Some remarks on axiomatic definition of entropy measure

1 Introduction

In the general case, information imprecision is composed of ambiguity and indistinctness [e.g. 12]. Information ambiguity is interpreted as a lack of clear recommendation for one alternative among various given. We interpret information indistinctness as a lack of explicit distinction of given information and its negation. Among other things, an increase of information imprecision decreases its usefulness. This leads to a problem of indistinctnees evaluation.

The basic model for imprecise information is the concept of fuzzy set. Colloquially understood, the entropy measure of fuzzy set can be defined as a mapping that increases with the increase in fuzziness of evaluated set. Thus, information indistinctness can be calculated by means of entropy measure [3].

The relation “is less fuzzy than” is interpreted as an inequality fulfilled by membership functions. This inequality is systematically repeated across the literature.

The main goal of this paper is to reconsider an answer to the following question: What does it mean that one set “is less fuzzy than” the other? Obtained conclusions will be applied in defining the entropy measure in a more precise way.

2 Previous definitions of entropy measure

Let 𝕏 be an arbitrary space. The family of all crisp subsets of space 𝕏 is denoted by B ( 𝕏 ) . Symbol F ( 𝕏 ) indicates a family of all fuzzy subsets in 𝕏 . Each fuzzy subset B ∈ F ( 𝕏 ) ) is described by its membership function μ A : 𝕏 → [ 0 , 1 ] . In terms of multi-valued logic, the value μ _A  ( x ) is interpreted as the truth value of a sentence x  ∈  A . Basic set theory operations and relations are defined in the usual way, suggested by Zadeh [28].

For any fuzzy subset, its ambiguity is evaluated by an energy measure d : F ( 𝕏 ) → [ 0 ; 1 ] defined in general case [17] as a function fulfilling the conditions d ( ∅ ) = 0 ,(1) d ( A ) ≤ d ( 𝕏 ) = 1 ,(2) A ⊂ B ⇒ d ( A ) ≤ d ( B ) .(3)

In the original paper, in place of the condition (2), de Luca and Termini have only assumed that the energy measure is finite. A more detailed definition of energy measure depends on the properties of space 𝕏 . The following types of energy measures are most often used in practical applications.

In case of 𝕏 = { x i } i = 1 n d ( A ) = 1 n · ∑ i = 1 n μ A ( x i ) .(4)

In case of 𝕏 = { x i } i = 1 + ∞ d ( A ) = lim n → + ∞ ∑ i = 1 n μ A ( x i ) 1 + ∑ i = 1 n μ A ( x i ) .(5)

If -∞ < a < b < + ∞ then for the case of ( a , b ) ⊆ 𝕏 ⊆ 〈 a , b 〉 d ( A ) = ∫ a b μ A ( x ) dx b - a .(6)

In case of 𝕏 = ℝ d ( A ) = lim y → + ∞ ∫ - y y μ A ( x ) dx 1 + ∫ - y y μ A ( x ) dx .(7)

In addition, an energy measure can be used to determine a metric on the family F ( 𝕏 ) . In correspondence with suggestions given in [11], this metric is defined as a Hamming’s distance δ : F ( 𝕏 ) × F ( 𝕏 ) → [ 0 ; 1 ] given by the identity δ ( A , B ) = d ( A ∪ B ) - d ( A ∩ B ) .(8)

Therefore, value ς ( A , B ) = 1 - δ ( A , B )(9) can be interpreted as a degree of similarity between compared sets A , B ε F ( 𝕏 ) . The subsethood measure γ : F ( 𝕏 ) × F ( 𝕏 ) → [ 0 ; 1 ] is defined [14] by the identity γ ( A , B ) = d ( A ∩ B ) d ( A ) .(10)

The value γ (A, B) is interpreted as a degree of “fuzzy inclusion” of the set A ε F ( 𝕏 ) is in the set B ε F ( 𝕏 ) .

