In the introduction, it has been stated the importance of the aggregation process for decision making processes and how a generic aggregation framework could be applicable to a wider set of decision making contexts through the modeling and interpretation of the attached parameters of the aggregation model, as per the demand of the decision context. In this section, we propose extending the IOWA operator in order to deal with ELICIT information. However, we do not aim to make a simple and straightforward extension of the classical IOWA operator; we intend to develop suitable and necessary models of the IOWA operator for dealing with ELICIT information in decision making. Therefore, our proposal consists of defining the following operators and the order inducing variable:
3.1. Elicit-Iowa Operator
Here, we will replace the argument variable
of the OWA pair
[
27] with ELICIT expression
, and then the aggregation of the new OWA pairs
can be defined as follows:
Definition 5. Let be the set of all possible ELICIT expressions and Ω be the set that is equipped with a linear order. An ELICIT-IOWA operator of dimension n is a mapping that has an associated weighting vector of dimension n with and , then where × represents the multiplication between the scalar and fuzzy numbers. Let be an index function, such that is the index of the argument variable (see Definition 1 and Example 1), which is paired with the largest element of the set . is the equivalent trapezoidal fuzzy number corresponding to the ELICIT expression and ζ is the inverse function of .
The ELICIT-IOWA operator is also the OWA based operator, so that the weight can be obtained from a basic unit-interval monotonic (BUM) function [
28]
, which is characterized as: (i)
; (ii)
; (iii)
; then, the weight
In the IOWA situation, there is a straight forward extension of the OWA pair
with an additional variable, the so-called weight variable
[
35] that is the associated weight of each argument
, noted as a triple
. The weight variable
helps to obtain the crisp weight that is attached to the ordered position of the ELICIT-IOWA operator, which is called the functional generated weight [
35], and this crisp weight
can be computed as follows:
where
. Additionally, let
be an index function, such that
is the index of the argument variable
and the weight variable
, which are paired with the
largest element of the set
.
Nevertheless, if the weight variable is added to OWA pairs of the ELICIT-IOWA operator to obtain triples , and the BUM function Q is used, then we will obtain the definition of the important ELICIT-IOWA (ELICIT-I-IOWA) operator. That is:
Definition 6. Let be the set of all possible ELICIT expressions and Ω be the set that is equipped with a linear order. An ELICIT-I-IOWA operator is a mapping that has an associated weighting vector of dimension n with and computed by the BUM function Q and the ELICIT-I-IOWA operator is defined as follows: where within . Additionally, × represents the multiplication between scalar and fuzzy numbers. Let be an index function such that is the index of the argument variable and the weight variable, which is paired with the largest element of the set . is the equivalent trapezoidal fuzzy number corresponding to the ELICIT expression and ζ is the inverse function of .
Noticeably, if we apply the ELICIT-I-IOWA operator with the BUM function
, then we will obtain the functional weight as:
and
hence, it is the weighted average (WA) operator for ELICIT information.
Remark 1. Taking into account the fuzzy induced quasi-arithmetic OWA (QFIOWA) operator [36], which is a mapping that has associated weights , such that and where Ω is the set equipped with a linear order and Ψ is the set of all fuzzy numbers. g is a strictly continuous monotonic function.
If , then the QFIOWA is the fuzzy induced generalized OWA (FIGOWA) operator, defined as In a nutshell, because the ELICIT expression is equivalent to a trapezoidal fuzzy number through the function , i.e., , where is the set of all possible ELICIT expressions and Ψ is the set of all fuzzy numbers, if we add a strictly continuous monotone function g to the ELICIT-IOWA operator, the quasi-ELICIT-IOWA (ELICIT-QIOWA) operator will be defined as follows: Furthermore, let g be an exponential function where with parameter , then the quasi-ELICIT-IOWA operator is the generalized ELICIT-IOWA (ELICIT-GIOWA) operator and Several special cases, depending on the various values of the parameter λ, are as follows:
- (1)
If , then the ELICIT-GIOWA operator reduces to the ELICIT-IOWA operator;
- (2)
- (3)
If , then the ELICIT-GIOWA operator is close to the Harmonic average as - (4)
- (5)
Remark 2. Some special cases of the ELICIT-IOWA operator by using different crisp weighting vectors W are briefly detailed.
