Received December 29, 2016, accepted February 6, 2017, date of publication March 14, 2017, date of current version April 24, 2017.
Digital Object Identifier 10.1109/ACCESS.2017.2682231
A Review on Soft Set-Based Parameter Reduction
and Decision Making
SANI DANJUMA1,2 , TUTUT HERAWAN1,3 , MAIZATUL AKMAR ISMAIL1 ,
HARUNA CHIROMA5 , (Member, IEEE), ADAMU I. ABUBAKAR4 , (Member, IEEE),
AND AKRAM M. ZEKI4
1 Department
of Information Systems, Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia
University Kano, 234 Kano, Nigeria
Research Center, 54000 Yogyakarta, Indonesia
4 Kulliyah of Information and Communication Technology, International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia
5 Federal College of Education (Technical), 234 Gombe, Nigeria
2 Northwest
3 AMCS
Corresponding author: S. Danjuma ([email protected])
This work was supported by the International Islamic University Malaysia Research Grant through the Fundamental Research Grant
Schemes under Grant FRGS15-245-0486 and Grant FRGS15-226-0467s.
ABSTRACT Many real world decision making problems often involve uncertainty data, which mainly
originating from incomplete data and imprecise decision. The soft set theory as a mathematical tool that
deals with uncertainty, imprecise, and vagueness is often employed in solving decision making problem.
It has been widely used to identify irrelevant parameters and make reduction set of parameters for decision
making in order to bring out the optimal choices. In this paper, we present a review on different parameter
reduction and decision making techniques for soft set and hybrid soft sets under unpleasant set of hypothesis
environment as well as performance analysis of the their derived algorithms. The review has summarized
this paper in those areas of research, pointed out the limitations of previous works and areas that require
further research works. Researchers can use our review to quickly identify areas that received diminutive or
no attention from researchers so as to propose novel methods and applications.
INDEX TERMS Parameter reduction, decision making, soft set, hybrid soft sets, review.
I. INTRODUCTION
Many mathematical theories exist to deal with uncertainties, imprecise and vagueness involving collection of data
in information systems. One of the most popular and recent
mathematical theories to handle the problem of uncertainties
is soft set theory which is introduced by Molodtsov [1]. The
soft set theory is defined by Molodtsov as a tuple which is
associated with a set of parameters and a mapping from a
parameter set onto the power set of a universal set [1]. Unlike
the other existing mathematical theories for dealing with
uncertainties like probability theory [2], Fuzzy set theory [3],
intuitionistic fuzzy set theory [4], Rough set theory [5], grey
set theory [6], vague set [7] and etc. that all these theories lack
parameterization tools.
Although soft set having parameterization tool, still however in some cases involving non-Boolean datasets it requires
hybridization which could establish larger paradigms, so
that one can choose any type of parameters, which enormously simplifies the decision-making process and make the
procedure more proficient from available data. The major
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advantage of the soft set theory is that it need not bother with
any addition information about the data such as the probability in statistic or possibility value in fuzzy set theory. The
theory use parameterization as its main vehicle in developing
theory and its applications. The applications of the soft set are
progressing rapidly and many researchers are developing the
algorithm to solve many practical problems. The first and no
doubt application of soft set theory is decision making problem based on parameter reduction. The important problem of
parameter reduction and decision making is becoming fascinating to be useful approaches in dealing with uncertainties.
Parameterization reduction and decision making problems in
soft set theory are interesting to be explored. However, a
review that summarized recent advances on the applications
of the soft set theory in parameter reduction is scarce in the
literature despite the significance of the subject matter.
In this paper, we present a review on soft set-based parameter reduction and decision making which aims to provide
interested readers, who may be expert researchers in this area,
with the depth and breadth of the state of the art issues in
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S. Danjuma et al.: Review on Soft Set-Based Parameter Reduction and Decision Making
this research area. Additionally, it can be used as a starting
point for novice researchers in soft set theory. We also intend
for new soft setters to use this review as a starting point
for further advancement as well as an exploration of other
research problems that have received little or no attention
from researchers. The summary of the contribution of this
paper is as follows:
a. To summarize recent advances on soft set-based parameter reduction and decision making.
b. To highlight the advances of hybrid soft sets in parameter reduction and decision making.
c. To provide analysis and synthesis of the published
research outputs with insight on findings. Additionally,
we present a summary of applications of soft set and
hybrid soft set in real-life problems.
d. To point out unresolved problems and future research
direction in the applications of soft set and hybrid
soft sets in parameter reduction and decision making.
Finally, to highlight open research problems in those
research areas for easy future research identification.
The rest of the paper is organized as follows. Section 2
reviews of the basic concept of rough and soft set theories.
Section 3 recalls the hybridization of soft set with other set
theories. Section 4 presents a review on parameter reduction and decision making of the standard soft set. Section 5
presents parameter reduction and decision making of the
hybrid soft set with other existing theories. Section 6 presents
a review on applications of soft set in real-life problems.
Finally, section 7 presents conclusion and highlight future
researches.
II. RUDIMENTARY
This section presents fundamental concept of rough set theory
via constructive approach and soft set theory via descriptive
approach.
f (u, a) ∈ Va , for every (u, a) ∈ E ×A, called the information
function.
In an information system S = (U , A, V , f ) if Va = {0, 1},
for every a ∈ A then S is called a Boolean-valued information
system i.e. S = U , A, V{0,1} , f .
An information system is a finite data table and many concepts like object-attribute tuple, dependency of attribute, and
etc. are closely to the same concepts in relational database.
The following definition presents the notion of indiscernible
(similar) objects in an information system.
Definition 2 (See [10]): Let S = (U , A, V , f ), be an
information system and let B be any subset of A. Two objects
x, y ∈ U are said to B-indiscernible if and only if both x and
y have the same feature with respect to B i.e.
f (x, a) = f (y, a) ,
∀a ∈ B
From the notion of indiscernible objects above, then we can
construct the rough-approximating set via lower and upper
approximations of a subset X of U . The following definition
presents the notion of them.
Definition 3 (See [10]): Let S = (U , A, V , f ), be an
information system and let B be any subset of A and let X
be any subset U. The B-lower and B-upper approximations
of X, respectively denoted by B (X ) and B (X ), are defined by
B (X ) = {x ∈ U |[x] B ⊆ X }
and
B (X ) = {x ∈ U |[x] B ∩ X 6 = ϕ} .