De Luca and Termini [16] have proposed the following axiomatic definition of entropy measure:

Definition 1. Any mapping e : F ( 𝕏 ) → [ 0 ; 1 ] fulfilling the conditions A ε B ( 𝕏 ) ⇒ e ( A ) = 0 ,(11) A = A C ⇒ e ( A ) = 1 ,(12) A ε F ( 𝕏 ) is less fuzzy than B ε F ( 𝕏 ) ⇒ e ( A ) ≤ e ( B ) ,(13) e ( A ) = e ( A C ) .(14) is called an entropy measure. □

What does it mean that set “is less fuzzy than” the other? In [17] it was explained that “ A ε F ( 𝕏 ) is less fuzzy than B ε F ( 𝕏 ) ” when for each x ∈ 𝕏 the distance between value μ _B  (x) and the complement fixed point x C = 1 2 is less or equal than the distance between values μ _A  (x) and x C = 1 2 . Therefore, the condition “ A ε F ( 𝕏 ) is less fuzzy than B ε F ( 𝕏 ) ″(15) was replaced here by the conditionwas replaced here by the condition ∀ x ∈ 𝕏 : μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ 1 2 ≤ μ B ( x ) ≤ μ A ( x ) .(16)

This way, the initial definition of entropy measure has been replaced by the following definition:

Definition 2. Any mapping e : F ( 𝕏 ) → [ 0 ; 1 ] fulfilling conditions (11), (12), (14) and ∀ x ∈ 𝕏 : μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ 1 2 ≤ μ B ( x ) ≤ μ A ( x ) ⇒ e ( A ) ≤ e ( B ) .(17) is called an entropy measure. □

So far, the Definition 2 of entropy measure is being commonly used as a theoretical model for fuzziness assessment. Spectacular examples of considerations about this definition can be found in the works [5–9, 26].

Several proposals of fuzzy entropy measures are described in literature. These proposals can be roughly divided into two groups. In the first one we find the entropy measure, which was defined with the use of some specific functions variability. Introduction of these measures was justified only by the fact of fulfilling the conditions (11), (12), (14)and (17).

The second group consists of measures which are defined based on intuitive reasons. Those premises arise from a belief that the entropy measure should be treated as an assessment of a relationship between some sets.

Kaufmann [11] proposes to determine entropy measure as a distance between evaluated set and its nearest crisp set. In agreement with above suggestion and due to (8), the entropy measure is then given by the identity e ( A ) = inf { δ ( A , B ) : B ∈ B ( 𝕏 ) } = inf { d ( A ∪ B ) - d ( A ∩ B ) : B ∈ B ( 𝕏 ) }(18)

Another proposition was given by Yager [26], who defined entropy measure as a degree of similarity between fuzzy set and its complement. Then, due to (8) and (9), the entropy measure is given by an identity e ( A ) = ς ( A , A C ) = 1 - d ( A ∪ A C ) + d ( A ∩ A C )(19)

Kosko [14] proposed an entropy measure of A ε F ( 𝕏 ) defined as a degree in which fuzzy subset A ∪ A ^C is “fuzzily included” in the fuzzy subset A ∩ A ^C . Then, due to (10), the entropy measure is given by e ( A ) = γ ( A ∪ A C , A ∩ A C ) = d ( A ∩ A C ) d ( A ∪ A C )(20)

Let us consider once again the answer to the question: What does mean that a set “is less fuzzy than” some other? It is obvious that in accordance to [17] we can say that “ A ε F ( 𝕏 ) is less fuzzy than B ε F ( 𝕏 ) ” when for each x ∈ 𝕏 the distance between value μ _B  (x) and the complement fixed point x C = 1 2 is less or equal than the distance between the values μ _A  (x) and x C = 1 2 . In my opinion, this answer should be rewritten by replacing condition (15) by the condition ∀ x ∈ 𝕏 : | μ A ( x ) - 1 2 | ≥ | μ B ( x ) - 1 2 | .(21)

It is necessary to determine the logical relationship between the conditions (16) and (21).

