Interval Entropy of Fuzzy Sets and the Application to Fuzzy Multiple Attribute Decision Making

A series of new concepts including interval entropy, interval similarity measure, interval distance measure, and interval inclusion measure of fuzzy sets are introduced. Meanwhile, some theorems and corollaries are proposed to show how these definitions can be deduced from each other. And then, based on interval entropy, a fuzzy multiple attribute decision making (FMADM) model is set up. In this model, interval entropy is used as the weight, by which the evaluation values of all alternatives can be obtained.Then all alternatives with respect to each criterion can be ranked as the order of the evaluation values. At last, a practical example is given to illustrate an application of the developed model and a comparative analysis is made.


Introduction
Multiple attribute decision making (MADM) problems existed in the economic, management, and various social fields.And it refers to making preference decisions over the available alternatives that are characterized by multiple attributes.But for practical needs and peoples' more profound understanding, there is much uncertain information included in the decision making process.How to handle the uncertain information is the issue that decision makers must be concerned about.In 1965, the fuzzy set theory was proposed by Zadeh, which provided effective methods to solve the issue.Several years later, Bellman and Zadeh [1] put forward a fuzzy model based on MADM method by combining fuzzy set and decision making.In the model, the attributes which cannot be defined exactly will be expressed as some proper fuzzy sets and converted into classical decision making problems by use of level sets.Because of its great flexibility and adaptability, the model has been widely viewed as the basis of fuzzy decision making.The FMADM method, proposed by Bass and Kwakernaak in 1977 [2], is regarded as the classical method in fuzzy decision making.In recent years, many new methods have been applied to FMADM problems (such as hesitant fuzzy theory [3][4][5][6], TOPSIS method [7,8], some operators [9][10][11], preference relations [12], and intuitionistic fuzzy decision making [13]).Hadi-Vencheh and Mirjaberi [14] developed an approach to solve MADM problems considering distances both to the positive ideal solution and to the negative ideal solution.Zhang and Xu [15] proposed an extended technique for order preference based on Pythagorean fuzzy set.In this approach, a score function based comparison method is proposed to identify the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution.And a distance measure is defined to calculate the distances between each alternative and the Pythagorean fuzzy positive ideal solution as well as the Pythagorean fuzzy negative ideal solution, respectively.And the comparative analysis had been made among these methods [16].
As Zimmermann [17] pointed out, the existing methods can be divided into two steps: the first is to determine the weights of all alternatives and then combine them into the evaluation values with fuzzy operator; the second is to rank the order of all alternatives according to the evaluation values.For MADM problems, when the weight of alternative is defined, results of decision making depended on the values 2 Mathematical Problems in Engineering of alternatives.And the alternative weights in MADM can be classified as subjective and objective alternative weights based on the information acquisition approach.The subjective alternative weights are determined by preference information on the alternatives given by the decision maker.The objective alternative weights are determined by the decision making matrix.In terms of determining objective alternative weights, one of the most famous approaches is the entropy method, which expresses the relative intensities of alternative importance to signify the average intrinsic information transmitted to the decision maker.Entropy concept was used in various scientific fields, especially in information theory, and used to refer to a general measure of uncertainty [18].In MADM, the greater the value of the entropy corresponding to a special attribute is, which means the smaller attribute's weight, the less the discriminate power of that attribute in decision making process is.So far, a lot of literature pertaining to MADM analysis has been published using entropy weights, for instance, the cross entropy [19,20], fuzzy entropy [21], Shannon entropy [22], maximizing fuzzy entropy [23], and sine entropy [24].Jin et al. [25] proposed the interval value intuitionistic fuzzy continuous weighted entropy which generalizes intuitionistic fuzzy entropy measures defined by Szmidt and Kacprzyk on the basis of the continuous ordered weighted averaging operator.Zhang et al. [26] investigated the MADM problem with completely unknown attribute weights in the framework of interval value intuitionistic fuzzy sets.Using a new definition of interval value intuitionistic fuzzy entropy and some calculation methods for interval value intuitionistic fuzzy entropy, an entropy-based decision making method to solve interval value intuitionistic FMADM problems with completely unknown attribute weights is proposed.Besides the above results, there are also some papers focused on the relationships between entropy and other concepts (such as distance measure [27], similarity measure [28,29], and inclusion measure [30][31][32]).
For the above methods, the common point is to use different entropies as weights.That is to say, the weight value is one specific number.But in practice, because of uncertainty of people's cognition, the data of the decision making processes cannot be measured precisely and there may be some other types of data, for instance, interval value data.In other words, the decision maker would prefer to express the point of view in this form rather than a real number because of the uncertainty and the lack of certain data, especially when data are known to lie within bounded variables, or when facing missing data, judgment data, and so forth.In MADM it is most probable that we confront such a case.So when the weight is an interval value, how to use entropy to represent it is worthy of discussion.
In this paper, according to the definition of fuzzy set entropy, the concept of interval entropy on fuzzy sets is proposed, and its application in MADM is introduced.There are four sections in the paper.Firstly, Section 2 is preliminary, and interval entropy of fuzzy set is proposed in Section 3. Secondly, the relationships among interval entropy, interval similarity measure, interval distance measure, and interval inclusion measure are discussed in Section 4. Finally, the FMADM analysis has been conducted in risk assessment of Taiwan railway reconstruction project in Section 5.