From Definition 3, the B (X ) is a collection of all elements of
U which can be certainly classified as a member of X using
B. Meanwhile, the B (X ) is the set of all elements in U which
can be possibly classified as X using B. It is clear that a roughapproximating set is not a (crisp) set, when B (X ) 6 = B (X ).
The following sub-section, we present the notion of soft set
theory via a descriptive approach.
A. ROUGH SET THEORY
Rough set theory proposed by Pawlak and Skowron [8] is a
result of long term fundamental research project in developing a new mathematical model for computer science. It can
be referred as one of alternative set theories for analyzing
data. The idea of rough set theory is about approximation
of imprecise set using lower and upper approximations.
In this sub-section, rough set theory is presented from the
point of view via constructive approach. The starting point
of rough set theory for data analysis is due to the occurrence of uncertainty in an information system. The notion
of an information system (or information table) is defined as
follow.
Definition 1 (See [9]): An information system is a 4-tuple
(quadruple)
S
where
(U , A, V , f ),
=
U = u1 , u1 , · · · , u|U| is a non-empty finite set of objects,
A = aS
1 , a1 , · · · , a|A| is a non-empty finite set of attributes,
V = a∈A Va , Va is the domain (value set) of attribute a,
f = U × A → V is an information function such that
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B. SOFT SET THEORY
A soft set is defined as a parameterization of the subset of
the universal set which is introduced by [1]. Maji et al. [11]
studied the theory of soft sets initiated by [1], they defined
equality of two soft sets, subset and super set of a soft set,
complement of a soft set, null soft set and absolute soft set
with examples. Let U be an initial universe set and let Ebe
set of parameters in relation to object in U. If the set P (U )
denote the power set of U, then formally the definition of a
soft set is given as follow:
Definition 4 (See [1]): A pair (F, E) is called a soft set
over U, where F is a mapping given by F : E → P (U ).
In other words, the soft set is parameterized family of
subsets of the set U . Every set F (e), for e ∈ E from this
family may be considered as the set of e-element the soft set
(F, E) or as the set of e-approximate elements of the soft set.
It is clear that a standard soft set is not a crisp set. As an
illustration of a soft set, let us consider the following example.
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Example 1: A soft set (F, E) describes the attractiveness
of houses which Mr. X is going to buy. The set U is the set of
houses and there are 6 houses under consideration. Let
U = {h1 , h2 , h3 , h4 , h5 , h6 }
TABLE 1. Representation of a soft set of example 1.
and E = {e1 , e2 , e3 , e4 , e5 } .
The set E is the set of parameters describing the condition of all houses. Each parameter is a word or sentence,
i.e. E = {Expensive; Beautiful; Wooden; Cheap; in the
green surrounding} which can be represented a s E =
{e1 , e2 , e3 , e4 , e5 }. We consider a mapping F : E → P (U )
which is defined as follows:
F (e1 ) = {h2 , h4 } , F (e2 ) = {h3 , h5 } , F (e3 ) = {h1 , h3 } ,
F (e4 ) = {h4 , h6 } , and F (e5 ) = {h3 }
Thus, we have the soft set as follow:


Expensive Houses = {h2 , h4 }



Beautiful Houses = {h3 , h5 }

(F, E) = Wooden Houses = {h1 , h3 }



Cheap Houses = {h4 , h6 }



In the Green Surrounding Houses = {h }
3
The soft set is a mapping from parameter to the clearly
defined subset of the universe U . We may also represent a soft
set (F, E) in a finite complete Boolean-valued information
system.
Proposition 1 (See [12]): If (F, E) is a soft set over the
universe U, then (F, E) is a Boolean-valued information
system S = U , A, V{0,1} , f .
Proof: Let (F, E) be a soft set over the universe U , we
define mapping
f = {f1 , f2 , · · · , fn } ,
where
(
1, x ∈ F (ai )
1 ≤ i ≤ |A| .
1, x ∈
/ F (ai )
S
Hence, if A = E, V =
a∈A Va , where Va = {0, 1},
then a soft set (F, E) can be considered as a Boolean-valued
information system S = U , A, V{0,1} , f
Thus, a soft set (F, E) in Example 1 can be represented in
the form of Boolean-valued information system as shown in
the Table 1 below:
In the following section, we present a review on parameter
reduction of soft set.
fi : U → Vi
and f (x) =
III. HYBRID SOFT SET THEORY
This section presents some definitions of the marriage of soft
set theory with other existing set theories. The notion of fuzzy
set is given as follow:
Definition 5: Let U be a non-empty set of universe. A fuzzy
set is a pair (U , m), where m : U → [0, 1], for each x ∈ U
the value m (x) is called the grade membership of x in (U , m).
From Definition 5, the notion of a hybrid fuzzy set and soft
set is given in Definition 6 as follow:
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Definition 6 (See [13]): Let F (U ) be the set of all
subsets
in universe U and E be a set of parameters. A pair F̃, E is
called a fuzzy soft set over U, where F̃ is a mapping given by
F̃ : E → F (U ).
From Definition 6, it is clear that a fuzzy soft set is a special
case of a soft set. The Definition 7 below presents the notion
of an interval-valued fuzzy set.
Definition 7 (See [14]): An interval-value fuzzy set Y on a
universe U is a mapping Y : U → Int [0, 1], where Int [0, 1]
stands for the set of all closed subintervals of [0, 1] the set of
all interval-valued fuzzy set on U is denoted by I (U ).
From Definition 7, the notion of a hybrid internal-value
fuzzy set and soft set is given in Definition 8 as follow:
Definition 8 (See [14]):Let Ube an initial universe and E
set of parameters. A pair G̃, E is called an interval-value
fuzzy soft set over U, where G̃ is a mapping given by G̃ : E →
I (U ).
From Definition 8, it is clear that an interval-value fuzzy
soft set over a universe U is a special case of a soft set. The
Definition 9 below presents the notion of an intuitionistic
fuzzy set.
