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

Authors: Miin-Shen Yang; Zahid Hussain; Mehboob Ali
Journal: Complexity
 (2020)

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## 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.


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## Body

## 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 A in X is a pair grade, denoted by μ,ν, with the condition 0≤μx+νx≤1,∀x∈X, and a degree of nondeterminacy πx=1−μx−νx. 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 beliefBel 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 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 Section2, 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.


## 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 Ω,M,P as a probability space and a multivalued mapping, denoted by Ψ, from Ω to Θ which assigns a subset Ψω⊂Θ to every ω∈Ω. That is, Ψ is a set-valued function from Ω to the power set 2Θ of Θ. For any subset S of Θ, we have S∗=ω∈Ω:Ψω⊂S,Ψω≠ϕ and S∗=ω∈Ω:Ψω∩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. Then, the lower probability P∗ and upper probability P∗ of S in Κ are defined as(1)P∗S=PS∗PΘ∗,P∗S=PS∗PΘ∗,where P∗S and P∗S can only be defined if PΘ∗≠0. It can be shown that P∗S=1−P∗Sc, P∗S+P∗Sc≤1, P∗S+P∗Sc≥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 thatΘ 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:2Θ⟶0,1, with the conditions: imϕ=0 and ii∑Si∈2ΘmSi=1. The 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 2Θ with mS≠0 is called a focal element. A belief Bel measure is a function Bel:2Θ⟶0,1, and a plausibility Pl measure is a function Pl:2Θ⟶0,1 in which they are defined as any element S in 2Θ, BelS=∑T⊂SmT=P∗S, and PlS=∑T∩S≠ϕmT=P∗S, where BelS=1−PlSc and PlS=1−BelSc.


### 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) S˜ in X is defined by the form S˜=x,μS˜x,νS˜xx∈X with the condition 0≤μS˜x+νS˜x≤1, where the function μS˜x:X⟶0,1 denotes the degree of membership of x∈S˜, and νS˜x:X⟶0,1 denotes the degree of nonmembership of x∈S˜. For any x∈X, πS˜x=1−μS˜x+νS˜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:(i)
S˜⊆T˜ if and only if ∀x∈X, μS˜x≤μT˜x and νS˜x≥νT˜x(ii)
S˜=T˜ if and only if ∀x∈X, μS˜x=μT˜x and νS˜x=νT˜x(iii)
S˜∪T˜=maxμS˜x,μT˜x,minνS˜x,νT˜x, ∀x∈X(iv)
S˜∩T˜=minμS˜x,μT˜x,maxνS˜x,νT˜x, ∀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: (i) xj∈S˜:mYes=μS˜xj, (ii) xj∉S˜:mNo=νS˜xj, and (iii) mYes,No=πS˜xj represents that both xj∈S˜ and xj∉S˜ hypotheses cannot be rejected. Consider the fact μS˜x+νS˜x+πS˜x=1, and so BelS˜xj=mYes=μS˜xj and PlS˜xj=mYes+mYes,No = μS˜xj+πS˜xj=1−νS˜xj. 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.


## 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 Ω,M,P as a probability space and a multivalued mapping, denoted by Ψ, from Ω to Θ which assigns a subset Ψω⊂Θ to every ω∈Ω. That is, Ψ is a set-valued function from Ω to the power set 2Θ of Θ. For any subset S of Θ, we have S∗=ω∈Ω:Ψω⊂S,Ψω≠ϕ and S∗=ω∈Ω:Ψω∩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. Then, the lower probability P∗ and upper probability P∗ of S in Κ are defined as(1)P∗S=PS∗PΘ∗,P∗S=PS∗PΘ∗,where P∗S and P∗S can only be defined if PΘ∗≠0. It can be shown that P∗S=1−P∗Sc, P∗S+P∗Sc≤1, P∗S+P∗Sc≥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 thatΘ 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:2Θ⟶0,1, with the conditions: imϕ=0 and ii∑Si∈2ΘmSi=1. The 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 2Θ with mS≠0 is called a focal element. A belief Bel measure is a function Bel:2Θ⟶0,1, and a plausibility Pl measure is a function Pl:2Θ⟶0,1 in which they are defined as any element S in 2Θ, BelS=∑T⊂SmT=P∗S, and PlS=∑T∩S≠ϕmT=P∗S, where BelS=1−PlSc and PlS=1−BelSc.


## 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) S˜ in X is defined by the form S˜=x,μS˜x,νS˜xx∈X with the condition 0≤μS˜x+νS˜x≤1, where the function μS˜x:X⟶0,1 denotes the degree of membership of x∈S˜, and νS˜x:X⟶0,1 denotes the degree of nonmembership of x∈S˜. For any x∈X, πS˜x=1−μS˜x+νS˜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:(i)
S˜⊆T˜ if and only if ∀x∈X, μS˜x≤μT˜x and νS˜x≥νT˜x(ii)
S˜=T˜ if and only if ∀x∈X, μS˜x=μT˜x and νS˜x=νT˜x(iii)
S˜∪T˜=maxμS˜x,μT˜x,minνS˜x,νT˜x, ∀x∈X(iv)
S˜∩T˜=minμS˜x,μT˜x,maxνS˜x,νT˜x, ∀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: (i) xj∈S˜:mYes=μS˜xj, (ii) xj∉S˜:mNo=νS˜xj, and (iii) mYes,No=πS˜xj represents that both xj∈S˜ and xj∉S˜ hypotheses cannot be rejected. Consider the fact μS˜x+νS˜x+πS˜x=1, and so BelS˜xj=mYes=μS˜xj and PlS˜xj=mYes+mYes,No = μS˜xj+πS˜xj=1−νS˜xj. 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.


