#### Abstract

Belief and plausibility measures in Dempster–Shafer theory (DST) and fuzzy sets are known as different approaches for representing partial, uncertainty, and imprecise information. There are several generalizations of DST to fuzzy sets proposed in the literature. But, less generalization of DST to intuitionistic fuzzy sets (IFSs), that can somehow present imprecise information better than fuzzy sets, was proposed. In this paper, we first propose a simple and intuitive way to construct a generalization of DST to IFSs with degrees of belief and plausibility in terms of degrees of membership and nonmembership, respectively. We then give belief and plausibility measures on IFSs and construct belief-plausibility intervals (BPIs) of IFSs. Based on the constructed BPIs, we first use Hausdorff metric to define the distance between two BPIs and then establish similarity measures in the generalized context of DST to IFSs. By employing the techniques of ordered preference similarity to ideal solution (TOPSIS), the proposed belief and plausibility measures on IFSs in the framework of DST enable us to construct a belief-plausibility TOPSIS for solving multicriteria decision-making problems. Some examples are presented to manifest that the proposed method is reasonable, applicable, and well suited in the environment of IFSs in the framework of generalization of DST.

#### 1. Introduction

In real world, there exists much uncertainty that is random, fuzzy, vague, imprecise, and ambiguous. Traditionally, probability has been used to model uncertainty under randomness. Belief and plausibility measures based on Dempster–Shafer theory (DST) [1, 2] are emerged as another way of measuring uncertainty in which they have been widely studied and applied in many areas [3, 4]. The theory of evidence, interpreted by Shafer [2], intended to generalize probability theory. On the contrary, two-value logic based on set theory is somehow difficult for handling complex systems, and so Zadeh [5] proposed fuzzy sets (FSs) as an extension of ordinary sets. Since then, fuzziness has been widely employed to handle a type of uncertainty that is different from probability/randomness. FSs are based on membership values between 0 and 1. However, in some real-life settings, it may not be always true that nonmembership degree is equal to one minus membership degree. Therefore, to get more purposeful reliability and applicability, Atanassov [6, 7] generalized FSs to intuitionistic fuzzy sets (IFSs) which include both membership degree and nonmembership degree with a degree of nondeterminacy. That is, an IFS in is a pair grade, denoted by , with the condition , and a degree of nondeterminacy . IFSs are an extension of FSs in which it can be utilized to model those uncertain and vague situations with more available information than FSs. Characterization of IFSs is important as it has many applications in different areas, such as a graphical method for ranking IFS values using entropy [8], hybrid aggregation operators for triangular IFSs [9], and IFS graphs with applications [10].

The generalization of DST to fuzzy sets was first given by Zadeh [11]. Subsequently, various generalizations of DST to fuzzy sets were proposed [12–15]. But, less generalization of DST to IFSs was proposed in the literature. Hwang and Yang [16] proposed a generalization of belief and plausibility functions to IFSs based on fuzzy integral where the generalization is able to catch more information about the change in intuitionistic fuzzy focal elements on DST. In [17, 18], they mentioned that there is a strong link between the theory of IFSs and the evidence theory of DST that makes it possible to aggregate local criteria without limitation. They also suggested an approach as the solution of these problems based on the interpretation of IFSs in the framework of DST. However, in [17, 18], they did not give consideration on a generalization of belief and plausibility measures on IFSs.

In this paper, we propose belief and plausibility measures on IFSs in the framework of DST. We first give a simple and intuitive way to construct the degrees of belief and plausibility in terms of degrees of membership and nonmembership in an IFS, respectively. We then propose belief and plausibility measures on IFSs and construct belief-plausibility intervals (BPIs) of IFSs. Distance and similarity measures are important tools for determining the degree of difference and degree of similarity between two objects. Various distance and similarity measures between fuzzy sets/IFSs had been widely studied and applied in the literature [19–22]. In this paper, we also create new distance and similarity measures between BPIs of IFSs by using Hausdorff metric. These similarity measures between BPIs of IFSs are then applied in multicriteria decision-making by constructing a belief-plausibility TOPSIS.