Definition 9 (See [15]): An intuitionistic fuzzy set (IFS)
A in E is defined as an object of the form A =
hx, µA (x) , VA (x) |x ∈ E i, where µA : E → [0, 1] and
VA : E → [0, 1] are respectively defined as the degree of
membership and non-membership of the element x ∈ E. For
every x ∈ E, the following property is hold
0 ≤ x, µA (x) , VA (x) ≤ 1.
From Definition 9, the notion of a hybrid intuitionistic fuzzy
set and soft set is given in Definition 10 as follow:
Definition 10 (See [15]): Consider U and E as a universe
set and a set of parameters respectively. Let P(U ) denotes
the
set of all intuitionistic fuzzy set of U. A pair H̃ , E is called
an intuitionistic fuzzy soft set over U, where H̃ is a mapping
given by H̃ : E → I (U ).
From Definition 10, it is clear that an intuitionistic fuzzy
soft set over a universe U is a special case of a soft set. In the
next section, we present a review on parameter reduction and
decision making of soft set.
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IV. PARAMETER REDUCTION AND DECISION
MAKING OF SOFT SET
Parameter reduction of soft set focuses on how to efficiently reduce the number of parameters from a dataset while
improving several aspect such as storage, speed, and accuracy
of the decision making process. In the literature, there are
number of successful implementation of parameter reduction
and decision making using soft set theory. Maji et al. [16]
firstly presented an application of rough set-based dimensionality reduction [8] in decision-making problem to a soft set.
They used few parameters after reduction to select optimal
choice of objects for decision making and the decision value
was computed with respect to condition parameters. Though,
the results of their techniques have some problems because
they firstly computed the reduction of rough set and then
compute the choice value to select the optimal object for
decision making. This shows that the optimal choice object
could be changed after the dimensionality reduction of the
rough set. Chen et al. [17] presented an approach based
on soft set for parameter reduction (PR) to find optimal
decisions on a general Boolean datasets as an improvement
of Maji et al. [16]. They improved the application of soft set
theory in a decision-making problem. However, Chen et al.
method faced the problem in determining all level of suboptimal choices. Consequently, Kong et al. [18] proposed
the concept of normal parameter reduction (NPR) to find
optimal decisions as well as all level of suboptimal decisions
and added parameter set of sets. This is because the methods
of [16] and [17] only considered the optimal choice, they
did not consider all level of suboptimal choices. In NPR,
the data of optimal objects can be deleted directly from the
normal parameter reduction, and the next optimal choice
can be obtained exactly from normal parameter reduction
in which the data of optimal objects were deleted. It was
evident from the performance analysis that NPR had shown
the ability to significantly reduce the dimensionality, but the
performance of their algorithm in term of computational time
is still an outstanding issue. Zou and Chen [19] proposed an
algorithm of parameter reduction based on invariability of
the optimal choice object, to efficiently decrease the number of the parameter used for evaluation and cut down the
workload. The authors applied basic operation of relational
algebra to realize the algorithm using Sequential Query Language (SQL). The result shows that it is effective as compared
to [17]. Herawan et al. [20] proposed an approach of attribute
reduction in multi-valued information system using soft set.
The AND operations were used for dimensionality reduction.
The result of the experiment found that the obtained reducts
are equivalent to the Pawlak’s rough reducts technique [8].
Rose et al. [21] presented technique of decision making by
parameterization reduction to determined maximal supported
sets from Boolean value information system based on soft
set theory and also provide consistency decision making. The
approach ensured that any process of attribute elimination in
convert complex database into a smaller database so as to
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make decision. The result of the experiment shows that this
technique is better than that method proposed by [16] and [17]
because it maintains the optimal and all level of sub-optimal
objects. Cagman and Enginoglu [22] initiated a novel soft
set based decision-making the scheme, called uni-int decision
making. This method could decrease an extensive set of alternatives to its subset of optimal objects based on the criteria
given by decision maker. It has achieved the optimal decision
from the experiment. Ma et al. [23] proposed a new efficient
normal parameter reduction algorithm based on soft sets
oriented parameter sum. The authors used oriented parameter sum without parameter reduction degree and decision
partition to reduce the parameter. The comparative results
on the dataset shows that the new efficient normal parameter
reduction algorithm based on the soft set was found to have
relatively less computational complexity, easy implementation and easy to understand than that the algorithms proposed
by [17] and [18]. Meanwhile, some researchers continues to
give their contribution in parameter reduction of a soft set and
decision making either directly or by means of expanding the
soft set to include other mathematical models. With the aim
of reducing parameter, attribute reduction and dimensionality
reduction. Ali [24] discussed the idea of reduction of parameters based on soft sets and studied approximation space of
Pawlak’s rough set model associated with the soft set. The
work deletes parameter only one at a time in order to avoid
the deep searching. The method of reduction of the parameter
is similar to the reduction of the attribute in the rough set.
The result of the experiment proved its efficiency. Jothi and
Inbarani [25] proposed a new soft set based on unsupervised
feature selection in order to reduce the amount of time to find
a minimal subset of the feature using soft set. The reduction
algorithm, when tested for the feature selection, shows that it
is close to rough set based on reduction and more effective
in terms of speed and performance compare with the rough
set. Hence, it reduces the computational time. Deng and
Wang [26] studied the relationship between any two objects
in a soft set, exploring sufficient and necessary conditions on
characterizing pseudo parameter reductions. Also, to normal
parameter reductions and notion of parameter significance,
in order to obtained pseudo parameter reductions and normal parameter reductions simultaneously. The result of the
experiment shows that it has generally low computational cost
compare with the PR and NPR. Feng et al. [27] extended the
work of [22] by proposing two new concepts of choice value
soft sets and k-satisfaction relations. They presented deeper
insights into the principle of uni–int decision making and
show its limitations. Finally, they proposed several new soft
decision making schemes. Yuan et al. [28] proposed an alternative approach of parameter reduction using soft set-based
data mining concept according to the importance degree of
each factor in decision making system. They claimed that
the proposed solution is more widely used in decision analysis systems and obtains a reduction parameter set. They
defined an impact factor of parameter and sub-sequentially
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presented a new indiscernibility relation and decision making priority. Their derived algorithm has been used in a
house purchase system to show some valuable information
and give a feasible result. Han and Geng [29] developed a
pruning method which filters out objects that cannot be an
element of any optimal decision set. The authors enhanced the
Feng et al. [27] int m-int n decision-making scheme and
the experimental result of the method shows that it has a
higher efficiency in computing the optimal solution. Kumar
and Rengasamy [30] studied parameter reduction using soft
set theory for better decision making. Hence, sample dataset
can be converted into binary valued data and shows that the
number of the parameter can be reduced without the loss of
any original information in order to make decision making
better. However, the dealing of optimal and suboptimal choice
was ignored. Xu et al. [31] presented a novel parameter
reduction method guided by soft set theory to select financial
ratios for business failure prediction. The method integrates
statistical logistic regression into a soft set and the result
demonstrates superior forecasting performance in terms of
accuracy and stability compare to NPR. Han and Li [32]
proposed a method of compiling all the normal parameter
reduction of a soft set into a proposition of parameter Boolean
variables. The method was used to provide an implicit representation of NPR and it improved the retrieval performance.
