A Multi-Attribute Decision-Making Method Using Belief-Based Probabilistic Linguistic Term Sets and Its Application in Emergency Decision-Making

1  Introduction

Emergencies of various types are occurring increasingly frequently in the modern interconnected world, which is a phenomenon that has attracted great public attention [1]. Emergencies are characterized by complexity, uncertainty and a high degree of destructiveness. Their occurrence causes great harm in terms of both public health and property damage [2]. Proper awareness of disasters and effective coping strategies can help the public respond such that they do not become life-threatening [3]. Therefore, actively engaging in disaster prevention education is of great significance in minimizing the damage caused by disasters [4].

Choosing a multiple-disaster risk reduction education plan is an emergency decision problem and therefore essentially a multi-attribute decision problem [5]. Due to the randomness, multi-dimensionality and uncertainty of emergency decision-making, a disaster risk reduction education plan should also consider the complexity of the decision environment [6]. Decision makers are the main participants in emergency decision-making, and their professional knowledge and experience play a key supporting role in evaluating events as they occur. Therefore, human subjectivity will inevitably be a factor in emergency decision-making. To avoid the decision error caused by fuzziness and uncertainty, a wide variety of information expression methods have been proposed to capture DMs’ subjectivity, such as the intuitionistic [7,8], hesitant [9,10], and Pythagorean fuzzy sets [11], among others.

Because of the lack of accurate information and the environmental uncertainty-as well as the urgency to make a decision-linguistic information and judgment are often employed as they represent the only means through which to represent DMs’ point of view in linguistic terms, such as {bad,medium,good}, which is commonly applied to decision problems [12]. In particular, probabilistic linguistic term sets [13], which have achieved great success in decision-making, have been proposed as an extension of hesitant fuzzy linguistic term sets as a means to capture information inaccuracy and event randomness.

Since PLTSs were first proposed in [13], the research on PLTS and MADM based on PLTS has attracted considerable attention. Liao et al. [14] proposed a comprehensive approach for multi-expert MADM problems that considers both quantitative and qualitative criteria and has been implemented to address green enterprise ranking issues. Jiang et al. [15] combined PLTS and the fuzzy least absolute regression to construct a fuzzy regression model capable of handling mixed types of inputs. Nie et al. [16] developed a group decision-making support model using prospect theory-based consistency recovery strategies based on PLTS. Wang et al. [17] extended TOPSIS, VIKOR and TODIM methods based on PLTS to be applicable to multi-expert MADM issues. An integrated MADM method based on PLTS was presented in [18] to determine the best medical product supplier. Li et al. [19] applied PLTS to determine variable weights in solving multi-attribute two-sided matching problems. To evaluate web celebrity shops, Liang et al. [20] employed long short-term memory and PLTSs to describe customers’ satisfaction. Several novel aggregation operators of PLTSs were defined in [21] to solve multi-attribute group decision-making problems. A MADM model was presented in [22] based on incomplete dual probabilistic linguistic preference relations to help enterprises select their 5G partners.

Although traditional PLTSs have been applied to different types of decision problems, the existing approaches show certain limitations and weaknesses, which are reflected in three aspects. The first is in terms of information expression. As probability information is often incomplete, normalization is required in order to fill any gaps in the data [13,23,24]. Second, it only allows for the distribution of probabilities on the singleton and cannot express situations in which the DM hesitates over two or more linguistic terms [24–26]. Third, the PLTS-based decision methods mostly fail to consider the missing attribute values, which are often unavoidable in practice [12,27,28]. In terms of information aggregation, aggregation operators based on the traditional Dempster’s fusion rule or ER method may suffer from information explosion caused by too many linguistic terms, which implies that the complexity of the algorithm needs to be reduced. In terms of decision-making processes, there is no research on the extension of belief-based PLTSs to emergency management issues, and any related studies do not provide a complete decision-making framework. By recognizing these limitations, the motivation of this study is to overcome these defects to make PLTSs more applicable to long-term decision-making. In particular, as decision environments in many applications, such as responding to major emergencies, are highly uncertain and complex, more powerful tools are needed in order to capture how DMs evaluate information. To achieve such a goal, this paper proposes and develops an uncertain information representation and processing method using belief function theory [29,30], which has been widely used in information fusion [31–34] and decision-making [35–37].

In short, the purpose of this study is to extend PLTS by incorporating belief function theory to make it more flexible in terms of expressing opinions. The innovations of this paper can be summarized as follows:

•   The new concept of belief-based PLTS is first introduced, and then the composition of its belief interval is proposed, which includes the belief and plausibility functions. Several examples are provided to facilitate understanding of how the extended belief-based PLTS is used.