- (1)
If , then - (2)
If , then - (3)
If , then It is the same as the fuzzy arithmetic mean operator that was introduced in [15];
3.2. Elicit-T1-Iowa Operator
Inspired by the type-1 OWA operator, which extends the OWA operator [
30] to the type-1 fuzzy sets on the basis of Zadeh’s extension principle [
12,
13,
14]. We will then follow the definition of the type-1 OWA operator that is based on the IOWA operator [
27] in order to obtain the induced type-1 OWA (t1-IOWA) operator as follows:
Definition 7. Let be the set of all type-1 fuzzy sets defined on the universe of discourse and Ω be the set that is equipped with a linear order. An t1-IOWA operator of dimension n is a mapping that has an associated weighting vector of dimension n with fuzzy weights , such that where ⊗ is the multiplication operation on fuzzy numbers and ∗ is a t-norm operator. Let be an index function, such that is the index of the argument variable, which is paired with the largest element of the set .
According to the predefined, if the t1-IOWA operator has an associated interval weight
; in other words, the membership of the interval weight equals 1, i.e.,
, we will easily obtain this formula
Therefore, the previous definition can be shown as follows:
Definition 8. An t1-IOWA operator of dimension n is a mapping that has an associated weighting vector of dimension n with interval weights and , such that where ⊗ is multiplication operation on fuzzy numbers and ∗ is a t-norm operator. Let be an index function, such that is the index of the argument variable, which is paired with the largest element of the set .
It is very complicated to calculate with the extended principle, because we need to discretize the domain of the fuzzy set. Therefore, we rely on
-cut based method to solve this problem, which is more computationally efficient and provide better approximations. To do this, we will follow the concept of the
-level type-1 OWA operator [
37] guided by
-cut of interval weights to obtain the definition of the
-level t1-IOWA operator with fuzzy weights as follows:
Definition 9. Let represents the -cut of fuzzy sets , and then denote the corresponding OWA pairs as . Let represent the -cut of fuzzy weights . For each , an -level t1-IOWA operator is defined as , such that where and . Let be an index function, such that is the index of the argument variable, which is paired with the largest element of the set .
The proposed t1-IOWA operator provides a basic IOWA aggregation framework for aggregating fuzzy information. Because each ELICIT expression is equivalent to a fuzzy number, we can define the aggregation of ELICIT expressions based on the t1-IOWA operator framework that we proposed. For this purpose, we replace the fuzzy parameter variable with the ELICIT expression. Therefore, we will extend the t1-IOWA operator in order to aggregate ELICIT information, where the fuzzy weight is induced by the type-2 linguistic quantifier [
32].
Definition 10. Let be the set of all possible ELICIT expressions and Ω be the set that is equipped with a linear order. An ELICIT-t1-IOWA operator of dimension n is a mapping that has an associated weighting vector of dimension n with fuzzy weights , such that where ⊗ is the multiplication operation on fuzzy numbers. Let be an index function, such that is the index of the argument variable, which is paired with the largest element of the set . is the equivalent trapezoidal fuzzy number that corresponds to the ELICIT expression and ζ is the inverse function of .
Similar to the crisp weight that was computed by the linguistic quantifiers
Q reviewed in
Section 2.2, the fuzzy weights are expressed by a fuzzy number to indicate higher uncertainty. Therefore, the fuzzy weights are computed by type-2 linguistic quantifiers
[
32] that are based on the type-2 fuzzy sets [
38]. Type-2 linguistic quantifier
guided the type-1 OWA operator by the fuzzy weight
. Especially, if the secondary membership function is equal to 1, the linguistic type-2 quantifier is called the interval-valued type-2 linguistic quantifier, and it then computed the interval weight as
where
is the primary membership of the variable
for all
.
Figure 2 depicts the footprint of uncertainty (FOU) of the interval-valued type-2 linguistic quantifier “
most”.
As aforementioned, the fusion of the ELICIT information is equivalent to the fusion of the trapezoidal fuzzy number and it considers the general mathematical multiplication and division of trapezoidal fuzzy numbers that are not trapezoid preserving [
34], which is not conducive to completing the ELICIT-CW approach. Therefore, it needs an approximation process to obtain the trapezoidal fuzzy number before the last re-translation process. Inspired by the type-1 OWA operator [
32,
37], we shall apply the EKM algorithm [
39,
40] upon the
-cut to implement the ELICIT-t1-IOWA operator with interval weight
. After that, we will obtain an approximated result of
.
For sake of clarity, let
and interval weight
is computed by the interval-valued type-2 linguistic quantifier “
most”, we shall compute
utilizing idea of the EKM algorithm [
39,
40] and
set theory.
In order to do this, simplify the process as the following one.
Step 1: Initialization
- (1)
Given the interval weights with for are two endpoints of the interval weight and ordered fuzzy arguments that are defined on the domain for all .