Preliminary
In this section, some definitions are introduced.Here, let  be a set.The mapping   :  → [0, 1] is called a fuzzy subset of .Let () and () denote the class of all fuzzy sets and crisp sets over , respectively.Definition 1 (see [33]).A fuzzy number is defined as a fuzzy subset with the membership function   (), for any  ∈  satisfies the following properties: (1)  is a normal fuzzy set; that is, ∃ ∈ ,   () = 1.
Definition 2 (see [7]).The fuzzy number  is called the trapezoidal fuzzy number and denoted by [, , , ], where the membership function can be expressed as the following: If  = , then  is called the triangular fuzzy number and denoted by [, , ].Therefore, a triangular fuzzy number is a special case of the trapezoidal fuzzy number.

Interval Entropy of Fuzzy Sets
In fact, entropy of fuzzy sets can be used to describe the general measure of fuzziness through the mapping between fuzzy numbers and real numbers on [0, 1], just like the process of the defuzzification, in which only one point is used to represent the fuzzy number.But, in many real life problems, the data of the decision making processes cannot be measured precisely.For instance, when the fuzziness of the fuzzy set can be expressed as a maximum of 0.8 and a minimum of 0.
), then we have We can construct some interval entropy formulas based on Definition 4 as follows.
For instance, let  = { If   () and   () can be integrated over the considered interval [, ], then The following theorem shows that the above formulas are all interval similarity measures.
Proof.When  = 2, we have Mathematical Problems in Engineering Namely, That is to say, The proof for for the reason that Namely, That is to say, IS 2 (, ) ≤ IS 2 (, ) and IS 2 (, ) ≤ IS 2 (, ).
And the proofs for other cases are similar.
is called interval distance measure on (), if ID satisfies the following properties: We can construct some interval distance measure formulas based on Definition 9 as follows.
Let  = { 1 ,  2 , . . .,   }; we set If   () and   () can be integrated over the considered interval [, ], then ] ; The following theorem shows that the above formulas are all interval distance measures.
The proofs for other cases are similar.
We can construct some interval inclusion measure formulas based on Definition 11 as follows.