•   The novel aggregation operator for belief-based PLTS is proposed using an OWA-based soft likelihood function, which can effectively describe DMs subjective attitudes, and the aggregation process is demonstrated using numerical examples. The aggregation operator is upgraded by considering the reliability of information sources, and the aggregation process is demonstrated using numerical examples.

•   To compare different belief-based PLTSs, a novel possibility degree method is defined based on the belief interval of the singleton. An illustrative example is provided to demonstrate how to determine the sizes of different belief-based PLTSs.

•   Based on the proposed belief-based PLTS and its corresponding operations, a multi-attribute decision-making framework is constructed and applied to emergency decision-making, and the detailed process is developed and explained step by step.

To illustrate the applicability of the proposed approach and verify its usefulness, a complete case study is presented based on the defined emergency decision methodology for selecting the best emergency risk education program. Decision results and comparative analysis are given to demonstrate the usefulness and effectiveness of the method proposed in this paper.

The study is carried out according to the following organization. In Section 2, we introduce some preliminaries, including probabilistic linguistic term set, belief function theory, and OWA operator. In Section 3, we explore extending PLTS with belief function theory, introducing basic concepts, aggregation operators, and comparison function. In Section 4, we provide a framework for multi-attribute decision making and apply it to emergency decision problems. In Section 5, we give a case study to demonstrate the usefulness and effectiveness of the proposed emergency decision method. In Section 6, we summarize this study and provide future research.

2  Preliminaries

Some basic prerequisites are introduced in this section, including linguistic term set, probabilistic linguistic term set, belief function theory, and OWA operator.

2.1 Linguistic Term Set (LTS)

In practical decision problems, the linguistic term set is very common and useful, because it is more in line with the DM’s habit of thinking [38]. A subscript-symmetric linguistic term set is defined as S={sα|α=−τ,…,−1,0,1,…,τ}, in which τ is a positive integer, and sα satisfies the following characteristics [39]:

•   sα indicates “positive” when α>0, sα indicates “neutral” when α=0, and sα indicates “negative” when α<0;

•   sα>sα^ iff α>α^;

•   the negation operator of sα is defined as neg(sα)=s−α.

2.2 Probabilistic Linguistic Term Set (PLTS)

Probabilistic linguistic term set was presented by Pang et al. [13] to address the weakness that all linguistic term sets in hesitant fuzzy linguistic term set (HFLTS) have the same weight. To demonstrate the shortcomings of HFLTS, and to emphasize the need to introduce PLTS, an illustrative example is presented.

Example 1. In a serious mine accident, the DMs need to evaluate the emergency plans. With regard to an emergency plan ERX, 100 experts use the LTS S={s−1=poor,s0=medium,s1=good} to evaluate its feasibility, among which 80 consider it is “good”, and 10 believe it is “medium”, so this case could be expressed as

Feasibility(ERX)={medium(0.1),good(0.8)},(1)

which is not appropriate for HFLTS.

From the above example it can be concluded that the assessment information is expressed not only as a set of possible linguistic terms but also as their corresponding probabilities. To effectively express the opinions of DMs, the probabilistic linguistic term set is proposed and defined as follows:

Definition 2.1. [13] Let S={s−τ,…,s−1,s0,s1,…,sτ} be a LTS, a PLTS is denoted as

L(p)={L(k)(p(k))|L(k)∈S,p(k)≥0,k=1,2,…,#L(p),∑k=1#L(p)⁡p(k)≤1},(2)

where p(k) means the probability of L(k), and #L(p) is the number of nonzero linguistic terms. Note that if ∑k=1#L(p)p(k)≠1, the normalization process is started. The normalized PLTS is defined as L˙(p)={L(k)(p˙(k))}, where p˙(k)=p(k)/∑k=1#L(p)p(k).

Based on the above definition, we can represent the evaluation information in Example 1 by a PLTS as L(p)={s0(0.1),s1(0.8)}. The comparison method of two PLTSs was provided in literature [13].

Definition 2.2. [13] Let two normalized PLTSs be represented as L1(p) and L2(p), we have

•   If E(L1(p))>E(L2(p)), then L1(p)≻L2(p);

•   If E(L1(p))=E(L2(p)), then

    –  If σ(L1(p))>σ(L2(p)), then L1(p)≺L2(p);

    –  If σ(L1(p))=σ(L2(p)), then L1(p)∼L2(p),

Definition 2.3. [13] Let a normalized PLTS be L(p)={L(k)(p(k)),k=1,2,…,#L(p)}, and r(k) be the subscript of L(k), the score of L(p) can be calculated as

E(L(p))=sα¯,(3)

where α¯=∑k=1#L(p)r(k)p(k).

