11] D. Denneberg. Non-additive measures and integral. Kluwer Academic, Dordrecht (The Netherlands),
1994.
[12] M. Grabisch. Alternative representations of discrete fuzzy measures for decision making. Inter-
national Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5:587{607, 1997.
[13] M. Grabisch. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and
Systems, (92):167{189, 1997.
[14] M. Grabisch. Set functions, games and capacities in Decision Making, volume 46 of Theory and
Decision Library. Springer, 2016.
[15] M. Grabisch and C. Labreuche. Bi-capacities. I. the Choquet integral. Fuzzy Sets and Systems,
151(2):236{259, 2005.
[16] M. Grabisch and C. Labreuche. Bi-capacities. II. De�nition, M�obius transform and interaction.
Fuzzy Sets and Systems, 151(2):211{236, 2005.
[17] S. Greco, B. Matarazzo, and S. Giove. The Choquet integral with respect to a level dependent
capacity. Fuzzy Sets and Systems, 175(1):1{35, 2011.
[18] I. Kojadinovic and J.L. Marichal. Entropy of bi-capacities. European Journal of Operational
Research, 178(1):168{184, 2007.
[19] F. Lange and M. Grabisch. New axiomatizations of Shapley interaction index for bi-capacities.
Fuzzy Sets and Systems, 176:64{75, 2011.
[20] J. Li, X. Yao, X. Sun, and D. Wu. Determining the fuzzy measure in multiple criteria decision
aiding from the tolerance perspective. Representation of associative functions. European Journal
of Operational Research, 264:428{439, 2018.
[21] P. Miranda and P. Garc��a-Segador. Order cones: A tool for deriving k-dimensional faces of cones
of subfamilies of monotone games. Annals of Operational Research, 295(1):117{137, 2020.
[22] P. Miranda and M. Grabisch. p-symmetric bi-capacities. Kybernetica, 40(4):421{440, 2004.
[23] Y. Narukawa, V. Torra, and M. Sugeno. Choquet integral with resepct to a symmetric fuzzy
measure of a function on the real line. Annals of Operations Research, 244:571{581, 2016.
[24] C.E. Osgood, G.J. Suci, and P.H. Tannenbaum. The measurement of meaning. University of
Illinois Press, Urbana (Il.), 1957.
[25] R. Stanley. Two poset polytopes. Discrete Comput. Geom., 1(1):9{23, 1986.
[26] M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of
Technology, 1974.
[27] L. Xie and M. Grabisch. The core of bicapacities and bipolar games. Fuzzy Sets and Systems,
158(9):1000{1012, 2007.