Lemma 1. Condition (21) is equivalent to the condition ∀ x ∈ 𝕏 : μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ 1 - μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ 1 2 ≤ μ B ( x ) ≤ 1 - μ A ( x ) ∨ 1 2 ≤ μ B ( x ) ≤ μ A ( x ) .(22)

Proof. For any x ∈ 𝕏 we have | μ A ( x ) - 1 2 | ≥ | μ B ( x ) - 1 2 | ⇔ ( | μ A ( x ) - 1 2 | ≥ μ B ( x ) - 1 2 ∧ μ B ( x ) ≥ 1 2 ) ∨ ( | μ A ( x ) - 1 2 | ≥ 1 2 - μ B ( x ) ∧ μ B ( x ) ≤ 1 2 ) ⇔ ( μ A ( x ) - 1 2 ≥ μ B ( x ) - 1 2 ∧ μ A ( x ) ≥ 1 2 ∧ μ B ( x ) ≥ 1 2 ) ∨ ( 1 2 - μ A ( x ) ≥ μ B ( x ) - 1 2 ∧ μ A ( x ) ≤ 1 2 ∧ μ B ( x ) ≥ 1 2 ) ∨ ( μ A ( x ) - 1 2 ≥ 1 2 - μ B ( x ) ∧ ∧ μ A ( x ) ≥ 1 2 ∧ μ B ( x ) ≤ 1 2 ) ∨ ( 1 2 - μ A ( x ) ≥ 1 2 - μ B ( x ) ∧ μ A ( x ) ≤ 1 2 ∧ μ B ( x ) ≤ 1 2 ) ⇔ ( μ A ( x ) ≥ μ B ( x ) ≥ 1 2 ) ∨ ( 1 - μ A ( x ) ≥ μ B ( x ) ≥ 1 2 ) ∨ ( 1 - μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ) ∨ ( μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ) ,

This, together with the alternative commutativity, completes the proof. □

Comparison of conditions (16) and (22) shows that condition (21) is necessary only for the condition (16). It implies that replacing the condition (16) by condition (21) narrows the definition of entropy measure. Moreover, we have here:

Lemma 2. Condition (21) is equivalent to the condition A ∩ A C ⊂ B ∩ B C .(23)

Proof. The condition (A.25) is equivalent to the condition stating, that for any x ∈ 𝕏 we have min { μ A ( x ) , 1 - μ A ( x ) } ≤ min { μ B ( x ) , 1 - μ B ( x ) } ⇔ min { μ A ( x ) , 1 - μ A ( x ) } ≤ μ B ( x ) ≤ 1 2 ∨ min { μ A ( x ) , 1 - μ A ( x ) } ≤ 1 - μ B ( x ) ≤ 1 2 ⇔ μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ 1 - μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ μ A ( x ) ≤ 1 - μ B ( x ) ≤ 1 2 ∨ 1 - μ A ( x ) ≤ 1 - μ B ( x ) ≤ 1 2 ⇔ μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ 1 - μ A ( x ) ≤ μ B ( x ) ≤ 1 2 ∨ 1 2 ≤ 1 - μ B ( x ) ≤ μ A ( x ) ∨ 1 2 ≤ μ B ( x ) ≤ μ A ( x )

This, together with the Lemma 1, completes the proof. □

Analysis stated above leads directly to a proposal of a new, more precise definition of entropy measure.

Definition 3. Any mapping e : F ( 𝕏 ) → [ 0 ; 1 ] fulfilling the conditions (A.17), (A.18), (A.20) and A ∩ A C ⊂ B ∩ B C ⇒ e ( A ) ≤ e ( B ) .(24) is called an entropy measure. □

All of the entropy measures proposed by Kaufman [11], Yager [26] or Kosko [14] satisfy the condition (24). This pose as another argument for accepting Definition 3 as axiomatic definition of entropy measure.

It is worth mentioning that the sets A ∩ A ^C and A ∪ A ^C are, respectively, a W-empty set and a W-universe described in [20]. This observation may be a suggestion for further studies on entropy measurement.

For membership function μ A : 𝕏 → [ 0 , 1 ] Atanassov [1] defines intuitionistic fuzzy set (for short IFS) A as the set of ordered triples A = { ( x , μ A ( x ) , ν A ( x ) ) : x ∈ 𝕏 } ,(25) where nonmembership function ν A : 𝕏 → [ 0 , 1 ] fulfills the condition ν A ( x ) ≤ 1 - μ A ( x )(26) for each x ∈ ℝ . In terms of multi-valued logic, value ν _A  (x) is interpreted as the truth value of a sentence x ∉ A. The family of all IFS in the space 𝕏 is denoted by symbol I ( 𝕏 ) .

