Belief and Plausibility Measures on Intuitionistic Fuzzy Sets with Construction of Belief-Plausibility TOPSIS

Belief and plausibility measures in Dempster–Shafer theory (DST) and fuzzy sets are known as different approaches for representing partial, uncertainty, and imprecise information. (ere are several generalizations of DST to fuzzy sets proposed in the literature. But, less generalization of DSTto 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 DSTto 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 DSTenable 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.


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]. e 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 reallife settings, it may not be always true that nonmembership degree is equal to one minus membership degree. erefore, 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. at is, an IFS A in X is a pair grade, denoted by (μ, ]), with the condition 0 ≤ μ(x) + ](x) ≤ 1, ∀x ∈ X, and a degree of nondeterminacy π(x) � 1 − μ(x) − ](x). 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]. e generalization of DST to fuzzy sets was first given by Zadeh [11]. Subsequently, various generalizations of DST to fuzzy sets were proposed [12][13][14][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. ey 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 (Bel) and plausibility (Pl) 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 beliefplausibility 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][20][21][22]. In this paper, we also create new distance and similarity measures between BPIs of IFSs by using Hausdorff metric. ese similarity measures between BPIs of IFSs are then applied in multicriteria decision-making by constructing a belief-plausibility TOPSIS. e 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 Bel and Pl measures on IFSs in the context of DST. We also give more properties of these proposed Bel and Pl measures on IFSs. In Section 4, we create BPIs of IFSs based on the proposed Bel and Pl 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 Bel and Pl 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.

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.

Dempster-Shafer eory.
e 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 (Ω, M, P) as a probability space and a multivalued mapping, denoted by Ψ, from Ω to Θ which assigns a subset Ψ(ω) ⊂ Θ to every ω ∈ Ω. at is, Ψ is a setvalued function from Ω to the power set 2 Θ of Θ. For any subset S of Θ, we have S * � ω ∈ Ω: Ψ(ω) ⊂ S, Ψ(ω) ≠ ϕ and S * � ω ∈ Ω: Ψ(ω) ∩ S ≠ ϕ . In particular, Θ * � Θ * � Ω. Let Κ be the class of subsets S of Θ so that S * and S * belong to Μ. en, the lower probability P * and upper probability P * of S in Κ are defined as where P * (S) and P * (S) can only be defined if P(Θ * ) ≠ 0. It can be shown that P * (S) � 1 − P * (S c ), P * (S) + P * (S c ) ≤ 1, P * (S) + P * (S c ) ≥ 1, and P * (S) ≤ P * (S). It is worthy to mention that if Ψ is a single valued function, then Ψ becomes a random variable with P * (S) � P * (S). On the contrary, if Ψ is a multivalued mapping, then an outcome ω may be mapped to more than one value. Furthermore, suppose that Θ 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., m: 2 Θ ⟶ [0, 1], with the conditions: (i) m(ϕ) � 0 and (ii) S i ∈2 Θm(S i ) � 1. e function m, also known as basic probability assignment (BPA), assigns an identical weight to every subset of S⊆Θ from the available evidence. An element S in the power set 2 Θ with m(S) ≠ 0 is called a focal element. A belief (Bel) measure is a function Bel: 2 Θ ⟶ [0, 1], and a plausibility (Pl) measure is a function Pl: 2 Θ ⟶ [0, 1] in which they are defined as any element S in 2 Θ , Bel(S) � T⊂S m(T) � P * (S), and Pl(S) � T∩S≠ϕ m(T) � P * (S), where Bel(S) � 1 − Pl(S c ) and Pl(S) � 1 − Bel(S c ).

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 where the function μ S (x): X ⟶ [0, 1] denotes the degree of membership of x ∈ S, and ] S (x): X ⟶ [0, 1] denotes the degree of nonmembership of x ∈ S. For any x ∈ X, π S (x) � 1 − (μ S (x) + ] S (x)) is called the intuitionistic fuzzy index of the element x to the IFS S for representing the degree of uncertainty.
Definition 2 (Atanassov [6,7]). If S and T are two IFSs of the set X, then the following holds: ∀x ∈ X 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: represents that both x j ∈ S and x j ∉ S hypotheses cannot be rejected. Consider the fact μ S (x) + ] S (x) + π S (x) � 1, and so and . Based on these notions, belief intervals can be constructed in Dymova and Sevastjanov [17,18]. Some Bel and Pl 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.