## 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 beliefBel 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 BelS˜+BelS˜c≤1. 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 BelS˜=BelS˜c=0. For this reason, the belief committed to a proposition S˜ may not be fully described by the belief function BelS˜. It is necessary to assign a doubt function DouS˜ committed to the proposition S˜, i.e., the belief committed to its complement S˜c. That is, DouS˜=BelS˜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 membershipμS˜x for any x in an IFS S˜ is analogous to the degree of belief Belx 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˜. Thus, we may assign BelS˜x=μS˜x, which actually reflects the connection between the degrees of membership and belief. In this sense, BelS˜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 μS˜x. We can also assign the degree of doubt about the occurrence of an element x in S˜ with BelS˜cx. Since, both degrees of membership and nonmembership are represented by μx,νx in IFSs, we can write the complement of an IFS S˜ in terms of degree of doubt for each element x in S˜ with BelS˜cx=νS˜x. In fact, both BelS˜x and BelS˜cx can be acted as personal judgement. In this sense, a complete description about an IFS S˜ should be better given both belief function BelS˜x and doubt function DouS˜x=BelS˜cx. On the contrary, the plausibility function PlS˜x can be expressed in terms of the doubt function as PlS˜x=1−BelS˜cx = 1−DouS˜x = 1−νS˜x=μS˜x+πS˜x. Thus, the plausibility function PlS˜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 μS˜x,νS˜x,πS˜x of an IFS S˜ has the property μS˜x+νS˜x+πS˜x=1, 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.
LetX 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 BelS˜x, is defined as a real-valued function with(2)BelS˜x=μS˜x,∀x∈X.Definition 4.
LetX 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 PlS˜x, is defined as a real-valued function with(3)PlS˜x=1−νS˜x,∀x∈X.Definition 5.
LetX=x1,x2,…,xn be a fine universe of discourses and let S˜ be an IFS on X. A belief measure Bel on S˜ under weights wi with ∑i=1nwi=1 is defined as(4)BelS˜=∑i=1nwiBelS˜xi.Definition 6.
LetX=x1,x2,…,xn be a universe of discourse, and let S˜ be an IFS on X. A plausibility measure Pl on S˜ under weights wi with ∑i=1nwi=1 is defined as(5)PlS˜=∑i=1nwiPlS˜xi.
In general, the weightswi 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.Theorem 1.
Bel and Pl measures on IFSs in equations (4) and (5) obey the following properties:(1)
0≤BelS˜≤PlS˜≤1(2)
IfS˜1⊆S˜2then BelS˜1≤BelS˜2and PlS˜1≤PlS˜2(3)
BelS˜=1−PlS˜cand PlS˜=1−BelS˜c(4)
BelS˜+BelS˜c≤1and PlS˜+PlS˜c≥1Proof.
(1)
LetS˜ be an IFS. We have 0≤μS˜x≤μS˜x+πS˜x≤1, ∀x∈X, πS˜xi≥0. This implies 0≤μS˜x≤1−νS˜x≤1, and then 0≤BelS˜x≤PlS˜x≤1. Thus, we have 0≤∑i=1nwiBelS˜xi ≤ ∑i=1nwiPlS˜xi≤∑i=1nwi=1, and then, 0≤BelS˜≤PlS˜≤1.(2)
LetS˜1 and S˜2 be any two IFSs with S˜1⊆S˜2. We have μS˜1x≤μS˜2x,∀x∈X. This implies BelS˜1x≤BelS˜2x, ∀x∈X, and then ∑i=1nwiBelS˜1xi≤∑i=1nwiBelS˜2xi. We obtain that BelS˜1≤BelS˜2. Similarly, we can prove that, if S˜1⊆S˜2, then PlS˜1≤PlS˜2.(3)
SinceBelS˜=∑i=1nwiBelS˜xi = ∑i=1nwiμS˜xi=1−∑i=1nwi+∑i=1nwiμS˜xi = 1−∑i=1nwi1−μS˜xi=1−PlS˜c, we have that BelS˜=1−PlS˜c. Similarly, we have that PlS˜=1−BelS˜c.(4)
Since0≤BelS˜≤1, 0≤PlS˜≤1, with 0≤BelS˜≤PlS˜≤1, we have BelS˜≤1−BelS˜c. Hence, BelS˜+BelS˜c≤1. Similarly, we can show that PlS˜+PlS˜c≥1.Example 1.
Let the universe of discourse beX=x1,x2,x3,x4,x5. Let S˜ be an IFS in X with S˜=x1,0.05,0.75,x2,0.35,0.55,x3,0.55,0.40,x4,0.65,0.05,x5,1.0,0.0. The degrees of belief and plausibility for elements xi in S˜ are found as follows:BelS˜x1=0.05, BelS˜x2=0.35, BelS˜x3=0.55, BelS˜x4=0.65, and BelS˜x5=1.0. Similarly, PlS˜x1=0.25, PlS˜x2=0.45, PlS˜x3=0.60, PlS˜x4=0.95, and PlS˜x5=1.0. Thus, the belief measure Bel on S˜ under the weights w1=0.1, w2=0.325, w3=0.175, w4=0.15, and w5=0.25 using Definition 5 is computed with BelS˜=∑i=15wiBelS˜xi=0.563. Similarly, we can find the plausibility measure Pl using Definition 6 with PlS˜=∑i=15wiPlS˜xi=0.669.
