#### Abstract

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.

#### 1. 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 [3–6], TOPSIS method [7, 8], some operators [9–11], 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 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 [30–32]).

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.

#### 2. Preliminary

In this section, some definitions are introduced. Here, let be a set. The mapping 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, , .(2) is a convex fuzzy set; that is to say, for any , and ,(3)The support of is bounded; that is to say, the set is bounded.

*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.

*Definition 3 (see [7, 34, 35]). *For , the -cut set of is a classic set defined as If is the trapezoidal fuzzy number, then can be denoted byLet be the interval value set; we set [25] (1);(2);(3).

#### 3. 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 , 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 0.2, 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.

*Definition 4. *A real function is called interval entropy on , if has the following properties:(EP1)If , then .(EP2)If, , then .(EP3)If , or , then(EP4).

*Remark 5. *It can be concluded that if , , then we have

We can construct some interval entropy formulas based on Definition 4 as follows.

Let ; If can be integrated over the considered interval , thenThe following theorem shows that the above formulas are all interval entropies.

Theorem 6. * () are interval entropies.*

*Proof. *When , by (7), we haveFirstly, if , then and , so Namely,SoTo (EP1): if , then or . SoNamely, .

(EP2) If, , then , .

Namely, .

(EP3) If , , thenFor , then we have Namely,When , the proof is similar.

(EP4) It can be easily concluded that .

The proofs for other cases are similar.

In particular, if is a trapezoidal fuzzy number and denoted bys , we have

#### 4. Property

##### 4.1. Other Definitions to Describe the Uncertainty

In fact, the conceptions to describe the general measure of uncertainty such as similarity measure, distance measure, and inclusion measure have also built the mapping between two fuzzy numbers and real numbers on , in which only one point is used to represent two fuzzy numbers (see appendix). But when the values are interval, it is necessary to extend these conceptions to interval values, and, in this section, the definitions of interval similarity measure, interval distance measure, and interval inclusion measure are proposed as follows.

*Definition 7. *A real function is called interval similarity measure on if satisfies the following properties:(SP1) if is a crisp set.(SP2).(SP3).(SP4)For all , , , if , then

We can construct some interval similarity measure formulas based on Definition 7 as follows.

For instance, let ; we setIf and can be integrated over the considered interval , then The following theorem shows that the above formulas are all interval similarity measures.

Theorem 8. * () are interval similarity measures.*

*Proof. *When , we haveIf , then ,Namely,That is to say,The proof for is similar.

(SP1) If , then or . Namely, .

(SP2) Consider(SP3) It is obvious that .

(SP4) If , then . Sofor the reason thatNamely,That is to say, and . And the proofs for other cases are similar.

*Definition 9. *A real function is called interval distance measure on , if satisfies the following properties: (DP1)* * if is a crisp set.(DP2)* *.(DP3).(DP4)For all , if , then

We can construct some interval distance measure formulas based on Definition 9 as follows.

Let ; we setIf and can be integrated over the considered interval , thenThe following theorem shows that the above formulas are all interval distance measures.

Theorem 10. * () are interval distance measures.*

*Proof. *When , we have If , thenNamely,That is to say, (DP1) If , then or ,Namely, .

(DP2) Consider(DP3) It is obvious that .

(DP4) If , then and for the reason that We haveNamely, , .

The proofs for other cases are similar.

*Definition 11. *A real function is called interval inclusion measure on , if satisfies the following properties: (IP1).(IP2).(IP3)For all , , , if , then

We can construct some interval inclusion measure formulas based on Definition 11 as follows.

Let , If and can be integrated over the considered interval , we set The following theorem shows that the above formulas are all interval inclusion measures.

Theorem 12. * () are interval inclusion measures.*

*Proof. *When , we haveIt is obvious that(IP1) Consider(IP2) Consider(IP3) If , then ,SoNamely, and . The proofs for other cases are similar.

##### 4.2. 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.

Theorem 13. *I*