Definition 2.4. [13] Let a normalized PLTS be L(p)={L(k)(p(k)),k=1,2,…,#L(p)}, r(k) be the subscript of L(k), and E(L(p))=sα¯, then the deviation degree of L(p) can be calculated as

σ(L(p))=∑k=1#L(p)⁡(p(k)(r(k)−α¯))2.(4)

2.3 Belief Function Theory

The theory of belief function is an uncertain reasoning one initiated by Dempster [29] and inherited and developed by Shafer [30], therefore also known as Dempster-Shafer theory (DST). After half a century of development, the theory has been well developed and played an important role in a wide variety of fields, such as decision-making [40–42] and information fusion [43–45]. Suppose that the answer to the question of concern consists of a finite set of mutually exclusive elements, called frame of discernement (FoD), expressed as Θ={θ1,θ2,…,θn}. The power set of Θ has all of the subsets of Θ, which is expressed as 2Θ={θ1,θ2,…,θn,θ1θ2,…,Θ}. A piece of evidence or information is defined as a mapping m:2Θ→[0,1] that satisfies ∑A∈2Θm(A)=1 and m(ϕ)=0, also known as the basic belief assignment (BBA). Two significant concepts in the belief function theory are introduced below, namely, belief function and plausibility function.

Definition 2.5. The belief function Bel:2Θ→[0,1] of propositions is defined as

Bel(A)=∑B⊆A⁡m(B),∀A⊆Θ.(5)

And we have Bel(ϕ)=0 and Bel(Θ)=1.

Definition 2.6. The plausibility function Pl:2Θ→[0,1] of propositions is defined as

Pl(A)=1−Bel(A¯)=∑B∩A=ϕ⁡m(B),∀A⊆Θ,(6)

where A¯=Θ−A, and Bel(A)≤Pl(A). BI(A)=[Bel(A),Pl(A)] forms the belief interval of proposition A [46,47].

2.4 OWA Aggregation Operator

OWA operator, originally introduced by Yager et al. [48–50], is an information aggregation method between the maximum and minimum operators.

Definition 2.7. [48] Let {a1,a2,…,an} be a data set. The ordered weighted average operator is defined as

OWA(a1,a2,…,an)=∑j=1n⁡wjbj,(7)

where ω={w1,…,wn} is the weighting vector that satisfies wj∈[0,1] and ∑j=1nwj=1, and bj is the jth largest value in {a1,a2,…,an}. Let λ be an index function, and λ(j) is the index of jth largest value, then the OWA operator can be described as

OWA(a1,a2,…,an)=∑j=1n⁡wjaλ(j).(8)

Definition 2.8. [48] OWA operators are highly dependent on weighting vector ω. When weight distributions are special, several types of OWA operators are expressed as follows.

•   ω∗:w1=1,wj,j≠1=0, the OWA operator can be denoted as

OWA(a1,a2,…,an)=aλ1=max(ai).(9)

•   ω∗:wj,j≠n=1,wn=0, the OWA operator can be denoted as

OWA(a1,a2,…,an)=aλn=min(ai).(10)

•   ωn:wj=1/n, the OWA operator can be denoted as

OWA(a1,a2,…,an)=1n∑j=1naj.(11)

•   ω⌈K⌉:wj,j≠K=0,wK=1, the OWA operator can be denoted as

OWA(a1,a2,…,an)=aλK.(12)

Definition 2.9. The weighting vector for OWA operator can be determined as

wj=f(jn)−f(j−1n),(13)

where f is a monotonic function that satisfies f(0)=0 and f(1)=1.

In [51], a useful function f(x)=xm, m≥0 was given, and then a parameter α was proposed to measure the degree of optimism satisfies m=1−αα. So for a given α, the weighting vector can be determined as

wj=(jn)1−αα−(j−1n)1−αα.(14)

Note that the larger attitudinal character α is, the more optimistic the DM is.

3  Belief-Based Probabilistic Linguistic Term Sets

In this section, the belief-based probabilistic linguistic term sets are developed to overcome the issues of PLTSs by expressing and dealing with more uncertainty. We first give the basic concept of belief-based PLTSs, then provide the aggregation operator and finally define the score function in order to effectively complete the emergency decision-making process.