However, the proposedBel and Pl measures on IFSs are entirely relied on memberships and nonmemberships. Therefore, 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 beX=x1,x2,x3,x4,x5. Let S˜, T˜1, T˜2, and T˜3 be IFSs with(6)S˜=x1,0.25,0.55,x2,0.45,0.35,x3,0.55,0.25,x4,0.7,0.15,x5,1.0,0.0,T˜1=x1,0.25,0.55,x2,0.45,0.35,x3,0.55,0.25,x4,0.85,0.05,x5,1.0,0.0,T˜2=x1,0.25,0.55,x2,0.45,0.35,x3,0.55,0.25,x4,0.65,0.25,x5,1.0,0.0,T˜3=x1,0.25,0.55,x2,0.45,0.35,x3,0.55,0.25,x4,0.95,0.0,x5,1.0,0.0.
Since the degrees of belief and plausibility for elementsxi in S˜ are entirely relied on memberships μx and nonmemberships νx, 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 μS˜x4<μT˜1x4 and νS˜x4>νT˜1x4. 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, respectively, as μS˜x4>μT˜2x4, νS˜x4<νT˜2x4, and μS˜x4<μT˜3x4, νS˜x4>νT˜3x4, in the IFSs S˜ and T˜3. Since the proposed belief and plausibility measures are constructed under weights wi with ∑i=1nwi=1, we can choose any arbitrary values of wi under the condition ∑i=1nwi=1. However, the responses of the proposed belief and plausibility measures remain the same for different values of wi. Assume that weights are assigned as w1=0.1, w2=0.3, w3=0.15, w4=0.25, and w5=0.2. The belief measures on the IFSs S˜, T˜1, T˜2, and T˜3 under the weights are BelS˜=0.6175, BelT˜1=0.6550, BelT˜2=0.6050, and BelT˜3=0.6800. We observe that the proposed belief measure of equation (4) from S˜⟶​T˜1 is increasing due to increasing membership values of μS˜x4<μT˜1x4. Similarly, S˜⟶​T˜2 is decreasing due to μS˜x4>μT˜2x4, and S˜⟶​T˜3 is increasing due to μS˜x4<μT˜3x4 . Similarly, the plausibility measure on the IFSs S˜, T˜1, T˜2, and T˜3 under the weights are PlS˜=0.7650, PlT˜1=0.7900,PlT˜2=0.7400, and PlT˜3=0.8025. We notice that the proposed plausibility measure of equation (5) from S˜⟶​T˜1 is increasing due to decreasing nonmembership values of νS˜x4>νT˜1x4. Similarly, S˜⟶​T˜2 is decreasing due to νS˜x4<νT˜2x4, and S˜⟶​T˜3 is increasing due to νS˜x4>νT˜3x4. 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.Table 1
Belief and plausibility responses on variations in IFSs.
Changes in IFSsBelief responsePlausibility responseS˜⟶T˜1IIS˜⟶T˜2DDS˜⟶T˜3IIU, I, and D represent unchanged, increased, and decreased, respectively.


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

In this section, we first construct belief-plausibility intervals (BPIs) of IFSs based on the proposedBel and Pl 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 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 dx,y=x−y be a Euclidean distance between x and y of two subsets S and T in the metric space S. Then, the forward distance and backward distance is given as follows:(7)hS,T=maxx∈Sminy∈Tx−y,hS,T=maxy∈Tminx∈Sx−y.Hausdorff metric is oriented, in other words, asymmetric as well, which means that most of the time forward distancehS,T is not equal to backward distance hT,S. Hausdorff metric is defined as(8)HS,T=maxhS,T,hT,S.Let us consider the real spaceℜ. For any two intervals S=s1,s2 and T=t1,t2, the Hausdorff metric HS,T is given by(9)HS,T=maxs1−t1,s2−t2.LetS˜ and T˜ be IFSs in X=x1,x2,…,xn. For each xi∈X, we define the belief-plausible intervals (BPIs) of xi in the IFSs S˜ and T˜ as i=1,2,…,n:(10)BPIS˜xi=BelS˜xi,PlS˜xi=μS˜xi,1−νS˜xi,BPIT˜xi=BelT˜xi,PlT˜xi=μT˜xi,1−νT˜xi.Thus,BPIS˜xi and BPIT˜xi are subintervals in 0,1. We then define the BPIs of the IFSs S˜ and T˜ as BPIS˜=BPIS˜xi:xi∈X and BPIT˜=BPIT˜xi:xi∈X. Let HBPIS˜xi,BPIT˜xi denote Hausdorff distance between BPIS˜xi and BPIT˜xi. Then, we define the Hausdorff distance dBPHBPIS˜,BPIT˜ between the two BPIs of BPIS˜ and BPIT˜ as(11)dBPHBPIS˜,BPIT˜=1n∑i=1nHBPIS˜xi,BPIT˜xi.We next give a definition for a distanced on BPIs.Definition 7.
