The paper considers new classes of spaces of finite, bounded, measurable multisets with different metrics, pseudometrics, quasimetrics, symmetrics, and some properties of these metrics. We discuss the possibilities to apply new types of metrics for estimating proximity of objects with many numerical and/or verbal attributes. We use the introduced indexes of similarity and dissimilarity of objects represented as multisets in new methods of group multiple criteria decision making.
M.R.Anderberg, Cluster Analysis for Applications, Academic Press, 1973.
2.
I.Batyrshin, Uncertainties with memory in construction of strict monotonic t-norms and t-conorms for finite ordinal scales: Basic definitions and applications, Applied and Computational Mathematics, 10(3) (2011), 498–513.
C.Brink, Some background on multisets, Automated Reasoning Project, Technical Report TR-ARP 2/87, RSSS, Australian National University, 1987.
5.
M.M.Deza and E.Deza, Encyclopedia of distances, Springer, 2009.
6.
J.C.Gower, Similarity, dissimilarity and distance, measures of, in: Encyclopedia of Statistical Sciences,John Wiley, 12, 2008, pp. 7730–7738.
7.
P.Jaccard, Etude comparative de la distribution florale dans une portion des Alpes et du Jura, Bulletin de la Société Vaudoise des Sciences Naturelles37(142) (1901), 547–579.
8.
W.A.Kosters and J.F.J.Laros, Metrics for mining multisets, in: Research and Development in Intelligent Systems XXIV, Proceedings of AI-2007,Springer-Verlag, 2008, pp. 294–303.
9.
M.Krawczak and G.Szkatula, Bidirectional comparison of multi-attribute qualitative objects, Information Sciences436-437 (2018), 367–387.
10.
L.A.Lyusternik and V.I.Sobolev, Elements of functional analysis, Nauka, 1965(in Russian).
11.
P.C.Mahalanobis, On the generalised distance in statistics, Proceedings of the National Institute of Science of India12 (1936), 49–55.
12.
E.Marczewski and H.Steinhaus, On a certain distance of sets and the corresponding distance of functions, Colloquium Mathematicum6 (1958), 319–327.
13.
M.O'Searcoid, Metric Spaces,SpringerLondon, 2009.
14.
J.Oxtoby, Measure and category,Springer-Verlag, 1971.
15.
A.B.Petrovsky, An axiomatic approach to metrization of multiset space, in: Multiple Criteria Decision Making,Springer-Verlag, 1994, pp. 129–140.
16.
A.B.Petrovsky, Metric spaces of multisets, Doklady of Academy of Sciences344(2) (1995), 175–177(in Russian).
17.
A.B.Petrovsky, Structuring techniques in multiset spaces, in: Multiple Criteria Decision Making,Springer-Verlag, 1997, pp. 174–184.
18.
A.B.Petrovsky, Multiattribute sorting of qualitative objects in multiset spaces, in: Multiple Criteria Decision Making in New Millennium,Springer-Verlag, 2001, 124–131.
19.
A.B.Petrovsky, Basis Concepts of Multiset Theory, Editorial URSS, 2002(in Russian).
20.
A.B.Petrovsky, Operations with multisets, Doklady Mathematics67(2) (2003), 296–299.
21.
A.B.Petrovsky, Spaces of Sets and Multisets, Editorial URSS, 2003(in Russian).
22.
A.Petrovsky, Multi-attribute classification of credit card-holders: Multiset approach, International Journal of Management and Decision Making7(2/3) (2006), 166–179.
23.
A.Petrovsky, Group verbal decision analysis, in: Encyclopedia of Decision Making and Decision Support Technologies,IGI Global, 1, 2008, pp. 418–425.
24.
A.Petrovsky, Group multiple criteria decision making: Multiset approach, in: Recent Developments and New Directions in Soft Computing. Studies in Fuzziness and Soft-Computing, 317, Springer International Publishing, 2014, pp. 19–33.
25.
A.B.Petrovsky and V.N.Lobanov, Multi-criteria choice in the attribute space of large dimension: Multi-method technology PAKS-M, Scientific and Technical Information Processing42(5) (2015), 76–86.
26.
A.B.Petrovsky, Indicators of similarity and dissimilarity of multi-attribute objects in metric spaces of sets and multisets, Artificial Intelligence and Decision Making (Iskusstvennyiy intellekt i prinyatiye resheniy)4 (2017), 78–94. (in Russian)
27.
A.Rebai, Canonical fuzzy bags and bag fuzzy measure as a basis for MADM with mixed non cardinal data, European Journal of Operational Research78 (1994), 34–48.
28.
L.Sheremetov, I.Batyrshin, D.Filatov, J.Martinez and H.Rodriguez, Fuzzy expert system for solving lost circulation problem, Applied Soft Computing8 (2008), 14–29.
29.
D.Singh, A.M.Ibrahim, T.Yohana and J.N.Singh, An overview of the applications of multisets, Novi Sad Journal of Mathematics37(2) (2007), 73–92.
30.
P.H.A.Sneath and R.R.Sokal, Numerical Taxonomy. The Principles and Practice of Numerical Classification, Freeman, 1973.