Li et al. [33] introduced soft coverings and investigated its
parameter reduction by means of knowledge on the attribute
reduction for covering information system. The algorithm of
the soft covering parameter reduction has been applied in
knowledge discovery and the result shows that the evaluation
system of the dataset used is efficient and feasible and thus,
save much time. Kong et al. [34] applied the particle swarm
optimization algorithm to reduce parameter in the soft set.
They defined dispensable core in the soft set and solve the
normal parameter reduction related to the dispensable core.
The experiment shows that the method is feasible and fast.
Xie [35] presented an alternative algorithm on the parameter
reduction of soft sets. The parameter reduction of soft sets
is investigated by means of the attribute reduction in information systems and its related algorithm is derived. Table 2
summarizes the various parameter reduction and decision
making methods of soft sets.
In the next section, we present a review on parameter
reduction and decision making of hybrid soft set with other
set theories.
V. PARAMETER REDUCTION AND DECISION MAKING OF
HYBRID SOFT SET
The soft set can be combined with other mathematical models
to deal with uncertainty and the combination is referred to
as hybridization. Numerous successful hybridizations of the
soft set, rough set, and fuzzy set have been studied. Also,
interval-valued fuzzy soft set and intuitionistic fuzzy soft set
where also considered by researchers to handle uncertainty
and imprecision in decision making and other applications.
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A. FUZZY SOFT SET-BASED APPROACH FOR PARAMETER
REDUCTION AND DECISION MAKING
With this point of view, Yang et al. [36] expanded the traditional soft set to fuzzy soft set to improve its quality, and
also discussed some immediate outcomes. To continue the
investigation on fuzzy soft sets, Roy and Maji [37] present
a novel method of object recognition from an imprecise
multi-observer data to decrease the problem of imprecision
in decision making using fuzzy soft set. The authors used a
method that involves the construction of a comparison table
from the resultant fuzzy soft set in a parametric sense, the
result of the experiment demonstrates the feasibility of the
methods. Kong et al. [38] argued that the algorithm proposed
in [37] was incorrect because using the algorithm will not
result to the right choice in general. Therefore, used a counter
example to illustrate the feasible of the method and proved the
efficiency of their method compare to [37]. Feng et al. [39]
presented an adjustable approach to fuzzy soft set based
decision making by using level soft sets. It is an improvement
of Roy and Maji method [37]. They weighted fuzzy soft set
is also introduced and applied to decision making problem.
Çağman et al. [40] present fuzzy parameterized fuzzy soft
sets (combination of fuzzy and soft set) and its operation
to create fuzzy parameterized fuzzy soft -decision making
method that allows development of decision processes. The
result of experiment demonstrates that the method can be
applied to many uncertainties problem. Jiang et al. [41] presented a fuzzy soft set theory (combination of fuzzy and soft
set) by applying the idea of fuzzy description logics to serve
as the parameters of fuzzy soft sets. The authors defined some
operations for the extended fuzzy soft sets. Additionally,
the authors proved the De Morgan’s laws in the extended
fuzzy soft set theory. Kong et al. [42] presented an algorithm
for multiple evaluation for fuzzy soft set decision problem
based on grey relational analysis, to combine multiple evaluation into a single evaluation value and use the evaluation
to make decision. The result of the experiment proved that
the new algorithm is efficient for solving decision problem.
Kong et al. [43] proposed a new parameter reduction which
combined fuzzy and soft set called fuzzy soft set to improve
the condition of redundant parameters because it is so strict
that the number of deleting parameters is very few. The
authors introduce some definitions of proximate parameter
reduction in fuzzy soft set. The result of the experiment
demonstrated the effectiveness of the proposed method. Basu
et al. [44] presented a new approach called mean potentiality
approach (MPA) to get a balanced solution of a fuzzy soft
set based decision making problem. And also, the reduced
parameter based on relational algebra with the help of MPA
algorithm. The method demonstrate that it is feasible compared to Feng’s method and NPR. Deng and Wang [45] introduced the concept of complete distance between two objects
and relative control the degree between two parameters, based
on an object-parameter to predict unknown data in incomplete fuzzy soft sets. The effectiveness of the method was
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TABLE 2. Parameter reduction and decision making of soft set.
demonstrated by given example and proved the feasibility of
the method compare with investigation of classical predicted
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method. Yang et al. [46] introduced multi-fuzzy soft set by
means of combining the multi-fuzzy set and soft set model.
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TABLE 2. (Continued) Parameter reduction and decision making of soft set.
Some operation and properties are defined and proved.
An illustrative example shows the validity of the method
as feasible. Li et al. [47] presented an approach to fuzzy
soft set in decision making by combining grey relational
analysis and Dempster-Shaper theory of evidence to calculate
grey mean relational degree which can be used to compute the uncertain degree of various parameters. The result
demonstrates the effectiveness and feasibility of the method.
Tao et al. [48] developed a novel concept of uncertain linguistic fuzzy soft set (ULFSS). The decision algorithm used
the external aggregation process that is TOPSIS technique for
collection decision information aggregate from individual.
The result demonstrate it feasibility. Tang [49] employed grey
relational analysis and Dempster–Shafer theory of evidence
in proposing a novel fuzzy soft set approach in decision making. Firstly, in calculating the grey mean relational degree, he
used the uncertain degrees of various parameters which are
determined via grey relational analysis. Secondly, according
to the uncertain degree, the suitable mass functions of different independent alternatives with different parameters are
given. Thirdly, Dempster’s rule of evidence combination is
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applied in order to aggregate the alternatives into a collective
alternative. Finally, the alternatives decisions are ranked and
the best alternatives decisions are obtained. The effectiveness
and feasibility of this proposed approach are demonstrated
by comparing with that the mean potentiality approach.