LetBPIS˜,BPIT˜, and BPIW˜ be the three BPIs corresponding to three IFSs S˜,T˜, and W˜ on X, respectively. A d on BPIs is called a distance if it satisfies the following conditions:(12)C1nonnegativity0≤dBPIS˜,BPIT˜≤1;C2separabilitydBPIS˜,BPIT˜=0,if and ifBPIS˜=BPIT˜;C3symmetricdBPIS˜,BPIT˜=dBPIT˜,BPIS˜;C4containmentIfBPIS˜,≤BPIT˜≤BPIW˜,thendBPIS˜,BPIT˜≤dBPIS˜,BPIW˜anddBPIT˜,BPIW˜≤dBPIS˜,BPIW˜;C5triangle inequalityFor anyBPIS˜,BPIT˜andBPIW˜,thendBPIS˜,BPIW˜≤dBPIS˜,BPIT˜+dBPIT˜,BPIW˜.
It is natural to ask “Is the defined distancedBPHBPIS˜,BPIT˜ reasonable?” To ensure the reasonability and validity of the proposed distance measure dBPH of equation (11), we give the following theorem.Theorem 2.
LetX=x1,x2,…,xn be a finite universe of discourses. The proposed distance dBPHBPIS˜,BPIT˜ for the two BPIs of the IFSs S˜ and T˜ satisfies all conditions in Definition 7.Proof.
LetBPIS˜ and BPIT˜ be any two BPIs on IFSs S˜ and T˜ in X=x1,…,xn. According to dBPHBPIS˜,BPIT˜=1/n∑i=1nHBPIS˜xi,BPIT˜xi, where 0≤HBPIS˜xi,BPIT˜xi≤1,∀xi∈X, due to all in absolute values. Thus, 0≤dBPHBPIS˜,BPIT˜≤1 and the condition (C1) of Definition 7 is satisfied. If BPIS˜=BPIT˜, then, for every xi∈X, BelS˜xi=BelT˜xi and PlS˜xi=PlT˜xi, and thus, dBPHBPIS˜,BPIT˜=0. Conversely, if dBPHBPIS˜,BPIT˜=0, then for every xi∈X, maxBelS˜xi−BelT˜xi,PlS˜xi−PlT˜xi=0. Hence, both terms BelS˜xi−BelT˜xi=0 and PlS˜xi−PlT˜xi=0. Then, we have that BPIS˜=BPIT˜. Thus, the condition (C2) is proved. Since maxBelS˜xi−BelT˜xi,PlS˜xi−PlT˜xi = maxBelT˜xi−BelS˜xi,PlT˜xi−PlS˜xi, dBPHBPIS˜,BPIT˜=1/n∑i=1nHBPIS˜xi,BPIT˜xi = 1/n∑i=1nHBPIT˜xi,BPIS˜xi=dBPHBPIT˜,BPIS˜. Thus, the symmetric property (C3) is satisfied. We next prove the condition (C4) of containment property. Let BPIS˜≤BPIT˜≤BPIW˜. Then, BelS˜xi≤BelT˜xi≤BelW˜xi and PlS˜xi≥PlT˜xi≥PlW˜xi, ∀xi∈X. Thus, we have HBPIS˜xi,BPIW˜xi=maxBelS˜xi−BelW˜xi,PlS˜xi−PlW˜xi,…, HBPIT˜xi,BPIW˜xi=maxBelT˜xi−BelW˜xi,PlT˜xi−PlW˜xi:(i)
IfBelS˜xi−BelW˜xi≥PlS˜xi−PlW˜xi, then HBPIS˜xi,BPIW˜xi=BelS˜xi−BelW˜xi. But, we have PlS˜xi−PlT˜xi≤PlS˜xi−PlW˜xi≤BelS˜xi−BelW˜xi and PlT˜xi−PlW˜xi≤PlS˜xi−PlW˜xi≤BelS˜xi−BelW˜xi. On the contrary, we have that BelS˜xi−BelT˜xi≤BelS˜xi−BelW˜xi and BelT˜xi−BelW˜xi≤BelS˜xi−BelW˜xi. On combining the above inequality equations, we obtain HBPIS˜xi,BPIT˜xi≤HBPIS˜xi,BPIW˜xi and HBPIT˜xi,BPIW˜xi≤HBPIS˜xi,BPIW˜xi. Hence, it follows that dBPHBPIS˜,BPIT˜≤dBPHBPIS˜,BPIW˜ and dBPHBPIT˜,BPIW˜≤dBPHBPIS˜,BPIW˜.(ii)
IfBelS˜xi−BelW˜xi≤PlS˜xi−PlW˜xi, then HBPIS˜xi,BPIW˜xi=PlS˜xi−PlW˜xi, but we have BelS˜xi−BelT˜xi≤BelS˜xi−BelW˜xi≤PlS˜xi−PlW˜xi and BelT˜xi−BelW˜xi ≤ BelS˜xi−BelW˜xi≤PlS˜xi−PlW˜xi. On the contrary, PlS˜xi−PlT˜xi≤PlS˜xi−PlW˜xi and PlT˜xi−PlW˜xi≤PlS˜xi−PlW˜xi. On combining the above inequalities, we have HBPIS˜xi,BPIT˜xi≤HBPIS˜xi,BPIW˜xi and HBPIT˜xi,BPIW˜xi≤HBPIS˜xi,BPIW˜xi. Hence, it follows that dBPHBPIS˜,BPIT˜≤dBPHBPIS˜,BPIW˜ and dBPHBPIT˜,BPIW˜≤dBPHBPIS˜,BPIW˜. Thus, (i) and (ii) completes the proof. Next, we prove triangle inequality. We show that, for any BPIS˜,BPIT˜,andBPIW˜, we have belief degrees BelS˜xi, BelT˜xi, and BelW˜xi and plausible degrees PlS˜xi, PlT˜xi, and PlW˜xi, respectively.(iii)