3.1 The Basic Concept

The probabilistic linguistic term set effectively solves the problem that the HFLTS cannot assign weight to the possible linguistic terms according to the DM’s preference. However, the PLTS has also been criticized for its limitations in decision-making [28]. According to Definition 2.1, PLTS allows probability information to be incomplete, but other operations of the PLTS can only be performed after the probability is normalized. Allowing incomplete information expression is conducive to contain more uncertainty, but the normalization for subsequent calculations directly distributes the uncertainty to each linguistic term, which artificially removes some unknown uncertainties and weakens the effectiveness of the modeling uncertainty.

In complex decision-making, especially emergency problems, there may be various uncertainties, including randomness, vagueness and incompleteness [52]. Randomness comes from non-deterministic conditions, ambiguity is caused by unclear classification of things, and incompleteness is caused by incomplete information. PLTS has some potential limitations in the expression of uncertainty, such as the aforementioned normalization problem, the inability to express simultaneity, and the lack of attribute values that are not considered. The belief function theory is a powerful tool for expressing and dealing with uncertainty, which has been widely declared and recognized by authoritative literature [29,34,35,46]. In order to solve the potential limitation of PLTS in MADM, the concept of belief-based PLTS is introduced.

For instance, in Example 1, the evaluation is represented by PLTS as Feasibility(ERX)={medium(0.1),good(0.8)}, in which 10 experts have not expressed their opinions. This part of information can be regarded as unknown or uncertain. Based on belief function theory, the evaluation can be represented as

Feasibility1(ERX)={medium(0.1),good(0.8),{poor,medium,good}(0.1)}={s0(0.1),s1(0.8),S(0.1)},(15)

where S is FoD, indicating unknown or uncertain.

It is worth noting that incompleteness of information is also seen in another scenario in decision making. For example, in multi-attribute emergency decision making, the DM has completed the evaluation of the performance of an emergency plan under other attributes, but only in the aspect of attribute “response speed” is unknown. The traditional PLTS-based emergency decision making method rarely considers this situation, which leads to the interruption of the emergency procedure. Within the frame of belief function theory, the missing attribute value can be represented as S(1). The ability of the belief function theory to express uncertain information can ensure that the decision can proceed even with the incomplete information.

Another motivation that we have to extend PLTS with belief function theory is to deal with a decision scenario where a DM is hesitant between two or more linguistic terms. In other words, the probability distribution on the single term no longer meets the expression requirements of DMs, but needs to allocate the belief on multiple subsets. Take the decision scenario in Example 1, where 80 experts evaluate it “good”, 10 evaluate it “medium”, and 10 hesitate between the terms “good” and “medium”. Obviously, PLTS is powerless to describe this case, but the theory of belief function can handle and represent such a case well with the expression

Feasibility2(ERX)={medium(0.1),good(0.8),{medium,good}(0.1)}={s0(0,1),s1(0.8),s0s1(0.1)}.(16)

Through the above analysis and discussion, we identify some limitations of PLTS in expressing DMs’ views, and analyze the advantages of belief function theory in solving these problems. This is therefore an appropriate time to present the belief-based probabilistic linguistic term set as below:

Definition 3.1. Let S={s−τ,…,s−1,s0,s1,…,sτ} be a LTS, a belief-based PLTS is defined as

ℒ={β(ι)|ι∈2S,β(ι)≥0,∑ι∈2S⁡β(ι)=1},(17)

where β(ι) means the basic belief assignment (BBA) of proposition ι (a subset of LTS S). Note that the belief-based PLTS ℒ is denoted by β for convenience.

Based on Definition 3.1, the evaluations in Eqs. (15) and (16) can be represented by belief-based PLTSs as ℒ1={s0(0.1),s1(0.8),S(0.1)} and ℒ2={s0(0.1),s1(0.8),s0s1(0.1)}, which can also be written as β1(s0)=0.1,β1(s1)=0.8,β1(S)=0.1 and β2(s0)=0.1,β2(s1)=0.8,β2({s0,s1})=0.1. According to the above definitions and examples, we can conclude that the belief-based PLTS has the following desired properties and advantages: (1) for PLTS with incomplete information, the operation of normalization is no longer necessary, and the uncertainty is represented by the universal set (i.e., LTS S, can also be interpreted as FoD), thus unifying the information expression mode based on linguistic term sets from the mathematical form (2) belief-based PLTS has the ability to capture DMs’ hesitations among possible linguistic terms, can more fully allow DMs to express opinions. It is worth noting that belief-based PLTS degrades to the traditional PLTS when the belief is all assigned to the singleton, i.e., it is backward compatible. Naturally, the belief function and plausibility function of belief-based PLTS can be defined.