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 (Bel) and plausible (Pl) measures on IFSs with the construction of belief-plausible intervals (BPIs). In the framework of DST, a Bel measure on a set S with its complement S c has Bel(S) + Bel(S c ) ≤ 1. erefore, 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 Bel(S) � Bel(S c ) � 0. For this reason, the belief committed to a proposition S may not be fully described by the belief function Bel(S). It is necessary to assign a doubt function Dou(S) committed to the proposition S, i.e., the belief committed to its complement S c . at is, Dou(S) � Bel(S c ) is used to achieve a full description of the belief committed to S. 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 μ S (x) for any x in an IFS S is analogous to the degree of belief Bel( x { }) in DST for the singleton x, it provides an insight to express the degree of belief in terms of the degree of membership for any x in S.
us, we may assign Bel S (x) � μ S (x), which actually reflects the connection between the degrees of membership and belief. In this sense, Bel S (x) is regarded as the degree of belief, credibility, or evidence of an element x in S, and so it can be also denoted by μ S (x). We can also assign the degree of doubt about the occurrence of an element x in S with Bel S c (x). Since, both degrees of membership and nonmembership are represented by 〈μ(x), ](x)〉 in IFSs, we can write the complement of an IFS S in terms of degree of doubt for each element x in S with Bel S c (x) � ] S (x). In fact, both Bel S (x) and Bel S c (x) can be acted as personal judgement. In this sense, a complete description about an IFS S should be better given both belief function Bel S (x) and doubt function Dou S (x) � Bel S c (x). On the contrary, the plausibility function Pl S (x) can be expressed in terms of the doubt function as Pl us, the plausibility function Pl S (x) associated with the IFS S indicates the extent to which one fails to doubt S or, in other words, the extent to which one finds that S is plausible. Since the ordered triple (μ S (x), ] S (x), π S (x)) of an IFS S has the property μ S (x) + ] S (x) + π S (x) � 1, it may represent the basic probability assignment (BPA) in the context of DST. erefore, we give the following definitions and propose these belief and plausibility measures on intuitionistic fuzzy sets.
Definition 3. Let X be a finite universe of discourses and let S be an IFS of X. A belief function of an element x in S, denoted by Bel S (x), is defined as a real-valued function with (2) Definition 4. Let X be a finite universe of discourses, and let S be an IFS of X. A plausibility function of an element x in S, denoted by Pl S (x), is defined as a real-valued function with Definition 5. Let X � x 1 , x 2 , . . . , x n be a fine universe of discourses and let S be an IFS on X. A belief measure Bel on S under weights w i with n i�1 w i � 1 is defined as . , x n be a universe of discourse, and let S be an IFS on X. A plausibility measure Pl on S under weights w i with n i�1 w i � 1 is defined as In general, the weights w i are assigned to be (1/n) in Definitions 5 and 6. We now discuss some properties of the proposed Bel and Pl measures on IFSs. (4) and (5) obey the following properties:

Theorem 1. Bel and Pl measures on IFSs in equations
(2) Let S 1 and S 2 be any two IFSs with S 1 ⊆S 2 . We have . Similarly, we have that e degrees of belief and plausibility for elements x i in S are found as follows: Similarly, we can find the plausibility measure Pl using Definition 6 with Pl(S) However, the proposed Bel and Pl measures on IFSs are entirely relied on memberships and nonmemberships. erefore, in the next example, we observe the response of the proposed Bel and Pl measures due to change in memberships and nonmemberships of IFSs.

Example 2. Let the universe of discourses be
Since the degrees of belief and plausibility for elements x i in S are entirely relied on memberships μ(x) and nonmemberships ](x), we can observe the response of the proposed belief and plausibility measures by taking three different IFSs T 1 , T 2 , and T 3 with respect to the IFS S based on some variation in their membership and nonmembership values. It is seen that the IFSs S and T 1 are with . 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 S and T 1 . Similarly, to notice the response of belief and plausibility measures on the IFSs T 2 and T 3 with respect to the IFS S, we consider the following variations in membership and nonmembership values of the IFSs S and T 2 ,  (4) the plausibility measure on the IFSs S, T 1 , T 2 , and T 3 under the weights are Pl(S) � 0.7650, Pl(T 1 ) � 0.7900, Pl(T 2 ) � 0.7400, and Pl(T 3 ) � 0.8025. We notice that the proposed plausibility measure of equation (5) from . ese 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.

