# Differentiating the Personalized Information of the Physician-Patient Communication for the Chronic Obstructive Pulmonary Disease with General Probabilistic Vector Linguistic Terms

Authors: Yuling Zhai; Zeshui Xu; Peijia Ren
Journal: Complexity
 (2019)

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## Abstract

To accurately determine the follow-up therapeutic schedules for the chronic obstructive pulmonary disease (COPD) patients, this paper aims to develop the analysis tools for the linguistic evaluation to improve the quality of the physician-patient communication. Firstly, we define the general probabilistic vector linguistic term (GPVLT), which is effective to depict people’s judgements from different sources. Then, we establish the multigranularity linguistic space and discuss the different forms of the probabilistic vector linguistic units (PVLUs) in it. Later on, we propose the nondirectional and the directional potentials of PVLUs, which can grasp the fuzziness and the development direction of the linguistic evaluations, respectively. Last but not least, the cases about the physician-patient communication for COPD and some comparisons with the other related methods are provided to illustrate the effectiveness and practicability of the PVLUs’ potentials.


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## Body

## 1. Introduction

Chronic obstructive pulmonary disease (COPD) [1], a life-threatening disease of the lungs, reduces the quality of life and causes premature death. With the deterioration of the environment in the 21st century, which is caused by construction dust, vehicle exhaust, and fumes, more and more people suffer from the COPD [2]. The related literature [3] and the Authoritative Medical Science Communication Network Platform release the fundamental knowledge about COPD, such as the symptoms, classifications, and stages. For this incurable disease, appropriate treatment can slow down its progress. Physician-patient communication, which can help patients develop self-management, is a new strongly recommended way to improve the clinical treatments of COPD [3, 4]. In this case, the quality and orientation of the physician-patient communication are considered to be two significant aspects of the care and treatment of COPD [5, 6].As an important component of the medical service, the physician-patient communication establishes a special interpersonal relationship between the medical care providers and the receivers. Physician-patient relationship, which is regarded as a type of complex multiattribute problems of social security system [7, 8], has been investigated from different angles in the past, such as the effects of the treatment [5, 8–10], the communication techniques [10], and so on. From the above researches, we may safely obtain the conclusions that (i) language is the most usual and available way in the physician-patient communication process for COPD [3, 7, 9] and (ii) good communication between the doctors and patients has positive impacts on the diagnosis and the treatment of COPD [4, 10]. It follows that distinguishing the quality and change trends of the linguistic evaluations accurately are very important for the physician-patient communication process.Fuzzy linguistic methods [11], which link the gap between the linguistic fuzziness and the numerical absoluteness, are the straightforward and suitable techniques to handle the decision-making problems with qualitative information. Linguistic evaluation scales (LESs) [11] are the products of fuzzy linguistic methods that each element is the combination symbol of a linguistic descriptor and a real number. Over the past few decades, many kinds of the LESs were presented and investigated, such as the subscript-symmetric LESs [12], the symmetrically and nonuniformly distributed LESs [13], the nonsymmetrically distributed LESs [14, 15], the multiplicative LESs [16], etc. When we use the LESs to evaluate objects and make a decision, the different kinds of linguistic term sets were proposed, like the interval hesitant linguistic term set [17], the hesitant linguistic term set [18], the extended linguistic term set [19], the hesitant fuzzy uncertain linguistic set [20], the dual hesitant fuzzy linguistic set [21], the probabilistic linguistic term set [22, 23], the linguistic 2-tuple model [24], the virtual linguistic model [25, 26], and so on. Moreover, to cope with the specific decision-making problems, some tools, such as the linguistic measures [27], the linguistic integrations [28], the linguistic preference relations [29–31], and the linguistic decision matrices [32, 33], were proposed to process the evaluation information. All the above researches with linguistic evaluations provide the effective procedures to portray the qualitative decision-making problems.However, in the practical physician-patient communication process of COPD, the same linguistic evaluation may indicate the different meanings for different doctors and patients. For example, suppose that a doctor has an individual LESSdoctor = {s−1 = “the most serious,” s−0.7 = “serious,” s−0.5 = “more serious,” s−0.2 = “slight serious,” s0 = “neutral,” s0.2 = “slight normal,” s0.5 = “more normal,” s0.7 = “normal,” s1 = “the most normal”} and a COPD patient has an individual LES Spatient = {s−1 = “the most serious,” s−0.7 = “serious,” s−0.4 = “slight serious,” s0 = “neutral,” s0.4 = “slight normal,” s0.7 = “normal,” s1 = “the most normal”}. The doctor gives “slight serious” as the diagnosis for the patient to explain that the illness is just