The remainder of the paper is organized as follows. In Section 2, we review some basic concepts of DST and also demonstrate briefly the interpretations of IFSs in the framework of DST. In Section 3, we exhibit these proposed and measures on IFSs in the context of DST. We also give more properties of these proposed and measures on IFSs. In Section 4, we create BPIs of IFSs based on the proposed and measures on IFSs. We then use Hausdorff metric to give new distance and similarity measures between BPIs of IFSs. Some examples are used to demonstrate that the proposed methods are reasonable and well suited in the environment of IFSs in the context of DST. In Section 5, we construct a belief-plausibility TOPSIS based on the proposed and measures on IFSs for solving multicriteria decision-making (MCDM) problems, which will lead us to rank and then select the best alternative among many alternatives. Finally, conclusions are stated in Section 6.

#### 2. Preliminaries

In this section, we first give a brief review of DST and then review some basic definitions of IFSs with previously defined belief and plausible measures on IFSs in the literature.

##### 2.1. Dempster–Shafer Theory

The lower and upper probabilities were initially proposed by Dempster [1] using a multivalued mapping. One decade later, Shafer [2] developed belief and plausibility measures on ordinary subsets based on Dempster’s lower and upper probabilities. Consider a triplet as a probability space and a multivalued mapping, denoted by , from to which assigns a subset to every . That is, is a set-valued function from to the power set of . For any subset of , we have and . In particular, . Let be the class of subsets of so that and belong to . Then, the lower probability and upper probability of in are defined aswhere and can only be defined if . It can be shown that , , , and . It is worthy to mention that if is a single valued function, then becomes a random variable with . On the contrary, if is a multivalued mapping, then an outcome may be mapped to more than one value.

Furthermore, suppose that is a finite set. A set function is defined by Shafer [2] as a mapping from the power set of the discernment space into unit interval, i.e., , with the conditions: and . The function also known as basic probability assignment (BPA), assigns an identical weight to every subset of from the available evidence. An element in the power set with is called a focal element. A belief measure is a function , and a plausibility measure is a function in which they are defined as any element in , , and , where and .

##### 2.2. Intuitionistic Fuzzy Sets

We next give a brief review of IFSs with these belief and plausible functions on IFSs proposed in the literature.

*Definition 1. *(Atanassov [6, 7]). An intuitionistic fuzzy set (IFS) in is defined by the form with the condition , where the function denotes the degree of membership of , and denotes the degree of nonmembership of . For any , is called the intuitionistic fuzzy index of the element to the IFS for representing the degree of uncertainty.

*Definition 2. *(Atanassov [6, 7]). If and are two IFSs of the set then the following holds:(i) if and only if , and (ii) if and only if , and (iii), (iv), Dymova and Sevastjanov [17, 18] used DST as an interpretation of IFS. In [17, 18], the following three hypotheses are implicitly used to make the connection between DST and IFSs: (i) , (ii) , and (iii) represents that both and hypotheses cannot be rejected. Consider the fact , and so and = . Based on these notions, belief intervals can be constructed in Dymova and Sevastjanov [17, 18]. Some and functions on IFSs were proposed by Hwang and Yang [16] using the concept of Sugeno integral defined by Ban [23]. However, the generalizations of belief and plausibility on IFSs defined by Hwang and Yang [16] are very complicated, and so they are hard to be followed and also difficult to be applied in MCDM. In next section, we propose a simple and intuitive approach for defining belief and plausibility measures on IFSs in the framework of DST.

#### 3. New Belief and Plausibility Measures on Intuitionistic Fuzzy Sets

In this section, we first give the interpretation of IFSs in the framework of DST. We then propose new belief and plausible measures on IFSs with the construction of belief-plausible intervals (BPIs). In the framework of DST, a measure on a set with its complement has . Therefore, DST is different from probability. In the context of DST, the degree of belief committed to a proposition and its complement may not be one but has a degree of ignorance. In the case of total ignorance, it is even with . For this reason, the belief committed to a proposition may not be fully described by the belief function . It is necessary to assign a doubt function committed to the proposition , i.e., the belief committed to its complement . That is, is used to achieve a full description of the belief committed to . Some discussion about doubt functions was also given by Salicone [24, 25].