Distance and Similarity Measures on Belief-Plausibility Intervals
In this section, we first construct belief-plausibility intervals (BPIs) of IFSs based on the proposed Bel and Pl measures. We then use Hausdorff metric to find the distance between BPIs of IFSs. is makes it to give similarity measures between BPIs. Literally, Hausdorff metric is a measure of how 4 Complexity far between two nonempty closed and bounded (compact) subsets S and T 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 d(x, y) � ‖x − y‖ be a Euclidean distance between x and y of two subsets S and T in the metric space S. en, 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 h(S, T) is not equal to backward distance h(T, S). Hausdorff metric is defined as Let us consider the real space R. For any two intervals S � [s 1 , s 2 ] and T � [t 1 , t 2 ], the Hausdorff metric H(S, T) is given by Let S and T be IFSs in X � x 1 , x 2 , . . . , x n . For each x i ∈ X, we define the belief-plausible intervals (BPIs) of x i in the IFSs S and T as i � 1, 2, . . . , n: us, BPI S (x i ) and BPI T (x i ) are subintervals in [0, 1]. We then define the BPIs of the IFSs S and T as BPI en, we define the Hausdorff distance d BPH (BPI S , BPI T ) between the two BPIs of BPI S and BPI T as We next give a definition for a distance d on BPIs.
It is natural to ask "Is the defined distance d BPH (BPI S , BPI T ) reasonable?" To ensure the reasonability and validity of the proposed distance measure d BPH of equation (11), we give the following theorem. Proof. Let BPI S and BPI T be any two BPIs on IFSs S and T in us, the symmetric property (C3) is satisfied. We next prove the condition (C4) of containment property. Let BPI S ≤ BPI T ≤ BPI W . en, On combining the above inequality equations, we obtain H(BPI S (x i ),BPI T (x i ))≤H(BPI S (x i ),BPI W (x i )) and us, (i) and (ii) completes the proof. Next, we prove triangle inequality. We show that, for any BPI S , BPI T , and BPI W , we have belief degrees Bel S (x i ), Bel T (x i ), and Bel W (x i ) and plausible degrees Pl and (iv), we obtain the property (C5).
In applications and ranking of alternatives, a weight vector w of an element x ∈ X is usually considered. erefore, we use equation (11) to establish a weighted Hausdorff distance between two BPIs BPI S and BPI T on IFS. Suppose that the weight of each element x i ∈ X is w i (i � 1, 2, 3, . . . , n) such that n i�1 w i � 1, where 0 ≤ w i ≤ 1, then the weight belief and plausible Hausdorff distance is given as follows: (11) if we replace w i � (1/n), for i � 1, 2, . . . , n. Consequently, equation (11) becomes a special case of equation (13). Property 3. For any two BPIs BPI S and BPI T of IFSs, the forward and backward Hausdorff distances obey either symmetric or asymmetric property, i.e.,

