Fuzzy Soft Expert Matrix Theory and Its Application in Decision-Making ()
1. Introduction
Most real-world problems in economics, social science, and the environment, for example, are fraught with uncertainty. Soft set theory [1] was initially given by Molodtsov in 1999 as a generic mathematical tool for dealing with ambiguous, fuzzily described, and uncertain things. Maji et al. [2] later investigated the notion of a fuzzy soft set. Fuzzy soft set theory can be widely used to solve decision-making problems [3] [4] . Majumdar et al. [5] extended the idea of fuzzy soft sets. Soft sets were extended to intuitionistic fuzzy soft sets by Maji et al. [6] . Matrices are useful in many fields of research and engineering. However, standard matrix theory does not always answer issues containing uncertainty. Yong et al. developed a matrix form of a fuzzy soft set and used it for particular decision-making issues [7] . Borah et al. expanded fuzzy soft matrix theory and its application in decision-making [8] . Cagman and Enginoglv [9] described fuzzy soft (fs) matrices and operations, as well as a fs-max-min decision-making approach.
Even though past models were effective, they frequently only engaged one expert. Many methods, such as joint and crossover, must be followed if you want to embrace the perspectives of many specialists. As a result, the user encounters challenges. To overcome this issue, Alkhazaleh and Salleh [10] [11] proposed the concepts of soft expert sets and fuzzy soft expert sets. The user may see all of the experts’ opinions in one model without any alteration. Even after any changes, the user may still access all expert perspectives. Serdar and Hilal [12] altered fuzzy soft expert sets in ways that were critical for the expansion of the concept of soft sets by eliminating their contradictions.
By combining matrices with fuzzy soft expert sets, we presented fuzzy soft expert matrices and specified several forms of fuzzy soft expert matrices as well as certain operations in this paper. Finally, we broaden our approach by applying these matrices to decision-making situations.
2. Preliminaries
In this section, we will review several fundamental concepts from fuzzy soft set theory, fuzzy soft matrix theory, and fuzzy soft expert theory.
Definition 2.1 Let
be a universe set,
a parameter set,
denote the power set of
and
. A pair
is called a soft set over
, where
is a mapping
.
In other words, a soft set over H is a parameterized family of subsets of the universe
. For
,
may be considered as the set of ε-approximate elements of the soft set
.
Definition 2.2 Let
be an initial universal set, and let
be a set of parameters. Let
denote the power set of all fuzzy subsets of
. Let
. A pair
is called a fuzzy soft set over
where
is a mapping given by
.
Definition 2.3 Let
be a fuzzy soft set over H, where
and
, for
and
, there exists the membership degree
, then we can present all membership degrees by a table as follows:
The fuzzy matrix
is said to be the fuzzy soft matrix of
over
. The set of all
soft matrices over
will be denoted by
.
Example 2.1 Let
,
and
be a fuzzy soft set over
given by
.
Hence the corresponding fuzzy soft matrix
is written by
.
Definition 2.4 Let
be a universe,
a parameters set,
an experts set and
an opinions set. Let
and
. Then
is said to be a fuzzy soft expert set over
, where
is a mapping given by
, where
denotes the power set of
.
Example 2.2 Let
be a universe set,
a parameters set,
be an experts set and
. Define the function
as follows:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
Then
is consists of the following approximate sets:
Definition 2.5 An agree-fuzzy soft expert set
which is also a fuzzy soft expert subset of
over
is defined by in the following
, where
such that
.
Definition 2.6 A disagree-fuzzy soft expert set
which is a fuzzy soft expert subset of
over
is defined by in the following
, where
such that
.
3. Several Matrix Types of Soft Expert Matrices
The definition of a fuzzy soft expert matrix is presented first in this section. Following that, many varieties of fuzzy soft expert matrices are introduced.
