Qual Quant (2011) 45:1445–1457
Extracting cover sets from free fuzzy sorting data
Published online: 23 April 2011
© Springer Science+Business Media B.V. 2011
Abstract Assignment of items to multiple categories requires suitable statistical methods.
The present paper provides a new approach to solve this task. The concept of fuzzy sets
is extended to cover sets (sets of overlapping clusters) in a simple manner introducing a
vector of item membership sums. The application of the new concept is exempliﬁed by mod-
ifying the fuzzy cluster analysis algorithm of Kaufman and Rousseeuw (Finding groups in
data: an introduction to cluster analysis, 1990) to cover set cluster analysis appropriately.
Wide equivalence of the numerical problems is demonstrated from Lagrange multipliers
and Karush-Kuhn-Tucker conditions. Additionally, some extensions are introduced to the
algorithm to improve its behavior for suboptimal large or small numbers of clusters. The
adapted algorithm in most cases reproduces single sortings for correct numbers of clusters.
Two applications to empirical free fuzzy sorting data sets are provided. Limitations of the
algorithm are discussed.
Keywords Sorting · Cognitive structure · Knowledge elicitation · Fuzzy sorting ·
Information architecture · Web site structure · Cover set cluster analysis ·
Fuzzy cluster analysis
1.1 Fuzzy sorting and fuzzy set theory
Fuzzy sorting is a sorting variant allowing participants to assign items to different groups
simultaneously (Harloff and Coxon 2005; Harloff 2008a,b). It has for example been used to
denote the actual presence of items on several different pages of a web site (Capra 2005;
Electronic supplementary material The online version of this article (doi:10.1007/s11135-011-9497-y)
contains supplementary material, which is available to authorized users.
J. Harloff (
Rudolf-Wilke-Weg 10, 81477 Munich, Germany