It is observed that DST and IFSs are intuitively interlinked to each other and have some relations between them. Since the degree of membership for any in an IFS is analogous to the degree of belief in DST for the singleton , it provides an insight to express the degree of belief in terms of the degree of membership for any in . Thus, we may assign , which actually reflects the connection between the degrees of membership and belief. In this sense, is regarded as the degree of belief, credibility, or evidence of an element in , and so it can be also denoted by . We can also assign the degree of doubt about the occurrence of an element in with . Since, both degrees of membership and nonmembership are represented by in IFSs, we can write the complement of an IFS in terms of degree of doubt for each element in with . In fact, both and can be acted as personal judgement. In this sense, a complete description about an IFS should be better given both belief function and doubt function . On the contrary, the plausibility function can be expressed in terms of the doubt function as = = . Thus, the plausibility function associated with the IFS indicates the extent to which one fails to doubt or, in other words, the extent to which one finds that is plausible. Since the ordered triple of an IFS has the property , it may represent the basic probability assignment (BPA) in the context of DST. Therefore, we give the following definitions and propose these belief and plausibility measures on intuitionistic fuzzy sets.

*Definition 3. *Let be a finite universe of discourses and let be an IFS of . A belief function of an element in , denoted by , is defined as a real-valued function with

*Definition 4. *Let be a finite universe of discourses, and let be an IFS of . A plausibility function of an element in , denoted by , is defined as a real-valued function with

*Definition 5. *Let be a fine universe of discourses and let be an IFS on . A belief measure on under weights with is defined as

*Definition 6. *Let be a universe of discourse, and let be an IFS on A plausibility measure on under weights with is defined asIn general, the weights are assigned to be in Definitions 5 and 6. We now discuss some properties of the proposed and measures on IFSs.

Theorem 1. * and measures on IFSs in equations (4) and (5) obey the following properties:*(1)*(2)**(3)**(4)*

*Proof. *(1)Let be an IFS. We have , , . This implies , and then . Thus, we have ≤ , and then, .(2)Let and be any two IFSs with . We have . This implies , , and then . We obtain that . Similarly, we can prove that, if , then .(3)Since = = we have that . Similarly, we have that .(4)Since , , with , we have . Hence, . Similarly, we can show that .

*Example 1. *Let the universe of discourse be . Let be an IFS in with . The degrees of belief and plausibility for elements in are found as follows:, , , , and . Similarly, , , , , and . Thus, the belief measure on under the weights , , , , and using Definition 5 is computed with . Similarly, we can find the plausibility measure using Definition 6 with .

However, the proposed and measures on IFSs are entirely relied on memberships and nonmemberships. Therefore, in the next example, we observe the response of the proposed and measures due to change in memberships and nonmemberships of IFSs.

*Example 2. *Let the universe of discourses be . Let , , , and be IFSs withSince the degrees of belief and plausibility for elements in are entirely relied on memberships and nonmemberships , we can observe the response of the proposed belief and plausibility measures by taking three different IFSs , , and with respect to the IFS based on some variation in their membership and nonmembership values. It is seen that the IFSs and are with and . By considering such type of variations in memberships and nonmemberships, we can easily observe the responses of the proposed belief and plausibility measures of equations (4) and (5) on the IFSs and . Similarly, to notice the response of belief and plausibility measures on the IFSs and with respect to the IFS , we consider the following variations in membership and nonmembership values of the IFSs and respectively, as , , and , , in the IFSs and . Since the proposed belief and plausibility measures are constructed under weights with , we can choose any arbitrary values of under the condition . However, the responses of the proposed belief and plausibility measures remain the same for different values of . Assume that weights are assigned as , , , , and . The belief measures on the IFSs , , , and under the weights are , , , and . We observe that the proposed belief measure of equation (4) from is increasing due to increasing membership values of . Similarly, is decreasing due to , and is increasing due to . Similarly, the plausibility measure on the IFSs , , , and under the weights are , , and . We notice that the proposed plausibility measure of equation (5) from is increasing due to decreasing nonmembership values of . Similarly, is decreasing due to , and is increasing due to These descriptions about the responses of the proposed belief and plausibility measures are given in Table 1. From Table 1, the proposed method for measuring belief and plausibility responses on variations in IFSs gives reasonable results.

