1 INTRODUCTION
There are various real life problems involving uncertainties and classical mathematical tools are not
sufficient for handling them. There are many theories developed recently for dealing with them. Some
of them are probability theory, theory of fuzzy sets [16], theory of intuitionistic fuzzy sets [1,2], theory
of vague sets [4], theory of interval mathematics [11] and theory of rough sets [13]. All of these theories
have their own advantages and some limitations as well. For example, in the theory of fuzzy sets and
intuitionistic fuzzy sets, it is very difficult to choose the membership and non membership functions
that give us the desired result; in the theory of probability the outcomes of an event must be unbiased;
in the theory of rough sets, the indiscernibility relation may create a situation where two completely
different objects are same. One major common drawback of these theories is probably the inadequacy
of parameterization tools which was observed by Molodtsov in 1999. Consequently he introduced the
concept of soft set theory [10] that is free from the difficulties that have troubled the usual theoretical
approaches. The absence of any restriction on the approximate description in soft set theory makes
it easily applicable in practice. A soft set model requires no prior knowledge of data sets. Molodtsov
provided several applications of soft set theory in his work. Maji et al. [9] introduced fuzzy soft set
by allowing the parameters to be mapped to the fuzzy sets. Further allowing the parameters to be
mapped to the intuitionistic fuzzy sets, Maji et al. [8] introduced the concept of intuitionistic fuzzy soft
set which is a generalization of standard soft set and fuzzy soft set in the sense that it is a soft set
whose approximate values are the intuitionistic fuzzy sets.
Lots of research work are currently active in the field of theoretical and practical soft sets. The
major portion of these works is based on complete information. However, incomplete information widely
exists in real life due to mishandling data, mistakes in processing or transferring data, mistakes in
measuring and collecting data or any other factor. Soft set under incomplete information is referred to
as an incomplete soft set. Similarly fuzzy soft set and intuitionistic fuzzy soft set under incomplete
information are referred to as incomplete fuzzy soft set and incomplete intuitionistic fuzzy soft set
respectively.
The simplest approach to transform an incomplete data set to a complete one is to delete all objects
related to missing information. But in this process we may deduce wrong information from it. On the
other hand, predicting the unknown information gives more fruitful results. Zou et al. [17] initiated
the study of incomplete soft sets. For incomplete soft set, they computed decision values rather than
filling the empty cells in the corresponding incomplete information system. The decision values are
calculated by the weighted average of all the choice values and the weight of each choice value is decided
by the distribution of other available objects. Incomplete fuzzy soft set is completed by the method
of average probability. Zou’s method is too complicated and it does not fill the empty cells of the
corresponding information system. So the soft set obtained by this method is only useful in decision
making. Using average probability method we can predict individual unknown value of fuzzy soft set
but all the predicted values of a parameter for different objects are equal, so this method is also of low
accuracy. Kong et al. [6] proposed a simple method equivalent to that of Zou which fills the empty cells.
To fill the empty cells in the incomplete information system, Kong’s method uses the values of target
parameter (the parameter for which the cell is empty) on the objects other than target object (the
object for which the cell is empty). Qin et al. [14] presented a method called DFIS (data filling approach
for incomplete soft set). In that paper, empty cells are filled in terms of the association degree between
the parameters, when a strong association exists between the parameters, otherwise they are filled
in terms of probability of other available objects. Khan et al. [5] proposed an alternative data filling
approach for incomplete soft set (ADFIS) to predict the missing data in soft sets. In ADFIS, the value
of the empty cell whose corresponding parameter has strongest association is computed first. Unlike
the DFIS, before filling second empty cell, the value of the first is inserted in the information table.
But the drawback of DFIS and ADFIS is that a parameter can have strongest association or maximal
association with more than one parameters having opposite type of association. In that case empty cell
can’t be filled. This method considers only the relation between parameters and does not take the effect
of objects into account. However there may be some relationship between objects too. For example
the houses in same locality have nearly same price. Deng et al. [3] introduced an object-parameter
https://doi.org/10.17993/3cemp.2022.110250.104-114
approach which uses the full information between object and between parameters. Deng’s approach
has some drawbacks as: (i) the estimated value may not be in the interval [0,1]; (ii) the information
between objects and between parameters is not comprehensive. To overcome these drawbacks Liu et
al. [7] improved Deng’s approach by redefining the notion of distance and dominant degree.
It is a review paper. In paper [15] we have put forward an algorithm to predict missing data in
an incomplete soft set and incomplete fuzzy soft set. For this we have defined the notion of distance
(emerged from the concept of Euclidean distance in
Rn
) between two objects (parameters) and defined
the degree of interdependence between two objects (parameters). And thus we have taken account
of the effect of other objects (parameters) on the target object (parameter). This algorithm uses the
full available data to reveal the hidden relationship between objects (parameters). Moreover we have
introduced an approach to predict missing data in an incomplete intuitionistic fuzzy soft set with the
help of algorithm for incomplete soft set and incomplete fuzzy soft set.
Rest of the paper has been organized as follows. Section 2 recalls the basic definitions and concepts
of soft set theory and information system. In section 3 we have introduced an algorithm to predict
missing data in an incomplete soft set and incomplete fuzzy soft set and given an application through
an example. In section 4, we have given an algorithm to predict the missing data in an incomplete
intuitionistic fuzzy soft set and given an application through an example. Finally we have concluded
this paper in section 5.
2 PRELIMINARIES
Let U={u1,u
2,...,u
m}be a universe set of objects and E={e1,e
2,...,e
n}be a set of parameters.
Definition 1 (Fuzzy set).[16] A fuzzy set Aover Uis given by
A={⟨u, µA(u)⟩|u∈U}
where
µA
:
U→
[0
,
1] is called the membership function of the fuzzy set
A
.
µA
(
u
)is said to be the degree
of membership of uin A.
Definition 2 (Intuitionistic fuzzy set).[1] An intuitionistic fuzzy set (IFS) Aover Uis given by
A={⟨u, µA(u),ν
A(u)⟩|u∈U;µA(u),ν
A(u)∈[0,1] and µA(u)+νA(u)≤1}}
where
µA
:
U→
[0
,
1] and
νA
:
U→
[0
,
1] are said to be the membership and non membership functions
of the intuitionistic fuzzy set Arespectively.
Definition 3 (Soft set).[10] A pair
A
=(
F, E
)is said to be a soft set over
U
, where
F
is a mapping
from
E
to
P
(
U
)(set of all crisp subsets of
U
). Sometimes it is also called a crisp soft set to emphasize
the fact that F(e)is a crisp set for every e∈E.
Alternatively, a soft set Ais given by
A={F(e)|e∈E}
where Fis a mapping from Eto P(U).
Definition 4 (Fuzzy soft set).[9] A pair
A
=(
F, E
)is said to be a fuzzy soft set over
U
, where
F
is
a mapping from Eto F(U)(set of all fuzzy sets over U).
Alternatively, a fuzzy soft set Ais given by
A={F(e)|e∈E}
where Fis a mapping from Eto F(U).
https://doi.org/10.17993/3cemp.2022.110250.104-114
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
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