Wang et al. [50] presented hybrid hesitant fuzzy set and fuzzy
soft sets. They presented algebraic properties of the proposed
hybrid approach and further applied it to multi-criteria group
decision making problems. They showed the applicability of
their hybrid model to calculate a numerical example. Das and
Kar [51] presented a hybrid hesitant fuzzy set and soft set.
They studied distances measurement procedure and aggregate functions on their hybrid model. Finally, they applied
it to medical analysis involving multiple attribute decision
making. Aktaş and Çağman [52] presented a combination of
fuzzy sets and soft sets. They used a matrix representation
of the soft sets which is very useful for computations of the
proposed combination method in decision making involving
uncertainties. Majumdar [53] presented some hybrid soft sets
i.e. fuzzy parameterized soft set and vague soft set. They
studied algebraic properties of those two hybridizations and
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TABLE 3. Parameter reduction and decision making of hybrid fuzzy soft set.
presented their application in decision making. Table 3 below
summarizes the various parameter reduction and decision
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making methods of hybrid fuzzy and soft sets discussed
above.
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TABLE 3. (Continued) Parameter reduction and decision making of hybrid fuzzy soft set.
B. INTERVAL-VALUED FUZZY SOFT SET-BASED APPROACH
FOR PARAMETER REDUCTION AND DECISION MAKING
Interval-valued fuzzy soft set where also studied by
researchers to handle uncertainty and imprecision in decision
making. Yang et al. [54] introduced the concept of intervalvalued fuzzy soft set which combined interval-valued fuzzy
and soft set to obtain a new soft set model which is free
from the inadequacy of parameterized tools. They have also
defined many algebraic operations such as the complement,
‘‘AND’’, ‘‘OR’’ operations of interval-valued fuzzy soft set.
The authors, used example to prove the validity of the
interval-valued fuzzy soft set in decision making problem.
Khalid and Abbas [55] initiated the study of distance measures for interval valued fuzzy soft set (combination of interval valued fuzzy and soft set), and Hausdorff metric-based
measures for intuitionistic fuzzy soft set. Some operation and
properties are introduced and also proposed some application on medical diagnosis and prove that the performance is
effective. Feng et al. [56] introduced the concept of level soft
sets of fuzzy soft sets (combination of fuzzy and soft set)
and developed an adjustable decision-making approach using
fuzzy soft sets. The authors considered appropriate reduct
fuzzy soft sets and level soft set to reduce too much simpler
crisp soft set. It is evidently that in some cases Feng’s soft
rough set model provides best approximations than classical
rough sets. Qin et al. [57] presented an adjustable approach to
interval-valued intuitionistic fuzzy soft sets based on decision
making by using reduct intuitionistic. The authors compute
the reduct intuitionistic fuzzy soft set and interval value
intuitionistic fuzzy soft set and the converted it into crisp
soft set by using level soft set of intuitionistic fuzzy soft
set. The performance of the algorithm shows that it is more
efficient than the algorithm developed by Jiang et al. [41].
Alkhazaleh et al. [58] present the generalization of fuzzy
soft set to possibility fuzzy soft set and gave some application of this approach in decision making and also
the concept of parameterized interval-valued fuzzy soft
set where also introduced. The result shown is feasible.
Alkhazaleh et al. [59] present the concept of parameterized
interval-value fuzzy soft set where the mapping is defined
from the fuzzy soft set parameters to interval-value fuzzy subset of the universal set. Furthermore they gave an application
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of this approach in decision making. Finally the outcomes
demonstrated that the technique can be applied to many problems with uncertainties. Ma et al. [60] proposed the parameter
reduction of the interval- valued fuzzy soft set to deal with
uncertainty and imprecision in decision making. They developed heuristic algorithms of parameter reduction that reduces
redundant parameters. The result of the experiment demonstrated the effectiveness of the algorithms. Alkhazaleh [61]
introduced multi-fuzzy soft set by consolidating the concept
with interval-value fuzzy set. Some operation of complement, union and intersection operations were presented on
the multi-interval-valued fuzzy soft set and its properties
and finally gives the application of this concept in decision
making and the result were clear as feasible. Mukherjee and
Sarkar [62] presented some similarity measures of a hybrid
interval-valued fuzzy set and soft set. Further they applied
the hybridization of interval-valued fuzzy soft set in decision making problems. Tripathy et al. [63] presented a new
approach to interval-valued fuzzy soft sets using membership function approach. They also presented its application
in decision-making. Table 4 below summarizes the various
parameter reduction and decision making methods of hybrid
interval-valued fuzzy and soft sets discussed above.
C. INTUITIONISTIC FUZZY SOFT SET-BASED APPROACH
FOR PARAMETER REDUCTION AND DECISION MAKING
Subsequently, intuitionistic fuzzy soft set were also studied
to give more in the hybridization so as to achieve higher
efficiency in parameterization and decision making. Maji [64]
introduced new operation on intuitionistic fuzzy soft set based
on a combination of intuitionistic fuzzy set and soft set model
and studied its properties. Similarity measurement method
was used to made decision selection. A simple example was
used to show the feasibility of the method. Jiang et al. [65]
presented an adjustable approach to intuitionistic fuzzy soft
set based on decision making by using level soft sets of an
intuitionistic fuzzy soft set to extend the decision-making
approach. Decision criteria function was used as a threshold
on membership value and non-member ship value. It evidently forms the result of the experiment shows that the
adjustable feature makes the IFSS approach more appropriate
in many real life applications. Mao et al. [66] presented a
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TABLE 4. Parameter reduction and decision making of hybrid interval-valued fuzzy soft set.
group decision-making method based on Intuitionistic fuzzy
soft matrix and uses median and p-quantile to compute
threshold vectors. The authors introduced aggregate operators
in intuitionistic fuzzy soft matrix and establish a criterion
that the expert has a high weight if his evaluation value is
close to the mean value and small weight if it is far from
the mean value. An illustrative example demonstrates the
efficiency of the method. Das and Kar [67] proposed an
algorithmic approach on an intuitionistic fuzzy soft matrix to
investigate a specific disease that reflecting the agreement of
all experts. The approach is guided by group decision-making
model and uses cardinal intuitionistic fuzzy soft set. The
approach demonstrates its effectiveness in sample case study.
