3.1. Construct Knowledge Measure with Distance for IvIFSs
In this section, we use the idea of TOPSIS to construct the knowledge measure for IvIFSs. Firstly, we refer to [
20], which defined the two positive ideal solutions
and
, as well as the negative ideal solution
. Secondly, the distances from a single element
to positive and negative ideal solutions are calculated, and the parameter
is introduced by considering the proportion of positive and negative ideal solutions and the influence of human factors, thereby constructing the total expression of the knowledge measure for IvIFSs.
Definition 6. Let . The function is called the knowledge measure for based on the distance D if K meets the following conditions:
- (K1)
is a crisp set;
- (K2)
;
- (K3)
contains more knowledge than , i.e., , where ;
- (K4)
.
Theorem 1. A mapping is called the knowledge measure of an IvIFS and can be defined bywhere , . Here, θ refers to people’s subjective attitude coefficient, represents the distance between the single element and negative ideal solution , and and denote the distances between and positive ideal solutions and , respectively. Proof. Next, we only need to prove that satisfies the conditions (K1–K4) in Definition 6.
Referring to the definition of the distance measure, it is easy to have
and then we obtain
This means that or . Thus, is a crisp set.
From the perspective of the definition and distance of an IvIFS,
It is obvious that
. Then, we obtain
The derivative of Equation (
10) with respect to
is
The derivative of Equation (
10) with respect to
is
The derivative of Equation (
10) with respect to
is
Then, we find that increases monotonically with respect to , and decreases monotonically with respect to , .
If
contains more knowledge, we have
Finally, we obtain .
According to the definition of the complement of the IvIFS
, i.e.,
we can deduce
.
Let
; the distance in Equation (
10) is the Hamming distance, and then the distances between the IvIFS
and the positive and negative ideal solutions are as follows:
Thus, we easily obtain
where
,
When a new formula for the knowledge measure was constructed using the Euclidean distance, it was also necessary to calculate distances between
and the positive and negative solutions as follows:
We can obtain
where
. □
3.2. Construct Entropy Measure with Distance for IvIFSs
Entropy is an important tool for measuring the uncertainty of fuzzy variables and processing fuzzy information. An IvIFS is used to describe a set whose elements cannot be clearly defined in terms of whether they belong to a given set. In this section, we mainly take the advantage of the distance and abstract function to construct the entropy of IvIFSs.
Theorem 2. Let be a strictly monotonically decreasing real function. A mapping is called the entropy of an IvIFS and is defined as: Proof. For any , we shall demonstrate that satisfies the four conditions in Definition 5.
From the definition of the entropy
and the distance
, it is obvious that
and then we can obtain
or
. Thus,
is a crisp set.
According to the definition of distance, it is evident that
and therefore,
.
Since the function
is strictly monotonically decreasing, it is easy to understand that if
then
. Thus, we deduce
.
According to the definition of the complement of the IvIFS
, i.e.,
it is clear that
is obtained.
It is noteworthy that different formulas can be developed through theorems to calculate entropy measures between different strictly monotonically decreasing functions, such as ; .
Let
and
D be the above-mentioned distance measure equations between IvIFSs; then, for any
, the entropy of an IvIFS is as follows:
It is not difficult to find that Equation (
14) is consistent with the entropy defined in reference [
7]. Therefore, the total entropy expression in Equation (
13) proposed in this paper can not only cover the entropy constructed by other studies in the literature but also derive more entropies for IvIFSs. For example, let
and the distance in Equation (
13) be the Hamming distance; then, the specific expression is as follows:
Next, we compare our proposed entropy E and knowledge measure K with the existing entropy measures , , , , , , and and knowledge measures based on a specific numerical example. □
Example 1. We consider six single-element IvIFSs given byto compare calculations between the recalled entropy measures from Equations (1)–(9) and the proposed measure from Equation (11) when and according to Equation (15). The calculated results of the specific measures are summarized in Table 1. It is worth noting that the results of , , , , and are the same for two different sets, and , and there is a counterintuitive situation. In addition, the calculation result of for sets and , as well as , and , are still the same, so the entropy fails in this case. At the same time, has the same results for and . However, there is no counterintuitive case for the calculation results of E, and K, so they show their superiority over other entropy formulas. It is not difficult to see that our knowledge measure is consistent with the calculation results of , which illustrates the effectiveness of the entropy and knowledge measures proposed in this paper.
Example 2. We have chosen the real cloud service selection problem from reference [26]. Suppose there are four cloud service alternatives: SAP Sales on Demand , Salesforce Sales Cloud , Microsoft Dynamic CRM and Oracle Cloud CRM . The evaluation of the four schemes by experts needs to take into account the following five attributes: performance , payment , reputation , scalability and security . The experts’ comprehensive evaluation of the five attributes is listed in Table 2. The weighting vector of is . The goal is to select the best cloud services. The attribute are beneficial, so the attribute values in R do not need to be normalized.
The interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operator referred to in [
2] is used to set the comprehensive evaluation value
by aggregating the individual evaluation values
with the attribute weight vector
:
Based on the overall collective evaluation value
of the alternatives, we calculate the knowledge measure of IvIFSs by using Equation (
11) when
. Thus, we obtain:
These knowledge measures of IvIFSs are ranked as follows: . Therefore, according to the knowledge measure, we can derive the ranking order of the alternatives: . Thus, the best alternative is —Oracle Cloud CRM.