IfBelS˜xi−BelW˜xi≥PlS˜xi−PlW˜xi, then HBPIS˜xi,BPIW˜xi=BelS˜xi−BelW˜xi. HBPIS˜xi,BPIW˜xi=BelS˜xi−BelT˜xi+BelT˜xi−BelW˜xi ≤ BelS˜xi−BelT˜xi+BelT˜xi−BelW˜xi = maxBelS˜xi−BelT˜xi,PlS˜xi−PlT˜xi+maxBelT˜xi−BelW˜xi,PlT˜xi−PlW˜xi = HBPIS˜xi,BPIT˜xi+HBPIT˜xi,BPIW˜xi = dBPHBPIS˜,BPIT˜+dBPHBPIT˜,BPIW˜.(iv)
Similarly,ifBelS˜xi−BelW˜xi≤PlS˜xi−PlW˜xi, then HBPIS˜xi,BPIW˜xi=PlS˜xi−PlW˜xi. HBPIS˜xi,BPIW˜xi=PlS˜xi−PlT˜xi+PlT˜xi−PlW˜xi≤PlS˜xi−PlT˜xi+PlT˜xi−PlW˜xi = maxBelS˜xi−BelT˜xi,PlS˜xi−PlT˜xi+maxBelT˜xi−BelW˜xi,PlT˜xi−PlW˜xi = HBPIS˜xi,BPIT˜xi+HBPIT˜xi,BPIW˜xi = dBPHBPIS˜,BPIT˜+dBPHBPIT˜,BPIW˜. It follows that dBPHBPIS˜,BPIW˜≤dBPHBPIS˜,BPIT˜+dBPHBPIT˜,BPIW˜. From (iii) and (iv), we obtain the property (C5).
In applications and ranking of alternatives, a weight vectorw˜ of an element x∈X is usually considered. Therefore, we use equation (11) to establish a weighted Hausdorff distance between two BPIs BPIS˜ and BPIT˜ on IFS. Suppose that the weight of each element xi∈X is w˜ii=1,2,3,…,n such that ∑i=1nw˜i=1, where 0≤w˜i≤1, then the weight belief and plausible Hausdorff distance is given as follows:(13)dwBPHBPIS˜,BPIT˜=∑i=1nw˜iHBPIS˜xi,BPIT˜xi.Remark 1.
Equation (13) becomes equation (11) if we replace w˜i=1/n, for i=1,2,…,n. Consequently, equation (11) becomes a special case of equation (13).Property 1.
LetdBPHBPIS˜,BPIT˜ be the distance between two BPIs BPIS˜ and BPIT˜ of IFSs. The similarity measures s between them can be denoted by sBPIS˜,BPIT˜=1−dBPHBPIS˜,BPIT˜.Property 2.
For three BPIsBPIS˜,BPIT˜, and BPIW˜ of IFSs with BPIS˜⊆BPIT˜⊆BPIW˜, we have dBPHBPIS˜,BPIW˜≥maxdBPHBPIS˜,BPIT˜,dBPHBPIT˜,BPIW˜.Property 3.
For any two BPIsBPIS˜ and BPIT˜ of IFSs, the forward and backward Hausdorff distances obey either symmetric or asymmetric property, i.e.,(14)dBPHBPIS˜,BPIT˜=BelS˜xi−BelT˜xiorPlS˜xi−PlT˜xi,ifBelS˜xi−BelT˜xi=PlS˜xi−PlT˜xisymmetric,maxBelS˜xi−BelT˜xi,PlS˜xi−PlT˜xi,ifBelS˜xi−BelT˜xi≠PlS˜xi−PlT˜xiasymmetric.
We also define the linguistic terms like dilation (DIL) and concentration (CON) as follows.Definition 8.
LetS˜=x,μS˜x,νS˜x:x∈X be an IFS on X. For any positive real number n, the IFS S˜n is defined as S˜n=x,μS˜xn,1−νS˜xn:x∈X. The corresponding BPIs can be defined as(15)BPIS˜n=x,BelS˜xn,PlS˜xn:x∈X,n>0,where BelS˜x=μS˜x and PlS˜x=1−νS˜x.
Using Definition8, the concentration and dilation of a BPIS˜ corresponding to an IFS S˜ in the context of DST can be defined as follows:(16)CONBPIS˜=x,BelCONBPIS˜x,PlCONBPIS˜x:x∈X,where BelCONBPIS˜x=BelBPIS˜x2 and PlCONBPIS˜x=PlBPIS˜x2.(17)DILBPIS˜=x,BelDILBPIS˜x,PlDILBPIS˜x:x∈X,where BelDILBPIS˜x=BelBPIS˜x1/2 and PlCONBPIS˜x=PlBPIS˜x1/2.
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 BPIsBPIS˜ and BPIS˜ of IFSs to define similarity between them. Let f be a monotone decreasing function and BPIS˜ and BPIS˜ be two BPIs. Since 0≤dBPHBPIS˜,BPIT˜≤1, f1≤fdBPHBPIS˜,BPIT˜≤f0, this implies 0≤fdBPHBPIS˜,BPIT˜−f1/f0−f1≤1. Hence, the similarity measure between BPIS˜ and BPIT˜ can be defined as follows:(18)sS˜,T˜=fdBPHBPIS˜,BPIT˜−f1f0−f1.