Definition 3.1 Let
be a universe,
a parameters set,
an experts set and O = {1 = agree, 0 = disagree} an opinions set. Let
and
.
is a fuzzy soft expert set over
, where
is a mapping given by
. Then the expert matrix of fuzzy soft expert set
is denoted as
(1)
where
,
represents the level of acceptance of
in the soft expert set
,
represents the level non-acceptance of
in the soft expert set
.
Example 3.1 Consider Example 2.2. With three experts making the decision, three fuzzy soft expert matrices can be obtained.
,
,
.
Definition 3.2 A fuzzy soft expert matrix
is called absolute disagree fuzzy soft expert matrix, if all of its elements are
.
Example 3.2 Consider
,
,
,
,
,
.
Then the corresponding fuzzy soft expert matrix
is written by
.
Definition 3.3 A fuzzy soft expert matrix
is called absolute agree fuzzy soft expert matrix if all of its elements are
.
Example 3.3 Consider
,
,
,
,
,
.
Then the corresponding fuzzy soft expert matrix
is written by
.
Definition 3.4 Let
be a fuzzy soft expert matrix, then
(i)
is called a fuzzy soft expert rectangular matrix, if
.
(ii)
is called a fuzzy soft expert square matrix, if
.
(iii)
is called a fuzzy soft expert row matrix, if
.
(iv)
is called a fuzzy soft expert column matrix, if
.
(v)
is called a fuzzy soft expert diagonal matrix, if
and
, for all
.
(vi)
is called a fuzzy soft expert lower triangular matrix, if
,
, for all
.
(vii)
is called a fuzzy soft expert upper triangular matrix, if
,
, for all
.
4. Several Operations of the Soft Expert Matrices
In this section, operations such as addition, subtraction, product, complement, scalar multiple, transpose and trace of fuzzy soft expert matrices are given. Some related properties are also presented.
Definition 4.1 Let
and
be two fuzzy soft expert matrices, then
is a fuzzy soft expert sub-matrix of
, denoted by
, if
(2)
Example 4.1 Let
and
be two fuzzy soft expert matrices as follows:
,
.
Then
.
Definition 4.2 Let
and
be two fuzzy soft expert matrices, then
is a proper fuzzy soft expert sub-matrix of
, denoted by
, if
and
, for at least one term
,
,
.
Definition 4.3 Let
and
be two fuzzy soft expert matrices, then
is equal to
if
,
,
.
Proposition 4.1 Let
,
and
be three fuzzy soft expert matrices, then
(i)
,
(ii)
,
(iii)
,
(iv)
,
(v)
.
Definition 4.4 Let
and
be two fuzzy soft expert matrices, then
is defined as the addition of
and
, where
. (3)
Example 4.2 If
and
are two fuzzy soft expert matrices as follows:
.
Then
.
Proposition 4.2 Let
,
and
be three fuzzy soft expert matrices, then
(i)
,
(ii)
,
(iii)
,
(iv)
,
(v)
.
Definition 4.5 Let
and
be two fuzzy soft expert matrices, then
is defined as the subtraction of
and
, where
. (4)
Example 4.3 If
and
are two fuzzy soft expert matrices as example 4.2, then
.
Proposition 4.3 Let
,
and
be three fuzzy soft expert matrices, then
(i)
,
(ii)
,
(iii)
,
(iv)
,
(v)
,
(vi)
,
(vii)
.
Proof: Proof (vii) only, the others may be similarly certified.
Let
,
,
, then
,
,
,
,
Thus
.
Definition 4.6 Let
and
be two fuzzy soft expert matrices, then we define the product of
and
as
, where
. (5)
Example 4.4 Let
and
be two fuzzy soft expert matrices as follows:
,
.
Then
.
Remark 4.1 When
exists,
does not necessarily exist. Even if both
and
exist,
and
may not be equal.
Proposition 4.4 Let
,
and
be three fuzzy soft expert matrices, then
. (6)
Proposition 4.5 Let
,
and
be three fuzzy soft expert matrices, then
(i)
,
(ii)
.
Definition 4.7 Let
be a fuzzy soft expert matrix, where
. Then the complement of the matrix is denoted by
, where
. (7)
Example 4.5 Let
be a fuzzy soft expert matrices as follows:
.