Definition 3.2. The belief function of belief-based PLTS Belℒ:2S→[0,1] is defined as

Belℒ(A)=∑B⊆A⁡β(B),∀A⊆S.(18)

And we have Belℒ(ϕ)=0 and Belℒ(S)=1.

Definition 3.3. The plausibility function of belief-based PLTS Plℒ:2S→[0,1] is defined as

Plℒ(A)=1−Belℒ(A¯)=∑B∩A=ϕ⁡β(B),∀A⊆S,(19)

where A¯=Θ−A, and Belℒ(A)≤Plℒ(A). So BIℒ(A)=[Belℒ(A),Plℒ(A)] forms the belief interval of proposition A.

3.2 A Novel Aggregation Operator for Belief-Based PLTS

In decision problems, an aggregation operator is inevitably employed, which is an approach combining two or more single information expressions. The above manifests the advantages of the belief-based PLTS in representing uncertain information. To make it exert maximum in decision making, this section provides a novel aggregation operator for belief-based PLTS ground on soft likelihood function.

Soft likelihood function was originally proposed by Yager et al. [53] for calculating likelihood functions of probabilistic evidence in the context of forensic crime investigations. It breaks the limitations of traditional likelihood functions that use logical “anding” to aggregate elements, and has been extended to uncertain decision environments by D numbers [54], power OWA [55,56], and Pythagorean fuzzy sets [57,58]. With the flexible and reliable combination ability of soft likelihood function, in this section, an aggregation operator for belief-based PLTS is proposed.

Definition 3.4. Let ℒ={ℒ1,ℒ2,…,ℒn} be n belief-based PLTSs on S={s−τ,…,s−1,s0,s1,…,sτ}. The likelihood value of ιi⊆S is defined as

Γιi=∏j=1nβj(ιi).(20)

The above definition multiplies the belief of ι in each belief-based PLTS to determine the value of aggregated ι. It is easy to find that this definition is a logical “and”, that is, the aggregated ι depends on the minimum ι in each belief-based PLTS. This means, however, that no matter how many belief-based PLTSs have a large ι value, Γιi=0 as long as βj(ιi)=0 in one belief-based PLTS, which is obviously not true in most applications. This kind of likelihood function is called hard likelihood function. To fix the defect, a soft likelihood function based on OWA operator has been developed [53], which is used to define the aggregation operator for belief-based PLTS.

Definition 3.5. Let λιi be an index function of ιi⊆S, and λιi(k) indicate the index of the kth largest belief of ιi in ℒ. The product operation of the j largest belief of ιi can be defined as

Prodιi(j)=∏k=1jβλιi(k)(ιi),(21)

where βλιi(k)(ιi) is the kth largest belief of ιi. Note that when j=n, Eq. (21) degenerates into the hard likelihood function, i.e., Prodιi(n)=Γιi. In addition, it is easy to tell that Prod is a monotonically decreasing function, i.e., if j1≥j2, then Prod(j1)≤Prod(j2).

Definition 3.6. The OWA-based soft likelihood function of ιi⊆S is defined as

Γιi,OWA=OWA(Prodιi(1),Prodιi(2),…,Prodιi(n))=∑j=1n⁡wjProdιi(j),(22)

where wj∈[0,1] is the weighting vector that satisfies ∑j=1nwj=1.

According to Definition 2.8, we analyze the OWA-based soft likelihood function corresponding to the special forms of weight distribution.

•   For ω∗:w1=1,wj,j≠1=0, Γιi,OWA=Prodιi(1)=βλιi(1)(ιi). In this case, the aggregated belief of ιi is equal to the largest one in ℒ.

•   For ω∗:wj,j≠n=1,wn=0, Γιi,OWA=Prodιi(n)=∏j=1nβj(ιi), which degenerates into the hard likelihood function.

•   For ωn:wj=1/n, Γιi,OWA=1n∑j=1nProdιi(j)=1n∑j=1n(∏k=1jβλιi(k)(ιi)), which is a simple average of Prodιi(j).

Definition 3.7. Based on Definition 2.9, the attitudinal character α can be employed to calculate the soft likelihood value of ιi⊆S. Therefore the aggregated belief of ιi can be calculated as

Γιi,α=∑j=1n⁡((jn)1−αα−(j−1n)1−αα)∏k=1jβλιi(k)(ιi),ιi⊆S.(23)

To ensure that the aggregated result is still a belief-based PLTS, it needs to be normalized as

ιi(βα)=Γιi,α∑ιi⊆SΓιi,α.(24)

Therefore, the aggregation operator of n belief-based PLTSs can be defined as

ℒ1⊕ℒ2⊕⋯⊕ℒn={ιi(βα)|ιi∈2S,βα(ιi)≥0,∑ιi∈2S⁡βα(ιi)=1}.(25)

To demonstrate the usage of the above-proposed aggregation operator, the following example is given.