By using equation (18), different kinds of similarity measures can be obtained by choosing an appropriate function f. The most simplest function f may be chosen as a linear fx=1−x. Then, the similarity measure between BPIS˜ and BPIT˜ of IFSs can be obtained by equation (11) as follows:(19)sLS˜,T˜=1−dBPHBPIS˜,BPIT˜.
Again, we may also choose another suitable function as a simple rational functionfx=1/1+x. Then, similarity measure between BPIS˜ and BPIT˜ of IFSs can be defined as follows:(20)sQBPIS˜,BPIT˜=1−dBPHBPIS˜,BPIT˜1+dBPHBPIS˜,BPIT˜.
Now, we consider the well-known exponential functionfx=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]. Therefore, we can construct other similarity measures between BPIS˜ and BPIT˜ of IFSs by using exponential function as follows:(21)sES˜,T˜=e−dBPHBPIS˜,BPIT˜−e−11−e−1.
We next present some numerical examples to validate effectiveness of the proposed similarity measures of equations (19)–(21).Example 3.
LetX=x1,x2,x3 be the universe of discourses, and suppose that there are three patterns as S˜1=x1,0.7,0.2,x2,0.5,0.3,x3,0.2,0.5, S˜2=x1,0.2,0.4,x2,0.6,0.3,x3,0.7,0.2, and S˜3=x1,0.2,0.3,x2,0.4,0.2,x3,0.5,0.1. Assume that a sample T˜ is given as T˜=x1,0.2,0.3,x2,0.4,0.2,x3,0.6,0.1. Then, the corresponding BPIs BPIS˜1, BPIS˜2, BPIS˜3, and BPIT˜ of S˜1,S˜2,S˜3, and T˜ are as follows: BPIS˜1=x1,0.7,0.8,x2,0.5,0.7,x3,0.2,0.5, BPIS˜2=x1,0.2,0.6,x2,0.6,0.7,x3,0.7,0.8, BPIS˜3=x1,0.2,0.7,x2,0.4,0.8,x3,0.5,0.9, and BPIT˜=x1,0.2,0.7,x2,0.6,0.8,x3,0.6,0.9. By using equations (19)–(21), we have(22)sLBPIS˜1,BPIT˜=0.667,sLBPIS˜2,BPIT˜=0.867,sLBPIS˜3,BPIT˜=0.967,sQBPIS˜1,BPIT˜=0.500,sQBPIS˜2,BPIT˜=0.765,sQBPIS˜3,BPIT˜=0.936,sEBPIS˜1,BPIT˜=0.552,sEBPIS˜2,BPIT˜=0.803,sEBPIS˜3,BPIT˜=0.949.
Based on above results, we can see that the sampleBPIT˜ is similar to the pattern BPIS˜3 according to the principle of maximum degree of similarity between two BPIs on IFSs. The above results reflect that the belief of BPIS˜3 and BPIT˜ is more similar on certain objects than the belief of BPIS˜1 and BPIT˜ and the belief of BPIS˜2 and BPIT˜, respectively.Example 4.
Suppose that there are three IFS patterns in the universeX=x1,x2,x3 of discourses as follows: S˜1=x1,0.4,0.4,x2,0.3,0.3,x3,0.2,0.2; S˜2=x1,0.3,0.3,x2,0.3,0.3,x3,0.3,0.3; S˜3=x1,0.2,0.2,x2,0.2,0.2,x3,0.2,0.2. Assume that a sample is given as T˜=x1,0.4,0.4,x2,0.3,0.3,x3,0.2,0.2. Then, the corresponding BPIs BPIS˜1,BPIS˜2,BPIS˜3, and BPIT˜ of S˜1,S˜2,S˜3, and T˜ are displayed as follows: BPIS˜1=x1,0.4,0.6,x2,0.3,0.7,x3,0.2,0.8,BPIS˜2=x1,0.3,0.7,x2,0.3,0.7,x3,0.3,0.7,BPIS˜3=x1,0.2,0.8,x2,0.2,0.8,x3,0.2,0.8, and BPIT˜=x1,0.4,0.6,x2,0.3,0.7,x3,0.2,0.8. By using equations (19)–(21), we have sLBPIS˜1,BPIT˜=1,sLBPIS˜2,BPIT˜=0.933,sLBPIS˜3,BPIT˜=0.900,sQBPIS˜1,BPIT˜=1, sQBPIS˜2,BPIT˜=0.874, sQBPIS˜3,BPIT˜=0.818, sEBPIS˜1,BPIT˜=1, sEBPIS˜2,BPIT˜=0.898, and sEBPIS˜3,BPIT˜=0.850. From above results, the sample BPIT˜ is the same as the pattern BPIS˜1 according to principle of the maximum degree of similarity between two BPIs. The 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 Examples3 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.