The Relationships with the above Definitions.
In this section, some conclusions about the relationships with those definitions are presented.From the follow theorems and corollaries, any two can be transformed into each other.( The proofs for other cases are similar. (2) When  = 1, let ID(, ) = IE  ( 1 (, )).
Proof.(1) It can be easily concluded by Theorem 15.
(2) It can be easily concluded by Theorem 13.
(3) It can be easily concluded by Theorems 13 and 15.
For example, if then is an interval similarity measure.And Mathematical Problems in Engineering 13 is an interval entropy.And are interval inclusion measures.
(3) It can be easily concluded by Theorems 15 and 17.For example, if then is an interval similarity measure.And is an interval entropy.And is an interval distance measure.Above all, it can be concluded that, among interval entropy, interval similarity measure, interval distance measure and interval inclusion measure, any two can be transformed into each other.

The Application to Fuzzy Multiple Attribute Decision Making
Given  alternatives and  criteria, a typical FMADM problem can be expressed in matrix format as where  1 ,  2 , . . .,   are the alternatives to be chosen,  1 ,  2 , . . .,   denote the evaluation criteria, x represents the value of alternative   with respect to criterion   evaluated, and the set  = { 1 ,  2 , . . .,   } is the weight of criterion [4-6, 8, 9, 11, 12].Then the alternatives can be ranked based on the given information [36][37][38].In this section, the steps of the proposed method are described with an example.How to determine the weights of all alternatives is described in Section 5.1.And the method to compute the evaluation values of all alternatives with respect to criteria is listed in Section 5.2.The method to rank the order of all alternatives named as possibility-based comparison relation is expressed in Section 5.3.Finally, a FMADM problem with thirty-one alternatives and twenty-three criteria is used to explain the process of the decision making.

Internal Entropy Weight.
It is well known that entropy is a method in obtaining the weights for a MADM problem especially when obtaining suitable weights based on the preferences and decision making experiments are not possible [22].The original procedure of entropy weight can be expressed in a series of steps [39]: (1) Compute entropy   .

The Evaluation Values
Set ( ≻ ) is used to represent the degree of interval values " ≻ " which is named as possibility-based comparison relation on interval values: Let  = { 1 ,  2 , . . .,   } be a set which contains finite fuzzy numbers to be ranked; we can select function  as the evaluation of comparisons [42].Then the fuzzy numbers can be ranked as the values of (  ≻ ).Here,  can be seen as an attitude which comes from the decision maker.Usually, the attitude of the decision maker can be classified into three groups: neutral, optimistic, and pessimistic.The optimistic attitude reflects the preferences of decision makers to the maximal payoff, and it is often used as follows: The neutral attitude reflects the preferences of decision makers for not only the payoff which could be obtained but also the risk which could be faced, and it is often used as follows: And the last attitude is the pessimistic one reflecting the preferences of decision makers for not only the payoff which could be gained but also the risk which could be faced.It is often used as follows: For example, the ranking results with respect to three attitudes are shown in Table 1.

A Numerical Example.
In this section, the steps of the proposed method are described with an example.Suppose that there is a FMADM problem with thirty-one alternatives and twenty-three criteria.Data are presented in Tables 2-3 [43] which can be represented as seven linguistic variables to evaluate degree of impact for railway reconstruction projects and their corresponding trapezoid fuzzy numbers are listed in Table 4.The decision making can be conducted as follows.
Step 1. Compute interval entropy weight of all alternatives according to the corresponding the -cut sets.Here, to

𝐴 11 S S VS VS A A A U S A U A 𝐴 12 A AS AS AS VS A VS A VS S S VS 𝐴 13 VS VS VS VS S VS S S AS S VS VS 𝐴 14 VS VS VS A A S A A S U A VS 𝐴 2 𝐴 21 A S S S S A A VS A U S S 𝐴 22 A A S VS S U A S S U S S 𝐴 23 A A S S A U A VS A U S S 𝐴 24 S A S AS A A S S A A VS S 𝐴 3 𝐴 31 VS U AS AS VS S A AS A S S S 𝐴 32 A U VS S S S A VS A A S VS 𝐴 33 A A S VS VS A U VS A A A VS 𝐴 34 S A VS A VS A A AS S A S S
4