We define hesitation function π A : 𝕏 → [ 0 , 1 ] by the identity π A ( x ) = 1 - μ A ( x ) - ν A ( x ) .(27)

Value π _A  (x) indicates the degree of our hesitation in the assessment of a relationship between element x ∈ 𝕏 and IFS A. The hesitation function describes information undecidability, which is interpreted as the impossibility to decide if each elementary state fullfills its postulated requirements. This undecidability is a cause of knightian uncertainty [13]. For this reason, the hesitation function π _A may be interpreted as an image of the knightian uncertainty.

Let us consider B ∈ F ( 𝕏 ) , described by its membership function μ B ∈ [ 0 ; 1 ] 𝕏 . This fuzzy subset can be identified with IFS represented by set of ordered triples B * = { ( x , μ B ( x ) , 1 - μ B ( x ) ) : x ∈ 𝕏 } ,(28)

The hesitation function of abovementioned IFS identically fulfills the condition π B ( x ) = 0 .(29)

This implies that the application of fuzzy sets in creating a real object model suggests an implicit acceptance of a strong assumption proclaiming, that we are always able to decide if each elementary state fulfills its requirements. However, as we know from everyday observations, usually they do not, and our considerations are burdened with a noticeable hesitation margin. This means that the class extension of fuzzy set to IFS expands the capabilities of a reliable description of imprecision.

For any A , B ∈ I ( 𝕏 ) set theory operations and relations are defined in the following way A ⊂ B ⇔ μ A ≤ μ B ∧ ν A ≥ ν B ,(30) A C = { ( x , ν A ( x ) , μ A ( x ) ) : x ∈ 𝕏 } ,(31) A ∪ B = { ( x , μ A ( x ) ∨ μ B ( x ) , ν A ( x ) ∧ ν B ( x ) ) : x ∈ 𝕏 } ,(32) A ∩ B = { ( x , μ A ( x ) ∧ μ B ( x ) , ν A ( x ) ∨ ν B ( x ) ) : x ∈ 𝕏 } .(33)

According to [2], for any IFS A ε I ( 𝕏 ) we determine the biggest fuzzy subset □A ⊂ A and the smallest fuzzy subset ◊A ⊃ A. We have □ A = { ( x , μ A ( x ) , 1 - μ A ( x ) ) : x ∈ 𝕏 }(34) ◊ A = { ( x , 1 - ν A ( x ) , ν A ( x ) ) ; x ∈ 𝕏 } .(35)

Then we can extend the domain of the energy measure to the space I ( 𝕏 ) in the following way.

The energy measure d : I ( 𝕏 ) → [ 0 ; 1 ] can be defined by an identity d ( A ) = d ( □ A ) .(36)

Moreover, we have:

the Hamming’s distance δ : I ( 𝕏 ) × I ( 𝕏 ) → [ 0 ; 1 ] is determined by the identity (8),

The Definition 2 of entropy measure has been generalized to IFS case by Szmidt and Kacprzyk [22]. They have proposed the following definition.

Definition 4. Any mapping e : I ( 𝕏 ) → [ 0 ; 1 ] fulfilling conditions (11), (12), (14) and [ ∀ x ∈ 𝕏 : μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) ∨ ν A ( x ) ≤ ν B ( x ) ≤ μ B ( x ) ≤ μ A ( x ) ] ⇒ e ( A ) ≤ e ( B )(37) is called an entropy measure of IFS. □

The above mentioned definition is obtained as in Definition 3 with condition (16) generalized to ∀ x ∈ 𝕏 : μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) ∨ ν A ( x ) ≤ ν B ( x ) ≤ μ B ( x ) ≤ μ A ( x ) .(38)

The Definition 4 is commonly considered as a theoretical model for fuzziness assessment of IFS. Some examples of this definition usage can be found in [7, 29].