## 5. 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, decision-making is a sort of daily activity. In multicriteria decision-making (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. Therefore, 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. The proposed similarity measures between BPIs of IFSs are used to measure the similarity between each alternativeAi from belief-plausible negative-ideal solution (BPNIS) and belief-plausible positive-ideal solution (BPPIS), respectively. Then, 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=A1,A2,…,Am and the set of criteria of the alternative Aii=1,2,…,m be represented by Cjj=1,2,…,n. The purpose of this problem is to choose the best alternatives out of alternatives. The steps for the proposed BP-TOPSIS are as follows:Step 1: construction of belief-plausible decision matrixSuppose thatA=A1,A2,…,Am is a set of alternatives on criteria C=C1,C2,…,Cn. Assume that the decision matrix in IFSs is D˜∗=a˜ijm×n=uij,νij with its corresponding belief-plausible decision matrix (BPDM) D˜=b˜ijm×n that is constructed by using BPIs of IFSs for properly handling MCDM problems as follows:(23)D˜=b˜ijm×n=Belij,Plij,whereBelij=uij and Plij=1−νij.The valuesBelij=uij denote degrees of belief, and νij represent degrees of doubt. We use Plij=1−νij to represent degrees of plausibility against the alternative Ai to the criteria Cj such that the conditions of 0≤Belij≤1, 0≤Plij≤1, and 0≤Belij≤Plij≤1, with i=1,2,…,m and j=1,2,…,n, are satisfied. Therefore, the BPDM D˜=b˜ijm×n can be constructed as follows:(24)D˜=b˜ijm×n=A1A2⋮AmBel11,Pl11Bel21,Pl21⋮Belm1,Plm1C1Bel12,Pl12Bel22,Pl22⋮Belm2,Plm2C2⋯⋯⋮⋯Bel1n,Pl1nBel2n,Pl2n⋮Belmn,PlmnCn.Step 2: determination of the weights of criteriaIn this step, we calculate criteria weights. The weights can be obtained by different ways. Suppose that the weights of criteriaCjj=1,2,…,n are w˜j,j=1,2,3,…,n with ∑j=1nw˜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:(25)w˜j=3Belj+Plj/2∑j=1n3Belj+Plj/2.Step 3: belief-plausible positive-ideal solution and belief-plausible negative-ideal solutionIn the TOPSIS method, the evaluation criteriaC can be divided into two disjoint sets, i.e., the benefit criteria P1 and the cost criteria P2, where P1⊆C, P2⊆C, and P1∩P2=ϕ. According to BPIs of IFSs and keeping in view the principle of the TOPSIS method, belief-plausible positive-ideal solution (BPPIS) and belief-plausible negative-ideal solution (BPNIS) can be defined as follows:(26)A+=Cj,Belj+,Plj+:Cj∈C,A−=Cj,Belj−,Plj−:Cj∈C,whereBelj+,Plj+=1,1 and Belj−,Plj−=0,0 if j∈P1 and Belj+,Plj+=0,0 and Belj−,Plj−=1,1 if j∈P2.Step 4: calculation of similarity measures from BPPIS and BPNISWe use equations (13), (19), (20), and (21) to obtain the following weighted similarities: swLS˜,T˜=1−dwBPHBPIS˜,BPIT˜, swQBPIS˜,BPIT˜=1−dwBPHBPIS˜,BPIT˜/1+dwBPHBPIS˜,BPIT˜, and swES˜,T˜=e−dwBPHBPIS˜,BPIT˜−e−1/1−e−1. These similarities will assist us to construct similarity measures of each alternative Ai from positive-ideal solution A+ and negative-ideal solution A−, respectively, as(27)E˜+Ai=sA+,Ai,E˜−Ai=sAi,A−,i=1,2,…,m.Step 5: calculation of relative closeness coefficient and ranking of alternativesThe relative closeness coefficient (RCC)N˜kAi of each alternative Ai with respect to the belief and plausible ideal solution is obtained by the following expression:(28)N˜Ai=E˜−AiE˜−Ai+E˜+Ai,i=1,2,…,m.Finally, the alternatives are ordered according to the relative closeness degrees. The ranking order of all alternatives can be determined according to ascending order of the relative closeness degrees. The most preferred alternative is the one with the highest relative closeness degree.Example 5.
Assume that a customer wants to purchase a car. The following five car companies as alternativesA=A1,A2,…,A5 are taken into consideration. Before buying a car, the customer takes into account the following four criteria: (1) price C1, (2) fuel economy and safety C2, (3) remote keyless entry system (RKES) C3, and (4) comfort and electronic stability. We notice that C1 is cost criteria, and so C1∈P2, C2, C3, and C4 are benefit criteria. Thus, C2, C3, and C4∈P1. The evaluation values of five possible alternatives A=A1,A2,…,A5 under the above four criteria can be denoted by the following IFSs:(29)A1=C1,0.5,0.2,C2,0.3,0.1,C3,0.4,0.5,C4,0.6,0.2,A2=C1,0.5,0.4,C2,0.1,0.6,C3,0.3,0.2,C4,0.5,0.1,A3=C1,0.7,0.2,C2,0.5,0.3,C3,0.1,0.6,C4,0.3,0.2,A4=C1,0.3,0.5,C2,0.6,0.3,C3,0.4,0.1,C4,0.6,0.0,A5=C1,0.2,0.3,C2,0.9,0.0,C3,0.5,0.3,C4,0.6,0.1.Step 1: the construction of corresponding BPDM is displayed in Table2.Step 2: the following weights of criteria are calculated by using equation (25) as follows:(30)w˜1=0.3048,w˜2=0.2524,w˜3=0.2141,w˜4=0.2288.Step 3: in this step, we calculate BPPIS and BPNIS as follows:(31)A+=0.0,0.0,1.0,1.0,1.0,1.0,1.0,1.0,A−=1.0,1.0,0.0,0.0,0.0,0.0,0.0,0.0.Step 4: similarity measures from BPPIS and BPNIS are shown in Table3.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 Table5, 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. That is, there is no conflict in the preference ordering of all the alternatives for BPIs of IFSs in which the belief-plausible TOPSIS gets the right choice of the alternative A3.