Das et al. [68] proposed an approach for making multiple
attribute group decision making problems. Using intervalvalued intuitionistic fuzzy soft matrix and confident weight
to the expert based on cardinals of intuitionistic fuzzy soft
set. The confident weight is assigned to each expert based
on their opinion. The result shows the effectiveness of using
weight assigning method. Deli and Çağman [69] proposed
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intuitionistic fuzzy parameterized soft sets for dealing with
uncertainties, based on both soft sets and intuitionistic fuzzy
sets. Thus, the decision acquired by using the operations
of intuitionistic fuzzy sets and soft sets that make this sets
useful and applicable in practically. The result of numerical
example demonstrated that the method is effective. Shanthi
and Naidu [70] proposed a similarity measure of hybrid
interval valued intuitionistic fuzzy set and soft set of root
type. Further, they applied it for decision making. Chen [71]
presented the inclusion-based technique for order preference
by similarity to ideal solution method with interval-valued
intuitionistic fuzzy sets. They also addressed multiple criteria group decision-making problems in the framework of
interval-valued intuitionistic fuzzy sets. Tripathy et al. [72]
introduced a hybrid intuitionistic fuzzy set and soft sets.
By using notion of a characteristic function, they applied it
for decision-making problem. Tripathy et al. [73] presented
another work on hybrid intuitionistic fuzzy set and soft sets
and its application in group decision making. Jia et al. [74]
presented a sequence of hybrid intuitionistic fuzzy soft sets
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and its related properties. Further they explored it for decision making problem. Karaaslan and Karataş [75] presented
algebraic operations i.e. OR and AND-products of intuitionistic fuzzy parameterized fuzzy soft sets. Furthermore,
they proposed a decision making method so-called ∧-aggr
based on intuitionistic fuzzy parameterized fuzzy soft sets.
Deli and Karataş [76] presented a hybrid interval valued
intuitionistic fuzzy parameterized set and soft set. By using
soft level sets, they constructed a parameter reduction method
to obtain a better decision making. They gave a numerical
example to show that the proposed hybrid method working
successfully for decision making problems containing uncertain data. Wu and Su [77] presented a hybrid group generalized interval-valued intuitionistic fuzzy soft sets which
contains the basic description by interval-valued intuitionistic
fuzzy soft set on the alternatives and a group of experts’
evaluation of it. Furthermore, a multi-attribute group decision
making model based on group generalized interval-valued
intuitionistic fuzzy soft sets weighted averaging operator is
built. It is employed to solve the group decision making
problems in the more universal interval-valued intuitionistic
fuzzy environment. Table 5 below summarizes the various
parameter reduction and decision making methods of hybrid
intuitionistic fuzzy and soft sets discussed above.
D. A HYBRID SOFT SET WITH FUZZY AND ROUGH SET
FOR PARAMETER REDUCTION AND DECISION MAKING
Another important aspect in parameter reduction and decision
making of a hybrid soft set is consolidating the soft fuzzy
rough set to tackle uncertainty and imprecision in decisionmaking Feng et al. [12] presented a hybrid model called rough
soft sets to provide the framework for consolidating fuzzy
sets, rough set, and soft sets. The study presents a preliminary
concept of hybridization of the soft fuzzy rough set which
could help researchers to establish whether the notion could
lead to a fruitful output or not. Hu et al. [78] developed a
new model called soft fuzzy rough sets (combination of soft
set, fuzzy and rough set) to reduce the influenced of noise
because datasets in a real-world application are contaminated
by noise. The membership is calculated by k th sample, where
k is determined by the tradeoff between the number of misclassified samples and the argumentation membership. The
result of their experiment conducted to test the robustness
of model in feature selection and evaluation have proved to
be effective. Meng et al. [79] discussed the combination of
fuzzy, rough and soft sets and present the new algorithm
soft rough set model. They have proven some theorems and
algebraic properties of fuzzy, rough and soft sets. Subsequently, the fuzzy soft set was employed to granulate the
universe of discourse. Ali [80] presented the concept of an
approximation space associated with each parameter in a soft
set and an approximation space associated with the soft set is
defined. The properties of the model are proved. Zhang [81]
introduced two concepts of reduct and exclusion which can
be used to find the reduct or exclusion of a set of parameters.
The authors use "smallest" fuzzy soft set that produces the
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same lower soft fuzzy rough approximation operators and the
same "upper" soft fuzzy rough approximation operators. As
such, the necessary and sufficient conditions were obtained
which demonstrate the effectiveness of their approach using
approximation approach. Zhang and Shu [82] presented a
generalization of a hybrid interval-valued fuzzy set and rough
set which is based on constructive and descriptive (axiomatic)
approaches. For constructive approach they employed an
interval-valued fuzzy residual implicator and its dual operator, meanwhile for descriptive approach, generalized intervalvalued fuzzy rough approximation operators are defined by
axioms. Finally, they illustrated the proposed model in decision making. Zhang et al. [83] presented a hybrid hesitant
fuzzy set and rough set over two universes. It is based on
a constructive approach and the union, the intersection and
the composition of hesitant fuzzy approximation spaces are
proposed, and some properties are also investigated. They
further applied the proposed hybrid hesitant fuzzy set and
rough set in decision making problem. Zhan et al. [84] presented a hybrid novel soft set with rough set and further they
combined with hemirings, so-called soft rough hemirings.
They use the proposed hybrid soft rough hemirings for corresponding multi-criteria group decision making. Table 6 below
summarizes the various parameter reduction and decision
making methods of hybrid soft set with fuzzy and rough sets
discussed above.
VI. REAL WORLD APPLICATIONS OF SOFT SETS
Following the previous successful existing set theories in
solving real world problems e.g. [85]–[87], soft set theory
has also been applied successfully in many areas. In order
to improve efficiency of parameter reduction in soft sets,
many approaches have been developed. This section shows
the various applications of soft set theory.