Example 2. Let S={s−1,s0,s1} be a LTS, and five belief-based PLTSs on S be represented by Definition 3.1 as ℒ1={s−1(0.7),s0(0.2),s0s1(0.1)}, ℒ2={s−1(0.55),s0(0.25),s1(0.2)}, ℒ3={s−1(0.8),s0s1(0.2)}, ℒ4={s−1(0.5),s0(0.2),s0s1(0.3)}, and ℒ5={s−1(0.5),s0(0.2),s0s1(0.3)}. We take s−1 as an example to show the solution of OWA-based soft likelihood function.

According to Definition 3.5, the product of the j largest belief of s−1 can be calculated as Prods−1(1)=βλs−1(1)(s−1)=β3(s−1)=0.8, Prods−1(2)=βλs−1(1)(s−1)×βλs−1(2)(s−1)=β3(s−1)×β1(s−1)=0.8×0.7=0.56, Prods−1(3)=βλs−1(1)(s−1)×βλs−1(2)(s−1)×βλs−1(3)(s−1)=β3(s−1)×β1(s−1)×β2(s−1)=0.8×0.7×0.55=0.3080, Prods−1(4)=βλs−1(1)(s−1)×βλs−1(2)(s−1)×βλs−1(3)(s−1)×βλs−1(4)(s−1)=β3(s−1)×β1(s−1)×β2(s−1)×β4(s−1)=0.8×0.7×0.55×0.5=0.1540, Prods−1(5)=βλs−1(1)(s−1)×βλs−1(2)(s−1)×βλs−1(3)(s−1)×βλs−1(4)(s−1)×βλs−1(5)(s−1)=β3(s−1)×β1(s−1)×β2(s−1)×β4(s−1)×β5(s−1)=0.8×0.7×0.55×0.5×0.5=0.0770.

Now we discuss the soft likelihood value of s−1 with different α.

•   When α=0.1, based on Eq. (14), the weighting vector is ω=(0,0.0003,0.0098,0.1241,0.8658)T, the soft likelihood value can be obtained by using Eq. (23) as Γs−1,0.1=0×0.8+0.0003×0.56+0.0098×0.3080+0.1241×0.1540+0.8658×0.0770=0.0890.

•   When α=0.5, based on Eq. (14), the weighting vector is ω=(0.2,0.2,0.2,0.2,0.2)T, the soft likelihood value can be obtained by using Eq. (23) as Γs−1,0.5=0.2×0.8+0.2×0.56+0.2×0.3080+0.2×0.1540+0.2×0.0770=0.3798.

•   When α=0.9, based on Eq. (14), the weighting vector is ω=(0.8363,0.0670,0.0416,0.0306,0.0245)T, the soft likelihood value can be obtained by using Eq. (23) as Γs−1,0.9=0.8363×0.8+0.0670×0.56+0.0416×0.3080+0.0306×0.1540+0.0245×0.0770=0.7260.

According to the above process, soft likelihood values of other propositions (s0, s1, and {s0s1}) under different α can be calculated and normalized by using Eq. (31). The results are manifested in Table 1. The conclusion is that (1) when α=0.1, ℒ1⊕⋯⊕ℒ5={s−1(0.9911),s0(0.0045),s0s1(0.0045)}, (2) when α=0.5, ℒ1⊕⋯⊕ℒ5={s−1(0.6732),s0(0.1106),s1(0.0709),s0s1(0.1453)}, (3) when α=0.9, ℒ1⊕⋯⊕ℒ5={s−1(0.5500),s0(0.1613),s1(0.1268),s0s1(0.1619)}.

3.3 The Aggregation Operator Considering the Reliability of Belief-Based PLTS

In practical decision problems, different belief-based PLTSs generally have different importance, i.e., weight, also known as reliability. The aggregation operator proposed in the previous section assumes that all belief-based PLTSs have the same reliability, which leads to certain limitations in decision making. To solve this problem, this section develops an aggregation operator considering reliability.

From the definitions in the previous section, it can be concluded that the advantages of soft likelihood function-based aggregation operator mainly come from two aspects: cumulative multiplication and OWA. Reliability will affect the order determination and weight vector generation.