Remark 1. Equation (13) becomes equation
We also define the linguistic terms like dilation (DIL) and concentration (CON) as follows. Using Definition 8, the concentration and dilation of a BPI S corresponding to an IFS S in the context of DST can be defined as follows: where Bel CON(BPI S ) (x) � (Bel BPI S (x)) 2 and Pl CON(BPI where 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. erefore, we next use the Hausdorff distance between two BPIs BPI S and BPI S of IFSs to define similarity between them. Let f be a monotone decreasing function and BPI S and BPI S be two BPIs. Since (1))) ≤ 1. Hence, the similarity measure between BPI S and BPI T can be defined as follows: By using equation (18), different kinds of similarity measures can be obtained by choosing an appropriate function f. e most simplest function f may be chosen as a linear f(x) � 1 − x. en, the similarity measure between BPI S and BPI T of IFSs can be obtained by equation (11) as follows: Again, we may also choose another suitable function as a simple rational function f(x) � (1/ (1 + x)).
en, similarity measure between BPI S and BPI T of IFSs can be defined as follows: Now, we consider the well-known exponential function f(x) � e − x due to its diverse applications and high usefulness in dealing with a similarity relation [28], Shannon entropy [29], correlation coefficient [30], cluster analysis [31], and multicriteria decision-making [32,33]. erefore, we can construct other similarity measures between BPI S and BPI T of IFSs by using exponential function as follows: We next present some numerical examples to validate effectiveness of the proposed similarity measures of equations (19)- (21).
e above results reflect that the belief of BPI S 3 and BPI T is more similar on certain objects than the belief of BPI S 1 and BPI T and the belief of BPI S 2 and BPI T , respectively.
Example 4. Suppose that there are three IFS patterns in the universe X � x 1 , x 2 , x 3 of discourses as follows: , BPI T ) � 0.850. From above results, the sample BPI T is the same as the pattern BPI S 1 according to principle of the maximum degree of similarity between two BPIs. e Complexity above results clearly indicated that the sets S 1 and T are exactly the same that actually match the true state.
We have shown the applicability and reliability of the proposed similarity measures by Examples 3 and 4. Furthermore, we find that the proposed similarity measures on BPIs of IFSs are suitable and well suited in the intuitionistic fuzzy environment using the context of DST. We next construct a novel belief-plausibility TOPSIS and then give its application to multicriteria decision-making.