A. SOFT SET THEORY FOR ASSOCIATION RULES MINING
Soft set theory has been used for mining interesting association rules from transactional databases. Herawan and
Deris [88] proposed the idea of association rules mining
based on soft set approach. It uses the concept of cooccurrence of parameters in defining support and confidence of association rules from Boolean-valued transactional
dataset. Vo et al. [89] presented another approach for mining maximal association rules in text data using soft set
theory. They have shown that the proposed method effectively used for mining interesting association rules which are
not obtained by using methods for regular association rule
mining. Recently, Feng et al. [90] presented a new model
of soft set for association rule mining as an improvement
of [88]. They refined several existing concepts to improve the
generality and clarity of former definitions.
B. SOFT SET THEORY FOR MEDICAL ANALYSIS
From the domain of medical analysis, soft set theory has
been extensively explored. Chetia and Das [91] presented a
hybrid interval-valued fuzzy and soft sets. The author used
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TABLE 5. Parameter reduction and decision making of hybrid intuitionistic fuzzy soft set.
Sanchez’s approach for medical diagnosis in interval valued fuzzy environment for analyzing patients with fever and
malaria. Herawan [92] presented an application of soft set
based on decision making through a Boolean valued information system from the patient suspected with ILI (InfluenzaLike Illness). The application explore how the technique
can be used to reduce the number of dispensable symptoms
and make an optimal decision. The result shows that the
technique can reduce the complexity of medical decision
without loss of original information. Kumar et al. [93] presented a novel approach based on bijective soft sets for the
generation of classification rules from the data set. The novel
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approach showed to be a valuable tool as compared the wellknown decision tree classifier algorithm and Naïve bayes.
Yuksel et al. [94] applied soft set theory to a medical diagnosis, to reduce parameter of the unnecessary biopsies in
patients undergoing medical evaluation for prostate cancer.
Later, the doctor can calculate the percentage of prostate
cancer risk.If the recommendation of the soft expert system
for the risk percentage is greater than 50% then biopsy is
necessary. Lashari and Ibrahim [95] developed a framework
for medical images classification using soft set to improve
the physician ability to detect and analyzed pathologies to
achieve better performance in terms of accuracy, precision,
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TABLE 6. Parameter reduction and decision making of hybrid soft set with fuzzy and rough sets.
and computational speed. The result demonstrated that the
algorithm can help to improve the physician ability to analyze
and detect pathology. Celik and Yamak [96] presented an
application of a hybrid fuzzy soft set theory through wellknown Sanchez’s approach to medical diagnosis. The data is
taken form collection of patients with symptoms temperature,
headache, cough and stomach problem. Alcantud et al. [97]
presented a soft set-based decision making procedure for
glaucoma diagnosis. They used an automated combination
and analysis of information from structural and functional
diagnostic techniques in order to obtain an enhanced Glaucoma detection in the clinic.
C. SOFT SET THEORY FOR INCOMPLETE DATA ANALYSIS
From Proposition 1, it is shown that a ‘‘standard’’ soft set
is equivalent to a Boolean-valued information system. In
real world application, we sometime find incomplete data
table. There are existing approaches based on soft set theory
for handling incomplete Boolean-valued information system.
Zou and Xiao [98] initiated data analysis approaches of soft
sets under incomplete information. For standard soft sets, the
decision value of an object with incomplete information is
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calculated by weighted-average of all possible choice values
of the object, and the weight of each possible choice value
is decided by the distribution of other objects. Qin et al. [99]
presented a novel data filling approach for an incomplete soft
set as an improvement of [98]. It is based on the association
degree between the parameters when a stronger association
exists between the parameters or in terms of the distribution
of other available objects when no stronger association exists
between the parameters. Recently, Khan et al. [100] presented
an alternative data filling approach for prediction of missing
data in soft sets. The technique is based on reliability of association among parameters in soft set. They have shown that
the proposed technique achieves better accuracy as compared
to [98] and [99]. Alcantud and Santos-García [101] presented
two related techniques in analyzing incomplete soft sets as
new solutions for decision making problems.
D. SOFT SET THEORY FOR DATA MINING
This sub-section presents applications of soft set theory for
data mining and knowledge discovery. Mushrif et al. [102]
classified texture using soft-set theory based classification algorithm. Experimental results show the superiority
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TABLE 7. The summary of real world applications of soft set and hybrid soft sets.
of the proposed approach compared with some existing
methods. Xiao et al. [103] use forecasting accuracy as a
criterion for fuzzy membership function and proposed a
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combined forecasting approach based on fuzzy soft set. The
result of the approach improves forecasting performance
as compared to combining forecasting rough set theory.
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TABLE 7. The summary of real world applications of soft set and hybrid soft sets.
Senan et al. [104] developed the application of soft set theory
for feature selection of traditional Malay musical instrument
sounds. This approach was tested and the obtained features
of the propose model are more efficient compare to the
traditional method. Qin et al. [105] presented a novel soft
set approach in selecting clustering attribute. It is based
on their proposed technique with is equivalent to the same
concept in rough set approximation. They proved that the
proposed technique achieves lower computational time and
higher accuracy compared to rough set-based approach [106].
Handaga et al. [107] improved the soft set-based classification technique in [102] by proposing an algorithm for classifying numerical data which is based on a hybrid fuzzy soft set
theory. Mamat et al. [108] improved the complexity and accuracy of the attribute selection techniques by [105] and [106]
using maximum attribute relative of soft set. Wang et al. [109]
proposed an application oriented mobile cloud computing
AMCCM (Adaptive Mobile Cloud Computing Middleware).
The author, used cloud computing to combine the hardware
resources control with the application layer software services in the system to automatically adjust the hardware
resources and uses fuzzy soft set as a virtual startup problem. The results were found to be effective and successful.
Kumar et al. [110] classified large gene expression data
based on an improvement of bijective soft set. To show
the efficiency of the proposed model, they compared
the performance to fuzzy-soft-set-based classification algoVOLUME 5, 2017
rithms, fuzzy KNN, and k-nearest neighbor approach.
Maand Qin [111] applied a new efficient parameter reduction algorithm into a real life e-shopping in Blackberry
mobile phone dataset. The result of the experiment proves
that the NENPR is feasible for dealing with e-shopping.