Definition 3.8. The reliability of {ℒ1,ℒ2,…,ℒn} is denoted as {r1,r2,…,rn} that satisfies rj∈[0,1] and ∑j=1nrj=1. Let γιi be an index function of β(ιi)×r, and γιi(k) indicate the index of the kth largest product of β(ιi)×r. The product operation of the j largest belief of ιi⊆S is defined as

Prod~ιi(j)=∏k=1jβγιi(k)(ιi),(26)

where βγιi(k)(ιi) is the kth largest belief of β(ιi)×r. Note that Prod~ is also a monotonically decreasing function, i.e., if j1≥j2, then Prod~(j1)≤Prod~(j2).

Definition 3.9. The OWA-based soft likelihood function of ιi⊆S considering reliability is defined as

Γ~ιi,OWA=OWA(Prod~ιi(1),Prod~ιi(2),…,Prod~ιi(n))=∑j=1n⁡w~jProd~ιi(j),(27)

where w~j∈[0,1] and ∑j=1nw~j=1 is the weighting vector.

Definition 3.10. To determine the OWA weighting vector, the sum of reliability of the j largest β(ιi)×r product is defined as

χj=∑k=1j⁡rγιi(k).(28)

Consequently, the OWA weighting vector of the soft likelihood function considering reliability can be defined as

w~j=(χj)1−αα−(χj−1)1−αα.(29)

Note that if the reliability of all belief-based PLTSs is equal, i.e., rj=1/n, ω~ is reduced to ω.

Definition 3.11. The aggregation operator of n belief-based PLTSs considering reliability is defined as

ℒ1⊕~ℒ2⊕~⋯⊕~ℒn={ιi(βα)|ιi∈2S,βα(ιi)≥0,∑ιi∈2S⁡βα(ιi)=1},(30)

where ιi(βα) is defined as

ιi(βα)=Γ~ιi,α∑ιi⊆SΓ~ιi,α.(31)

The case in Example 2 is used to demonstrate the aggregation operator considering reliability.

Example 3. Let the reliability of {ℒ1,…,ℒ5} be {r1=0.35,r2=0.25,r3=0.15,r4=0.2,r5=0.05}. Let’s take s−1 as an example, because β1(s−1)×r1=0.1050, β2(s−1)×r2=0.1375, β3(s−1)×r3=0.2800, β4(s−1)×r4=0.1000, and β5(s−1)×r5=0.0250, so γs−1(1)=3, γs−1(2)=2, γs−1(3)=1, γs−1(4)=4, and γs−1(5)=5. Based on Definition 3.8, the product of the j largest belief of s−1 can be calculated as Prod~s−1(1)=β3(s−1)=0.8, Prod~s−1(2)=β3(s−1)×β2(s−1)=0.44, Prod~s−1(3)=Prod~s−1(2)×β1(s−1)=0.3080, Prod~s−1(4)=Prod~s−1(3)×β4(s−1)=0.1540, and Prod~s−1(5)=Prod~s−1(4)×β5(s−1)=0.0770. According to Definitions 3.9 and 3.10, the soft likelihood values of s−1 with different α are calculated considering reliability, and the results are manifested in Table 2.

According to the above process, soft likelihood values of other propositions (s0, s1, and {s0s1}) under different α can be calculated and normalized by using Definition 3.11. Finally, the conclusion is that (1) when α=0.1, ℒ1⊕~⋯⊕~ℒ5={s−1(0.9802),s0(0.0134),s1(0.0036),s0s1(0.0028)} (2) when α=0.5, ℒ1⊕~⋯⊕~ℒ5={s−1(0.6661),s0(0.1197),s1(0.1132),s0s1(0.1010)} (3) when α=0.9, ℒ1⊕~⋯⊕~ℒ5={s−1(0.5335),s0(0.1291),s1(0.1560),s0s1(0.1814)}.

In Examples 2 and 3, we present several special cases to reflect the impact of parameter α on the result of aggregation. In order to further highlight the effect, the experiment is carried of aggregation operator based on soft likelihood function (with ans without) considering reliability. As can be seen from Fig. 1, with the increase of α, the belief of s−1 is decreasing while the other belief is increasing, from which we can see the influence of DM’s attitude on the aggregation result.

Figure 1: The belief distribution of the aggregated belief-based PLTS with different α

3.4 The Comparison between Belief-Based PLTSs

Belief-based PLTS is a new expression of uncertainty based on linguistic information. Due to its complexity, it is not easy to define score functions like PLTS. Therefore, by virtue of belief interval, we propose a novel possibility degree method for comparing belief-based PLTSs.