On Construction of Belief-Plausibility TOPSIS
Group decision making is considered as the cognitive process resulting in a selection of belief or a course of action among several alternatives. For human being, decisionmaking is a sort of daily activity. In multicriteria decisionmaking (MCDM) process [29], we rank and then select the best alternative from available finite set of alternatives. It plays a key role in most fields. Impreciseness is a real truth of daily life which requires close attention in matters of management and decisions. In real-life setting with decision-making process, information available is often uncertain, vague, or imprecise. We need a powerful tool to solve decision-making problems involving uncertain, vague, or imprecise information with high precision. erefore, we employ BPIs of IFSs to amicably tackle problems involving complex decision-making processes. In this section, we utilize similarity measures between BPIs of IFSs which provides us an opportunity to solve MCDM problems by using technique for order preference by similarity to an ideal solution (TOPSIS) [29] which is an approach to identify an alternative that is closest to the positive-ideal solution and farthest to the negative-ideal solution. We extend the concept of TOPSIS [29] to construct the belief-plausibility TOPSIS with BPIs of IFSs to solve MCDM problems.
In the following, we propose a new method to solve MCDM problems with unknown criteria weights. e proposed similarity measures between BPIs of IFSs are used to measure the similarity between each alternative A i from belief-plausible negative-ideal solution (BPNIS) and beliefplausible positive-ideal solution (BPPIS), respectively. en, the idea of the TOPSIS method is utilized to rank alternatives. We present a stepwise algorithm for the proposed belief-plausible TOPSIS method (BP-TOPSIS). Let the set of alternatives be denoted by A � A 1 , A 2 , . . . , A m and the set of criteria of the alternative A i i � 1, 2, . . . , m { } be represented by C j j � 1, 2, . . . , n . e purpose of this problem is to choose the best alternatives out of alternatives. e steps for the proposed BP-TOPSIS are as follows: Step 1: construction of belief-plausible decision matrix Suppose that A � A 1 , A 2 , . . . , A m is a set of alternatives on criteria C � C 1 , C 2 , . . . , C n . Assume that the decision matrix in IFSs is with its corresponding belief-plausible decision matrix (BPDM) D � [b ij ] m×n that is constructed by using BPIs of IFSs for properly handling MCDM problems as follows: where Bel ij � u ij and Pl ij � 1 − ] ij . e values Bel ij � u ij denote degrees of belief, and ] ij represent degrees of doubt. We use Pl ij � 1 − ] ij to represent degrees of plausibility against the alternative A i to the criteria C j such that the conditions of 0 ≤ Bel ij ≤ 1, 0 ≤ Pl ij ≤ 1, and 0 ≤ Bel ij ≤ Pl ij ≤ 1, with i � 1, 2, . . . , m and j � 1, 2, . . . , n, are satisfied. erefore, the BPDM D � (b ij ) m×n can be constructed as follows: Step 2: determination of the weights of criteria In this step, we calculate criteria weights. e weights can be obtained by different ways. Suppose that the weights of criteria C j j � 1, 2, . . . , n are w j , j � 1, 2, 3, . . . , n with n j�1 w j � 1 for 0 ≤ w j ≤ 1. Since the criteria weights are completely unknown, we propose a new weighting method on the basis of belief and plausibility values as follows: Step 3: belief-plausible positive-ideal solution and belief-plausible negative-ideal solution In the TOPSIS method, the evaluation criteria C can be divided into two disjoint sets, i.e., the benefit criteria P 1 and the cost criteria P 2 , where P 1 ⊆C, P 2 ⊆C, and P 1 ∩ P 2 � ϕ. According to BPIs of IFSs and keeping in view the principle of the TOPSIS method, belief-8 Complexity plausible positive-ideal solution (BPPIS) and beliefplausible negative-ideal solution (BPNIS) can be defined as follows: where Step 4: calculation of similarity measures from BPPIS and BPNIS We use equations (13), (19), (20), and (21)  ese similarities will assist us to construct similarity measures of each alternative A i from positive-ideal solution A + and negative-ideal solution A − , respectively, as Step 5: calculation of relative closeness coefficient and ranking of alternatives e relative closeness coefficient (RCC) N k (A i ) of each alternative A i with respect to the belief and plausible ideal solution is obtained by the following expression: Finally, the alternatives are ordered according to the relative closeness degrees.
e ranking order of all alternatives can be determined according to ascending order of the relative closeness degrees. e most preferred alternative is the one with the highest relative closeness degree.
Example 5. Assume that a customer wants to purchase a car.
e following five car companies as alternatives A � A 1 , A 2 , . . . , A 5 are taken into consideration. Before buying a car, the customer takes into account the following four criteria: (1) price (C 1 ), (2) fuel economy and safety (C 2 ), (3) remote keyless entry system (RKES) (C 3 ), and (4) comfort and electronic stability. We notice that C 1 is cost criteria, and so C 1 ∈ P 2 , C 2 , C 3 , and C 4 are benefit criteria. us, C 2 , C 3 , and C 4 ∈ P 1 . e evaluation values of five possible alternatives A � A 1 , A 2 , . . . , A 5 under the above four criteria can be denoted by the following IFSs: Step 1: the construction of corresponding BPDM is displayed in Table 2.
Step 2: the following weights of criteria are calculated by using equation (25) as follows: Step 3: in this step, we calculate BPPIS and BPNIS as follows: Step 4: similarity measures from BPPIS and BPNIS are shown in Table 3.
Step 5: the relative closeness coefficients (RCCs) using equation (28) are shown in Table 4. According to the increasing order of RCCs, the values of five alternatives are ranked, and the alternative with the highest value is considered as the ideal one, as shown in Table 5.
From Table 5, it is seen that the preference ordering of the alternatives by using the proposed similarity measures of equations (19)- (21) for BPIs of IFSs are all the same with the same best alternative.
at is, there is no conflict in the preference ordering of all the alternatives for BPIs of IFSs in Complexity 9

Conclusions
Many attempts had been made in the literature about the improvement in information measures, but there is still much room to improve these measures in a better way and use them to find new applications and novel directions. Although there are some generalizations of DST to fuzzy sets in the literature, there is less generalization of DST to IFSs. In this paper, we first use memberships and nonmemberships for an element in an IFS to express the degrees of belief and plausibility for the element in the IFSs and then propose a simple and intuitive way to compute belief and plausibility measures on IFSs. is gives a type of generalization of DST to IFSs. Based on the proposed belief and plausibility measures on IFSs, we construct belief-plausibility intervals (BPIs) of IFSs and then define Hausdorff distance between BPIs. is construction makes it possible to establish similarity measures of IFSs in the context of DST. e proposed approach to belief and plausibility measures on IFSs in the framework of DST enables us to solve multicriteria decisionmaking by extending TOPSIS to the belief-plausible TOP-SIS. Several examples are presented to demonstrate these suitability, reliability, and validity of the proposed methods. Based on computational results, it is seen that the proposed methods are reasonable and well-suited in the environment of IFSs in the content of DST.

Data Availability
All data are included in the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.