- (2)
For simplified the process, let
be a trapezoidal fuzzy number as
, and then the
-cut of them as:
and
- (3)
Let
:
where
and
are two endpoints of
.
Additionally, represent the lower and upper endpoints of interval , respectively.
- (4)
Let be a two tuple and let be a permutation that only acts on the first item of the two tuple, such that , then be the reordered two tuple with is the smallest elements in the set . It is the same to deal with . It helps to more easily implement the EKM algorithm.
Step 2: To obtain the initial lower bounded of
- (1)
Set
[
39,
40] (the nearest integer to
) and compute:
- (2)
Find
, such that
- (3)
Check if . If yes, stop and set . If no, go to Step 2 (4);
- (4)
compute
, and
and go to Step 2(2);
Step 3: To obtain the initial upper bounded of
- (1)
Set
[
39,
40] (the nearest integer to
) and compute:
- (2)
Find
, such that
- (3)
Check if . If yes, stop and set . If no, go to Step 3 (4);
- (4)
compute
, and
and go to Step 3(2);
Step 4: To obtain the final result , A and ;
where
and
.
Step 5: To execute an approximation of A as a trapezoidal fuzzy number
According to literature [
34,
41] regarding the operation of trapezoidal fuzzy numbers, we know that the result of
A is a generalized trapezoidal fuzzy number, so we need to approximate it with trapezoidal fuzzy number
.
Let
, then we will obtain
and
Let
, and then we can obtain
and
When considering that the area of the trapezoidal membership function graph can represent the size of the information carried, then
, where
is the granularity of the linguistic term set
S used for the ELICIT expression, such that
with
Let
, since
, then
Because
and
are defined on
, then
and
Therefore,
, such that
In general, the approximation of A is the trapezoidal function .
3.3. The Order Inducing Variable in the Form of Elicit Expression
Yager and Filev [
27] introduced the idea of obtaining the order inducing variable, which is,
is any target set containing linear order, which can be shown in detail as follows:
More generally, we see, that if Ω is any set of objectives such that there exists a linear ordering on Ω, for any distinct , then either or , but not both, then we can draw our the order inducing variable value from Ω.
As the name stipulates, the order inducing variables are used to reorder the argument variables, when considering that the arguments have no inherent order. There are various types of the order inducing variable, for instance, for is the set of all real numbers and it has an incontrovertible ordering, is the implicit lexicographic order and, even while using the linguistic term set , where is for , etc. However, and are not often chosen, where is the set of all possible ELICIT expressions, and is the set of all fuzzy numbers. The reason is that, when compared with , there is no standard ranking method for fuzzy numbers to obtain undisputed ordering results and, considering that, on the basis of the function , can be transformed into a subset of , i.e., . Therefore, we prefer to choose , which contains an inherent order.
There is much literature on fuzzy number ranking, but different fuzzy number ranking methods will provide different ranking results. Based on the work of Wang and Kerre [
42,
43], the ranking of fuzzy numbers can be roughly divided into the following three categories:
- (i)
the defuzzification-based method, for instance, centroid index (CI) method [
26,
44], the area method, so-called the magnitude function [
45], and so on;
- (ii)
method that measuring of the distance to the reference set [
42,
46,
47]; and,
- (iii)
pairwise comparison method [
43,
48,
49].
The purpose of all these methods is to obtain a numerical scale for ranking fuzzy numbers. Because there are many fuzzy numbers, such as interval fuzzy numbers, triangular fuzzy numbers, trapezoidal fuzzy numbers, Gaussian fuzzy numbers, and L-R type, etc., the order inducing variables only need to select one of the fuzzy numbers and one ranking method, and then the reordering of the argument variables is determined.
Inspired by the concept of numerical scale function
in [
50], in which the function
defines a numerical scale on the linguistic term set
. When assuming that the OWA pair belongs to the Cartesian Product set of dimension
n,
, is denoted as
, then the
can be obtained by using
[
16] to define the vertex of the triangle fuzzy numbers
without considering the adjustment parameter
[
15], where
is the granularity of the linguistic terms set
for all
, as proposed in [
16]. The definition is the following one:
Definition 11. Let be the set of all possible ELICIT expressions, then there is a mapping that is defined by the middle position of vertexes of ELICIT expressions, such that where is a set of linguistic terms and is the granular of the set S.
More generally, a mapping
can be defined as an extension of the numerical scalar function
as the following formula:
Example 3. Assume that OWA pairs under the linguistic terms set are noted as , then Thence, the decreasing sequence of the order inducing variable is , so that the reordered argument is .