Definition 3.12. Let ℒ be a belief-based PLTS on S={s−τ,…,s−1,s0,s1,…,sτ}, the belief interval of each singleton constitutes a belief interval vector of ℒ, expressed as

BIℒ⟶=(BIℒ(s−τ),…,BIℒ(s−1),BIℒ(s0),BIℒ(s1),…,BIℒ(sτ))T,(32)

where BIℒ(sα)=[Belℒ(sα),Plℒ(sα)] means the belief interval of linguistic term sα.

Definition 3.13. Let BIℒ→ be the belief interval vector of belief-based PLTS ℒ, the score of BIℒ→ can be calculated as

E(BIℒ→)=[∑α∈S⁡αBel(sα),∑α∈S⁡αPl(sα)]=[belℒ,plℒ].(33)

Definition 3.14. Let ℒ1 and ℒ2 be two belief-based PLTSs, BIℒ1→ and BIℒ2→ be their belief interval vector, and the corresponding scores be E(BIℒ1→)=[belℒ1,plℒ1] and E(BIℒ2→)=[belℒ2,plℒ2], the possibility degree for ℒ1≻ℒ2 can be defined as

p(ℒ1⪰ℒ2)=min{Iℒ1+Iℒ2,max{plℒ1−belℒ2,0}}Iℒ1+Iℒ2,(34)

where Iℒ1=plℒ1−belℒ1 and Iℒ2=plℒ2−belℒ2. The order relationship between ℒ1 and ℒ2 is denoted by ℒ1⪰ℒ2p.

Based on above definitions, under the assumption that ℒ1, ℒ2, ℒ3 are three belief-based PLTSs and their scores are E(BIℒ1→)=[belℒ1,plℒ1], E(BIℒ2→)=[belℒ2,plℒ2], E(BIℒ3→)=[belℒ3,plℒ3], four theorems as below can be easily deduced.

Theorem 3.1. p(ℒ1⪰ℒ2)∈[0,1].

Theorem 3.2. p(ℒ1⪰ℒ2)=1 iff plℒ2≤belℒ1.

Theorem 3.3. p(ℒ1⪰ℒ2)=0 iff plℒ1≤belℒ2.

Theorem 3.4. p(ℒ1⪰ℒ2)+p(ℒ1<ℒ2)=1. Particularly, p(ℒ1⪰ℒ1)=0.5.

The above theorems are obvious, so the proof is omitted. Definition 3.14 provides the possible degree calculation method for two belief-based PLTSs sorting, and we further extend it to the scenario of n belief-based PLTSs.

Definition 3.15. Let a set of belief-based PLTSs with n elements be represented as ℒ={ℒ1,ℒ2,…,ℒn}, and the score of ℒi is E(BIℒi→)=[belℒi,plℒi], i∈{1,…,n}. The possible degree matrix of ℒ can be obtained as P=(pij)n×n, where pij=p(ℒi⪰ℒj), ℒi,ℒj∈ℒ. Matrix P contains the possible degree relations among all belief-based PLTSs, so sorting ℒ is transformed to solve the sorting vector problem of matrix P. Based on [59], the sort value of belief-based PLTS ℒi can be calculated as

ρ(ℒi)=1n(n−1)(∑j=1n⁡pij+n2−1),ℒi∈ℒ,(35)

which can be employed to sort ℒ.

The belief-based PLTSs in Example 2 are used to be sorted by using the above method in this section.

Example 4. We take ℒ1 as an example to demonstrate the calculation of its score of belief interval vector. Based on Definitions 3.2, 3.3 and 3.12, the belief interval vector of ℒ1 can be calculated as

BIL1→=(BIL1(s−1),BIL1(s0),BIL1(s1))T=([BelL1(s−1),PlL1(s−1)],[BelL1(s0),PlL1(s0)],[BelL1(s1),PlL1(s1)])T=([0.7,0.7],[0.2,0.3],[0,0.1])T.(36)

Then the score of BIℒ1→ can be calculated by using Definition 3.13 as

E(BIℒ1→)=[belℒ1,plℒ1]=[(−1)∗0.7+0∗0.2+1∗0,(−1)∗0.7+0∗0.3+1∗0.1]=[−0.7,−0.6].(37)

So the scores of ℒ1,…,ℒ5 can be obtained as E(BIℒ2→)=[−0.35,−0.35], E(BIℒ3→)=[−0.8,−0.6], E(BIℒ4→)=[−0.5,−0.2], and E(BIℒ5→)=[−0.5,−0.2], respectively. The possible degree between two belief-based PLTSs is calculated based on Definition 3.14 below. Taking ℒ1 and ℒ2 as an example, p(ℒ1⪰ℒ2)=0 can be obtained. Then the possible degree matrix can be obtained as