We further make a comparison analysis to demonstrate the effectiveness of the proposed belief-plausible TOPSIS. As we know, except the TOPSIS, there are other ways to be used for handling MCDM problems. We next consider another way to solve the MCDM problem for buying a car of Example5. The construction is as follows:Step 1: construction of belief-plausible decision matrixBased on the decision matrixD˜∗=a˜ijm×n=uij,νij with its corresponding BPDM D˜=b˜ijm×n=Belij,Plij where Belij=uij and Plij=1−νij, we specify the options A=A1,A2,…,Am by the characteristic set Ai=Cj,Belij,Plij:Cj∈C.Step 2: determination of the weights of criteriaIn this step, we calculate criteria weights. The weights can be also obtained by the same way of equation (25) with w˜j=3Belj+Plj/2/∑j=1n3Belj+Plj/2.Step 3: construction of a belief ideal solutionThe belief ideal solutionA∗ is generally constructed with(32)A∗=Bel1∗,Pl1∗,Bel2∗,Pl2∗,…,Beln∗,Pln∗,whereBeli∗,Pli∗=maxjBelij∗,minjPlij∗.Step 4: calculation of similarity measuresEquations (19)–(21) are used to calculate the similarity measures sAi,A∗.Step 5: ranking of alternativesRank the alternativesAi in accordance with the degrees of similarities (19)–(21) and select the best alternative with maximum sAi,A∗.
We use the data of Example5 to rank the alternatives and then compare the ranking with the belief-plausible TOPSIS method:Step 1. The construction of corresponding BPDM is exactly the same as Table2.Step 2. The weights of criteria are calculated with(33)w˜1=0.3048,w˜2=0.2524,w˜3=0.2141,w˜4=0.2288.Step 3. The belief ideal solutionA∗ is constructed with(34)A∗=0.7,0.5,0.9,0.4,0.5,0.4,0.6,0.8.Step 4. Similarity measures from each alternativeAi to the ideal alternative A∗ are shown in Table 6.
The maximum similarity between alternativesAi to the ideal alternative A∗, as shown in Table 6, is considered as the best alternative. Thus, according to the maximum similarity rule, Table 7 shows the ranking of each alternative Ai in preferred order.
From Table7, it is seen that the preference ordering of alternatives by using this way all shows the alternative A1 as the best alternative. This method shows the conflict in ranking of alternatives. However, from Table 5, it is seen that the belief-plausible TOPSIS method actually ranks the alternatives according to our intuition, and there was no any conflict in ranking.Table 2
Belief-plausible decision matrix.
AlternativesCriteria evaluationC1C2C3C4A10.5,0.80.3,0.90.4,0.50.6,0.8A20.5,0.60.1,0.40.3,0.80.5,0.9A30.7,0.80.5,0.70.1,0.40.3,0.8A40.3,0.50.6,0.70.4,0.90.6,1.0A50.2,0.70.9,1.00.5,0.70.6,0.9Table 3
Similarity measure between each alternativeAi to BPNIS A− and BPPIS A+.
swLE˜−AiE˜+AiswQE˜−AiE˜+AiswEE˜−AiE˜+AiA10.33040.3595A10.19790.2191A10.22790.2518A20.36940.3257A20.22650.1945A20.26010.2241A30.46320.2771A30.30140.1607A30.34290.1858A40.18850.5267A40.10410.3575A40.12070.4035A50.14800.5628A50.07990.3916A50.09400.4397Table 4
Relative closeness coefficients.
swLN˜AiswQN˜AiswEN˜AiA10.4789A10.4746A10.4751A20.5314A20.5380A20.5372A30.6257A30.6521A30.6486A40.2636A40.2255A40.2303A50.2082A50.1695A50.1761Table 5
Ranking order of alternatives for different similarity methods.
MethodRankingBest alternativeswLA5≺A4≺A1≺A2≺A3A3swQA5≺A4≺A1≺A2≺A3A3swEA5≺A4≺A1≺A2≺A3A3Table 6
Similarity measures from each alternativeAi to the ideal alternative.
swLswQswEswLA1,A∗0.9339swQA1,A∗0.8760swEA1,A∗0.8988swLA2,A∗0.9072swQA2,A∗0.8301swEA2,A∗0.8597swLA3,A∗0.9133swQA3,A∗0.8404swEA3,A∗0.8686swLA4,A∗0.9124swQA4,A∗0.8389swEA4,A∗0.8528swLA5,A∗0.9212swQA5,A∗0.8220swEA5,A∗0.8220Table 7
Ranking order of alternatives for different similarity methods.
MethodRankingBest alternativeswLA2≺A4≺A3≺A5≺A1A1swQA5≺A2≺A4≺A3≺A1A1swEA5≺A4≺A1≺A2≺A3A1


## 6. 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. This 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. This construction makes it possible to establish similarity measures of IFSs in the context of DST. The proposed approach to belief and plausibility measures on IFSs in the framework of DST enables us to solve multicriteria decision-making by extending TOPSIS to the belief-plausible TOPSIS. 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.



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