𝐴 41 U VS VS AS VS S A AS A VS VS AS 𝐴 42 A VS A S S S S VS VS S A S 𝐴 43 A VS S VS S A A S VS S S S 𝐴 44 A S VS S VS S A S A A S S 𝐴 45 A VS AS AS VS S S AS A A A AS 𝐴 46 A S VS S VS S VS AS A A S VS 𝐴 5 𝐴 51 S U A VS S VS S AS VS S U AS 𝐴 52 VS A S S A A A S S A S AS 𝐴 53 VS A S U A A A A S A U S 𝐴 54 S S S A S S A VS AS S S VS 𝐴 6 𝐴 61 A S AS VS VS S S S S A A S 𝐴 62 A S VS S S S S S S A A S 𝐴 63 A VS VS S S A A A A A A S 𝐴 64 A VS VS A S S S VS A A A S 𝐴 65 A VS AS VS S S A AS A S S VS 𝐴 7 𝐴 71 A S AS VS S A S U A A A A 𝐴 72 A VS AS AS S A VS AS A S A S 𝐴 73 U A AS S VS U S S A A A VS 𝐴 74 U U AS S S A A S S S S VS
the trapezoid fuzzy number [, , , ], the interval entropy IE 7 () can be expressed as Then the corresponding -cut sets and interval entropy weights are listed in Table 5.
Step 2. Compute the evaluation values.
The alternative values (  )  can be drawn by (99) and the results are listed in Table 6.
Step 3. Ranking of all alternatives: the alternatives can be ranked by (105).(3) Normalize the degree of impact of risk factors for each risk aspect level: The results of local ranking and global ranking under different attitudes are listed in Tables 7 and 8, respectively.And there are some different results with Lu's method.The values of all linguistic variables are determined by interval entropy weight.Here, the values are the most in  and , the following are  and , and then the least are , , and .Among all these linguistic variables, , , and  are inclined to the optimistic attitude, and , , and  are inclined to the opposite.And the numbers of all linguistic variables are listed in Table 9.For example, to the second category, the numbers of  and  in  21 and  23 are 2 and 3, respectively.So, for the optimistic attitude, the decision maker would prefer  23 rather than  21 .The explanations to other cases are similar.

Conclusions
It is well known that entropy is a famous method in obtaining the weights for a MADM problem especially when obtaining suitable weights based on the preferences and decision making experiments are not possible.And most of the literatures pertaining to MADM analysis have been published using entropy weights.Among all these literatures, the result of entropy weight can be expressed as the certain number.But, in practice, because of uncertainty of people's cognition, when the weight of the alternatives cannot be measured precisely and be expressed as some uncertain types of data, how to use entropy to represent it is worthy of discussion.In this paper, the concept of interval entropy is proposed and the relationships among other conceptions such as interval entropy, interval similarity measure, interval distance measure, and interval inclusion measure are investigated in detail.And then, a FMADM model based on interval entropy is set up.In this model, interval entropy is used as the weight, by which the evaluation values can be obtained.And all alternatives with respect to each criterion can be ranked using the method based on the possibility degree.The achievements in the presented work will provide a new model for FMADM based on interval entropy weight.Because of the relationship between the interval entropy, interval similarity measure, interval distance measure, and inclusion measure, in the future, other types of uncertain weights will be researched in detail.

C. Distance Measure of the Fuzzy Sets
To indicate the degree of distance of two fuzzy sets, the concept of distance measure was proposed in 1992 [46].Real function : () × () → [0, 1] is called distance measure on (), if  satisfies the following properties:

D. Inclusion Measure of the Fuzzy Sets
Inclusion measure of fuzzy sets indicates the degree to which a fuzzy set is contained in another fuzzy set.In 1993, Sinha and Dougherty [47] proposed the concept as follows.
2, how about it?It is necessary to extend the value of entropy from the number to interval value, and then the definition of interval entropy is proposed as follows.

Table 1 :
The ranking results.

Table 6 :
The evaluation values of all alternatives.