On the other hand, we have:

Lemma 3. For any A , B ε I ( 𝕏 ) , the condition (23) is equivalent to ∀ x ∈ 𝕏 : μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) ∨ μ A ( x ) ≤ ν B ( x ) ≤ μ B ( x ) ≤ ν A ( x ) ∨ ν A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ μ A ( x ) ∨ ν A ( x ) ≤ ν B ( x ) ≤ μ B ( x ) ≤ μ A ( x ) .(39)

Proof. For any A , B ε I ( 𝕏 ) , the condition (23) is equivalent to the condition stating that for any x ∈ 𝕏 we have min { μ A ( x ) , ν A ( x ) } ≤ min { μ B ( x ) , ν B ( x ) } ∧ max { μ A ( x ) , ν A ( x ) } ≥ max { μ B ( x ) , ν B ( x ) } ⇔ min { μ A ( x ) , ν A ( x ) } ≤ min { μ B ( x ) , ν B ( x ) } ≤ max { μ B ( x ) , ν B ( x ) } ≤ max { μ A ( x ) , ν A ( x ) } ⇔ μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) ∨ μ A ( x ) ≤ ν B ( x ) ≤ μ B ( x ) ≤ ν A ( x ) ∨ ν A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ μ A ( x ) ∨ ν A ( x ) ≤ ν B ( x ) ≤ μ B ( x ) ≤ μ A ( x ) . □

The condition (39) is necessary only for the condition (38). It implies that replacing the condition (38) by (23) narrows the definition of entropy measure of IFS.

This leads directly to a proposal of a more detailed definition of entropy measure.

Definition 5. Any mapping e : I ( 𝕏 ) → [ 0 ; 1 ] fulfilling conditions (11), (12), (14) and (24) is called an entropy measure. □

Let us consider any information described by IFS A ∈ I ( 𝕏 ) . Information usefulness decreases with an increase in ambiguity. Thus, quality of considered information increases with a decrease in the energy measure d (A). Moreover, information usefulness decreases with an increase in ambiguity. Therefore, quality of considered information increases with an decrease in entropy measure e (A). These facts imply that energy and entropy measures can be applied as quality indicators. But here we encounter a following paradox.

Case study: Let be given any function g : 𝕏 → [ 0 , 1 ] . We define the function g : 𝕏 → [ 0 , 1 ] in following way h ( x ) = { g 2 ( x ) g ( x ) ≤ 1 2 g ( x ) g ( x ) > 1 2 .

Among other things, for any x ∈ 𝕏 we have here h (x) ≤ g (x). Let us consider the relationship between IFS A = { ( x , μ A ( x ) , ν A ( x ) ) ; x ∈ 𝕏 } = { ( x , g ( x ) , 1 - h ( x ) ) ; x ∈ 𝕏 } and fuzzy set B = { ( x , μ B ( x ) , ν B ( x ) ) ; x ∈ 𝕏 } = { ( x , g ( x ) , 1 - g ( x ) ) ; x ∈ 𝕏 } .

Both of these sets have identical membership functions. For hesitation function we obtain π B ( x ) = 0 ≤ g ( x ) - h ( x ) = π A ( x ) , which shows that information described by means of sets A and B differ in each case because of Knight’s uncertainty burdening only IFS A. This is a sufficient premise to recognize that the information reported by the IFS A is of lower quality than the one described by the fuzzy set B.

On the other hand, for any x ∈ 𝕏 we have:

for g ( x ) ≤ 1 2 μ A ( x ) = μ B ( x ) = g ( x ) ≤ 1 - g ( x ) = ν B ( x ) ≤ 1 - h ( x ) = ν A ( x ) . ( * )

Estimations (*) and (**) show, that the pair of sets A , B ε I ( 𝕏 ) satisfies the condition (23). Then, in agreement with (24), we obtain e ( A ) ≤ e ( B ) .

It is also easy to see that A ⊂ B. Thus, according to (3), we have d ( A ) ≤ d ( B ) .