Li and Xu [112] proposed a new method to measured airport
importance to seek out the most vulnerable nodes based
on fuzzy soft set. The result of experiment from the traffic data of the airport shows that the evaluation method
is accurate. Suhirmanet al. [113] presented an alternative
soft set-based approach based on maximum degree of domination for educational data mining. It uses for clustering student assessment datasets. Sutoyo et al. [114] used
multi-soft sets as a decomposition of multi-valued information system into many Boolean-valued information systems for conflict analysis. They proposed an alternative
approach for measuring support, strength, certainty and coverage of conflict situation. They showed through experimentation that their approach performs lower computational
time as compared to rough set-based conflict analysis model.
Khan, et al. [115] applied data analysis under incomplete
soft set for a virtual community detection. It is based on
the association between prime nodes in online social networks. Furthermore, they applied it to ranking algorithms
discussed above. Table 7 below summarizes the various realworld applications of soft set and hybrid soft sets discussed
above.
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In the following section, we conclude our review work and
highlights future researches.
VII. CONCLUSIONS AND FUTURE RESEARCH
Soft set is an alternative tool for dealing with uncertainty
problems. Its proficiency in dealing with uncertainty problems is as a result of its parameterized concept and its main
vehicle. In this work, we have presented and clarified the
voluminous works in this field within a short period of time
and several algorithms exist for parameter reduction, decision
making, applied research of soft set and hybrid soft set with
other set theories. We have reviewed different research on
parameter reduction and decision making in soft set theory
and hybrid soft set with other set theories. Several researchers
have contributed different type of algorithm for computing
parameter reduction by considering different cases like inconsistency, missing attribute value of decision making and information system. Researchers can use our review to quickly
identify areas that require improvement so as to proposed
novel approach. With the current advances in soft set theory,
a number of research issue is arising which requires attention
from researchers this include multiset in parameter reductions, multi-criteria decision making problem in an uncertain
environment. Moreover, it is clear that all the approaches
presented above so far in this area have their respective
advantages and disadvantages. This required attention from
researchers to develop more general approach for parameter
reduction of soft set. Moreover, the application of parameter
reduction of soft set to practical field has to be explored
more by researchers. Only few research work on extending
ontology based soft set were found in the literature. A very
limited practical application of soft set theory can be found
in existing literatures and many of them still facing problem
of performance of their models or algorithms. Thus, these
applied soft set-based researches also require more attention
to future soft setters.
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SANI DANJUMA received the B.Sc. degree from the Kano University of
Science and Technology, Wudil, and the M.Sc. degree from the Liaoning
University of Technology, China. He is currently pursuing the Ph.D. degree
in computer science with the University of Malaya. His research areas are
rough set soft set and data mining, and he has several publications.
TUTUT HERAWAN received the Ph.D. degree in information technology
from University Tun Hussein Onn Malaysia in 2010. He is currently an
Associate Professor with the Department of Information Systems, University
of Malaya. He is also a Principal Researcher with the AMCS Research Center, Indonesia. He has over 12 years of experience as an Academic and also
successfully supervised five Ph.D. students. He currently supervises 15 master’s and Ph.D. students and has examined master’s and Ph.D. theses. He has
edited five many books and published extensive articles in various book
chapters, international journals, and conference proceedings. His research
area includes soft computing, data mining, and information retrieval.
He delivered many keynote addresses, invited workshop, and seminars
and has been actively served as the Chair, a Co-Chair, a Program Committee Member, and a Co-Organizer for numerous international conferences/workshops. He has also guest edited many special issues in many
reputable international journals. He is an active Reviewer for various journals. He is the Executive Editor of the Malaysian Journal of Computer
Science (ISI JCR with IF 0.405).
VOLUME 5, 2017
S. Danjuma et al.: Review on Soft Set-Based Parameter Reduction and Decision Making
MAIZATUL AKMAR ISMAIL received the bachelor’s degree from the
University of Malaya (UM), Malaysia, the master’s degree from the University of Putra Malaysia, and the Ph.D. degree from UM. She has over
15 years of teaching experience. Since 2001, she has been a Lecturer with
the University of Malaya. He was involved in various studies, leading to
publication of a number of academic papers in the areas of Information
Systems specifically on Semantic Web and Educational Technology. She is
currently an Associate Professor with the Department of Information Systems, Faculty of Computer Science and Information Technology, UM. She
has been actively publishing over 40 conference papers in renowned local
and international conferences. A number of her works were also published
in reputable international journals. He has participated in many competitions
and exhibitions to promote her research works. She actively supervises at all
level of studies from undergraduate to Ph.D. To date, she has successfully
supervised three Ph.D. and numbers of masters’ students to completion. She
hopes to extend her research beyond Information Systems in her quest to
elevate the quality of teaching and learning.
HARUNA CHIROMA (M’14) received the B.Tech. degree in computer
science from Abubakar Tafawa Balewa University, Nigeria, the M.Sc. degree
in computer science from Bayero University Kano, and the Ph.D. degree in
artificial intelligence from the University of Malaya, Malaysia. He is currently a Faculty Member with the Federal College of Education (Technical),
Gombe, Nigeria. He has published articles relevance to his research interest
in international referred journals, edited books, conference proceedings, and
local journals. His main research interest includes metaheuristic algorithms
in energy modeling, decision support systems, data mining, machine learning, soft computing, human computer interaction, and social media in education, computer communications, software engineering, and information
security. He is a member of the ACM, NCS, INNS, and IAENG. He is currently serving on the Technical Program Committee of several international
conferences.
VOLUME 5, 2017
ADAMU I. ABUBAKAR (M’14) received the B.Sc. degree in geography, the
Pg.D. degree, the M.Sc. degree, and the Ph.D. degree in computer science.
He is currently an Assistant Professor with the International Islamic University Malaysia, Kuala Lumpur, Malaysia. He has authored over 80 articles
relevance to his research interest in international journals, proceedings, and
book chapters. His current research interest includes information science,
information security, intelligent systems, human computer interaction, and
global warming. He is a member of the ACM. He served in various capacities
in international conferences and currently a member of Technical Program
Committee of several international conferences. He received several medals
in research exhibitions.
AKRAM M. ZEKI received the Ph.D. degree in digital watermarking from
University Technology Malaysia. He is currently an Associate Professor
with the Information Systems Department, International Islamic University
Malaysia. His current research interest is on watermarking.
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