#### 4. Distance and Similarity Measures on Belief-Plausibility Intervals

In this section, we first construct belief-plausibility intervals (BPIs) of IFSs based on the proposed and measures. We then use Hausdorff metric to find the distance between BPIs of IFSs. This makes it to give similarity measures between BPIs. Literally, Hausdorff metric is a measure of how far between two nonempty closed and bounded (compact) subsets and in a metric space that looks like each other with respect to their positions in the metric space. Hausdorff metric is defined as the maximum distance of a set to the nearest point in the other set [26, 27]. Let be a Euclidean distance between and of two subsets and in the metric space . Then, the forward distance and backward distance is given as follows:

Hausdorff metric is oriented, in other words, asymmetric as well, which means that most of the time forward distance is not equal to backward distance . Hausdorff metric is defined as

Let us consider the real space For any two intervals and , the Hausdorff metric is given by

Let and be IFSs in . For each , we define the belief-plausible intervals (BPIs) of in the IFSs and as :

Thus, and are subintervals in . We then define the BPIs of the IFSs and as and . Let denote Hausdorff distance between and . Then, we define the Hausdorff distance between the two BPIs of and as

We next give a definition for a distance on BPIs.

*Definition 7. *Let , and be the three BPIs corresponding to three IFSs , and on , respectively. A on BPIs is called a distance if it satisfies the following conditions:It is natural to ask “Is the defined distance reasonable?” To ensure the reasonability and validity of the proposed distance measure of equation (11), we give the following theorem.

Theorem 2. *Let be a finite universe of discourses. The proposed distance for the two BPIs of the IFSs and satisfies all conditions in Definition 7.*

*Proof. *Let and be any two BPIs on IFSs and in . According to , where , due to all in absolute values. Thus, and the condition (C1) of Definition 7 is satisfied. If , then, for every , and , and thus, . Conversely, if , then for every , . Hence, both terms and . Then, we have that . Thus, the condition (C2) is proved. Since = , = . Thus, the symmetric property (C3) is satisfied. We next prove the condition (C4) of containment property. Let . Then, and , . Thus, we have ,…, :(i), then . But, we have and . On the contrary, we have that and . On combining the above inequality equations, we obtain and . Hence, it follows that and .(ii), then , but we have and ≤ . On the contrary, and . On combining the above inequalities, we have and . Hence, it follows that and . Thus, (i) and (ii) completes the proof. Next, we prove triangle inequality. We show that, for any , we have belief degrees , , and and plausible degrees , , and , respectively.(iii) then . ≤ = = = .(iv)Similarly, , then . = = = . It follows that . From (iii) and (iv), we obtain the property (C5).In applications and ranking of alternatives, a weight vector of an element is usually considered. Therefore, we use equation (11) to establish a weighted Hausdorff distance between two BPIs and on IFS. Suppose that the weight of each element is such that , where , then the weight belief and plausible Hausdorff distance is given as follows:

*Remark 1. *Equation (13) becomes equation (11) if we replace , for . Consequently, equation (11) becomes a special case of equation (13).

*Property 1. *Let be the distance between two BPIs and of IFSs. The similarity measures between them can be denoted by .

*Property 2. *For three BPIs , and of IFSs with , we have .

*Property 3. *For any two BPIs and of IFSs, the forward and backward Hausdorff distances obey either symmetric or asymmetric property, i.e.,We also define the linguistic terms like dilation (DIL) and concentration (CON) as follows.

*Definition 8. *Let be an IFS on . For any positive real number , the IFS is defined as . The corresponding BPIs can be defined aswhere and .

Using Definition 8, the concentration and dilation of a corresponding to an IFS in the context of DST can be defined as follows:where and .where and .

Similarity measures play a vital role to differentiate between two sets or objects. Similarity has a lot of applications in many areas. It is well known that the distance and similarity measures are dual concept. Therefore, we next use the Hausdorff distance between two BPIs and of IFSs to define similarity between them. Let be a monotone decreasing function and and be two BPIs. Since ,