Despite intuitively worse quality assessment of the information described by IFS A, we see that this information may be evaluated by more favorable values of quality indicators. Let us note that this conclusion was obtained for any energy measure and for any entropy measure determined by Definition 4. This conclusion can also be obtained in the situation when entropy measure is determined by a more precise Definition 5. In particular case of application of the energy measure determined by (4) or (5) or (6) or (7) we have d (A) < d (B) and e (A) < e (B). Thus, in each of these particular cases the quality of information reported by IFS A is higher than the quality of information described by the fuzzy set B. □

Results of the discussion above suggest that if information is described by IFS then its assessment should be used additionally third mapping characterizing undecidability of this information.

For IFS A ε I ( 𝕏 ) , any undecidability characteristic should be dependent on the relationship between the biggest fuzzy subset □A ⊂ A and the smallest fuzzy subset ◊A ⊃ A. Thus, undecidability may be evaluated by any entropy measure k : I ( 𝕏 ) → [ 0 ; 1 ] fulfilling axioms given by Burillo and Bustince [4] A ε F ( 𝕏 ) ⇒ k ( A ) = 0 ,(40) ∅ = □ A ⊂ ◊ A = 𝕏 ⇒ k ( A ) = 1 ,(41) □ B ⊂ □ A ⊂ ◊ A ⊂ ◊ B ⇒ k ( A ) ≤ k ( B ) .(42)

The Burillo’s and Bustince’s definition of entropy measure is not equivalent to de Luca’s and Termini’s. The Burillo’s and Bustince’s one is used to measure undecidability, which is a phenomenon different from indistinctness measured by de Luca’s and Termini’s entropy measure. To emphasize these facts, any Burillo’s and Bustince’s entropy measure –will be referred to as an ignorance measure.

For any fuzzy subset A ∈ F ( 𝕏 ) we have □A = ◊ A. This suggests that the decrease of similarity between sets □A and ◊A entails an increase in the undecidability of information described by A ε I ( 𝕏 ) . This implies that ignorance measure can be defined as a distance between sets □A and ◊A. In our considerations we will use the ignorance measure k : I ( 𝕏 ) → [ 0 ; 1 ] , given by identity k ( A ) = δ ( □ A , ◊ A ) = d ( □ A ∪ ◊ A ) - d ( □ A ∩ ◊ A ) = d ( ◊ A ) - d ( □ A ) .(43)

The starting point of our discussion was the space of fuzzy sets. The discussion outlined above is fully consistent with the intuitive understanding of the original Definition 1 presented by de Luca and Termini [16]. Definition 2, well known from the literature, was obtained from Definition 1 by replacing the relation “is less fuzzy than” (15) by an inequality (16) undecidability quantitative model of thisrelationship.

In this paper, meaning of the words “less fuzzy than” was reconsidered. Correspondingly, model (16) has been corrected. A new Definition 3 of entropy measure has been obtained from Definition 1 by replacing relation (15) with inclusion (23), which is, in fact, a corrected formal model of thisrelationship.

The inclusion (23) is a necessary condition only for inequality (16). This implies that the new Definition 3 of entropy measure is narrower than the old Definition 2.

In the next steps, our discussion was devoted to the case of intuitionistic fuzzy sets (IFS) space. The Definition 2 of entropy measure has been generalized to the case of IFS by Szmidt and Kacprzyk [22]. They have presented Definition 4 which has been obtained from the Definition 2 by generalizing the inequality (16) to the (37). On the other hand, the Definition 3 may be immediately generalized to the case of IFS space in a way that the domain of entropy measure is extended to IFS space. Definition 5 obtained in this way is narrower than the Definition 4, because of the inclusion (23) being a necessary condition for the inequality (37).

The entropy measure predefined by de Luca and Termini [16] may be applied for measurement of information indistinctness.

The entropy measure defined by Burillo and Bustince [4] may be applied for measurement of information undecidability. Therefore, this measure cannot replace the entropy measure proposed by de Luca and Termini.

An increase in the imprecision or in the undecidability significantly worsens the information quality. Thus, using vector-valued function (d (·) , e (·) , k (·)) allows for information quality management. Here, it is desirable to minimize the value of each coordinate. The more precise Definition 5 of entropy measure proposed in this paper should allow for improvement in the tools of such kind of management.