Concepts of solutions of uncertain equations with intervals,
probabilities and fuzzy sets for applied tasks
Received: 11 August 2016 / Accepted: 1 October 2016 / Published online: 18 October 2016
Ó Springer International Publishing Switzerland 2016
Abstract The focus of this paper is to clarify the concepts
of solutions of linear equations in interval, probabilistic,
and fuzzy sets setting for real-world tasks. There is a
fundamental difference between formal deﬁnitions of the
solutions and physically meaningful concepts of solution in
applied tasks, when equations have uncertain components.
For instance, a formal deﬁnition of the solution in terms of
Moore interval analysis can be completely irrelevant for
solving a real-world task. We show that formal deﬁnitions
must follow a meaningful concept of the solution in the real
world. The contribution of this paper is the seven formal-
ized deﬁnitions of the concept of solution for the linear
equations with uncertain components in the interval set-
tings that are interpretable in the real-world tasks. It is
shown that these deﬁnitions have analogies in probability
and fuzzy set terms too. These new formalized concepts of
solutions are generalized for difference and differential
equations under uncertainty.
Keywords Interval equations Á Fuzzy equations Á
Stochastic equations Á Quantiﬁers Á Difference equation Á
Granular Computing (Pedrycz and Chen 2011, 2015a, b)
approach focuses on analyzing data at multiple levels of
details or scales in a variety of granulation settings (Dubois
and Prade 2016; Xu and Wang 2016). Often each level has
its own uncertainties and needs speciﬁc methods for
modeling such uncertainties (Kovalerchuk et al. 2012).
Granular computing is an integral part of Computational
Intelligence (Pedrycz and Chen 2011, 2015a, b), Big Data
(Pedrycz and Chen 2015a, b) and Decision Making (Ped-
rycz and Chen 2015a, b). In all these domains solving
equations under uncertainty is common part of many tasks.
This includes equations with parameters represented by
probabilities, intervals or fuzzy sets.
At ﬁrst glance, it seems reasonable to translate the
concept of solution for equations without uncertainty to the
corresponding type of uncertainty. This is how many
researchers and practitioners solve the corresponding
uncertain problems. This paper stresses that the resulting
solutions are sometimes inadequate, because to come up
with a correct solution, we need to analyze the original
problem under uncertainty, not only its exact prototype.
Often, for exact data, different practical problems lead to
the same solution, while in the presence of uncertainty
these problems often lead to completely different solutions
as we show. This ambiguity cannot be resolved by simply
modifying the usual formal approach: e.g., even for the
simple case when the exact-case solution is just a sub-
traction, in the uncertainty case, there can be multiple
different deﬁnitions of a solution.
Most of the work in fuzzy equations has been concerned
with algorithms and theorems for solving fuzzy linear
equations, e.g., (Yager 1979; Sanchez 1984; Di Nola et al.
1985; Zhao and Govind 1991; Peeva 1991; Buckley and
& Boris Kovalerchuk
Department of Computer Science, Central Washington
University, Ellensburg, WA, USA
Department of Computer Science, University of Texas at El
Paso, El Paso, TX, USA
Granul. Comput. (2017) 2:121–130