Then
.
Proposition 4.6 Let
and
be two fuzzy soft expert matrices, then
(i)
,
(ii)
,
(iii)
,
(iv)
,
(v)
,
(vi)
,
(vii)
.
Definition 4.8 Let
be a fuzzy soft expert matrix, then the scalar multiple
is defined by
, where
.
Example 4.6 Let
be a fuzzy soft expert matrix as follows:
.
Then the scalar multiple of this matrix by a scalar is
.
Proposition 4.7 Let
and
be two fuzzy soft expert matrices, if
are two scalars such that
, then
(i)
,
(ii)
.
Definition 4.9 Let
be a fuzzy soft expert matrix, then the transpose matrix of
denoted by
is also a fuzzy soft expert matrix.
Example 4.7 Let
be a fuzzy soft expert matrix as follows:
.
Then the transpose matrix of
is
.
Proposition 4.8 Let
and
be two fuzzy soft expert matrices, if
is a scalar such that
, then
(i)
,
(ii)
,
(iii)
,
(iv)
,
(v)
,
(vi)
.
Definition 4.10 Let
be a fuzzy soft expert matrix, where
. Then the trace of
is defined as
. (8)
Example 4.8 Let
be a fuzzy soft expert matrix as follows:
.
Then the
trace of this matrix is
.
Proposition 4.9 Let
be a fuzzy soft expert matrix,
be a scalar such that
. Then
(i)
,
(ii)
.
5. Fuzzy Soft Expert Matrix Theory in Decision-Making
In this part, we define the value matrix, the scoring matrix, and the total score in relation to the fuzzy soft expert matrix. The fuzzy soft expert matrix theory is then applied to a decision-making situation.
Definition 5.1 Let
be a fuzzy soft expert matrix, then
(9)
is defined as the value matrix of
,
;
.
Definition 5.2 Let
and
be two fuzzy soft expert matrices, then
(10)
is defined as the score matrix of
and
.
Definition 5.3 Let
and
be two fuzzy soft expert matrices,
,
the corresponding value matrices, their score matrix is
. Then the total score for each
in
is
. (11)
Example 5.1 Consider a business that is prepared to invest in a factory in another nation. There are six additional addresses
available after the first discussions. Four decision factors
are chosen after thorough analysis. The parameters
stand for reasonable prices, easy access to transportation, a healthy environment, and sound public policy, respectively. The business determined that three experts
would make up a group of committee members to find the best candidate for the decision in order to make a fair selection. Let O = {1 = agree, 0 = disagree} an opinions set,
.
The committee members may use the following algorithm.
Step 1: input the fuzzy soft expert set
.
Following a thorough debate, the committee may develop the following fuzzy soft expert set
:
Step 2: find the fuzzy soft expert matrices of
and find the complement of these matrices, respectively.
The fuzzy soft expert matrices of
are
,
,
.
And the complement of these matrices are
,
,
.
Step 3: compute the addition of the fuzzy soft expert matrices
and
.
,
.
Step 4: compute the value matrices and the score matrix.
,
,
.
Step 5: compute the total score
for each
in
. The decision will select with highest score. If more than one maximum value is present, these can be used as reference options.
Now, the score of
can be computed by using the score matrix in step 4:
,
,
,
,
,
.
Because of
, then
can be used as optimal location.
6. Conclusion
The fuzzy soft expert theory is employed in a wide range of domains, from theory to practice. In this article, we define fuzzy soft expert matrices as matrix representations of fuzzy soft expert sets. Then, in order to obtain some fresh conclusions, many types of fuzzy soft expert matrices are introduced, and multiple processes are established. On this premise, a decision model is developed, and an example of its application is presented. This approach is based on the notion of fuzzy soft expert matrix operations, which are simple to implement. Each expert’s opinion can be expressed very explicitly in a matrix, and the decision result can be reached through some calculations. Therefore, this decision-making method is simpler and more feasible than previous methods.