a slight deviation from “neutral.” But the patient may interpret it as close to “serious” based on his individual LES. Thus, the physician-patient communication of COPD is a multigranularity linguistic decision-making [34–37], and there is a gap between the doctor and patient’s understandings of the same linguistic evaluation “slight serious.” If they fail to properly recognize and deal with the gap in the communication process, it easily leads to poor communication quality or arising conflict.Multigranularity linguistic decision-making is a kind of linguistic decision-making problem that a group of experts are invited to evaluate the same object together based on several different LESs. In the existing results of multigranularity linguistic evaluation, the LESs are assigned into the different hierarchies according to the characteristic granularities of them [34–37]. In the multigranularity linguistic decision-making process, there is an exploitation phase that should be taken before getting the final alternative solution. Although these results provided good ways to assign several LESs with different granularities in a linguistic evaluation process, they are hard to assign different LESs with different distributions. Besides, to fill up the above gap in the physician-patient communication of COPD, the expression of linguistic evaluation should be able to describe and distinguish the meanings of the same linguistic evaluation from different LESs with different distributions. Although the above-mentioned various linguistic terms can address linguistic evaluations well in decision-making, they are lacking in distinguishing the same linguistic evaluation for different experts based on different individual LESs. To address this issue, in 2016, Zhai et al. [38] introduced a probabilistic linguistic vector expression that can be directly driven from people’s own individual LESs. Through a numerical illustration of a personal hospital selection-recommender system, the probabilistic vector expression model of linguistic term has been verified that it can distinguish the different semantics of linguistic terms with the same numerical symbols better than the numerical symbol [38]. In 2017, Li et al. [39] proposed a personalized individual semantics model, which is driven based on a consistency optimization model of linguistic preference relation, by means of interval numerical scales and a 2-tuple linguistic model [24]. Furthermore, this model was extended to the hesitant fuzzy linguistic information environment [40] and was applied in a consensus process of the large-scale linguistic group decision-making [41]. Considering that the probabilistic vector expression contains the probability description of linguistic term and the numerical example in reference [38] is not related to the linguistic preference relation with the consistency problem and the consensus process, we choose the probabilistic vector linguistic term (PVLT) [38] as the research basis of this paper. This paper is the follow-up study of reference [38], which continues the study perspective of hierarchies in people’s cognitive conscious that proposed the PVLT in reference [38] and aims to improve the effectiveness of PVLT in algebraic calculations.PVLT [38] melts the different LESs with different distributions into a plane rectangular coordinate system and expresses them by vectors. It is a convenient technique to distinguish the same linguistic evaluation from different LESs in the multigranularity linguistic decision-making. PVLT can simultaneously take the values and the directions of the linguistic evaluation into account, based on which we can analyze the quality of physician-patient communication in more detail. For example, the values and the change directions of the linguistic evaluation are both important for doctors to determine the follow-up therapeutic schedules for each COPD patient. All linguistic evaluations in a time period (from the start time point to the terminal time point) make up an information flow. Based on PVLT, we can grasp the development direction of the linguistic evaluation flow, which can help us accurately determine the follow-up communication strategies for the treatments of COPD. It conforms to the needed techniques in the physician-patient communication of COPD treatment process.Even though the PVLT can comprehensively reflects people’s judgements based on their own individual LESs, it still has the limitation that the ordinate of the vector in the PVLT falls in the interval−∞,+∞. It may be inconvenient in the integral process of linguistic evaluations [38]. To improve the computational performance of the ordinate of the vector in PVLT, this paper proposes the general PVLT (GPVLT) to limit the ordinate of the vector in PVLT on the interval 0,1. This way of defining GPVLT can reflect people’s different modes of giving linguistic evaluations in decision-making. As a result, the paper aims to define the GPVLT to improve the computational capability of the ordinate of the vector in PVLT and to develop the tools based on GPVLT to handle the physician-patient communication problems for COPD. The main contributions of this paper are listed below:(1)
We introduce the individual vector linguistic system to describe the individual characteristic relations among the domain of discussion, the membership functions and the LES. Through the individual vector linguistic system, all numerical linguistic evaluations can be transformed into the vectors.(2)
We define the multigranularity vector linguistic space (MGVLS), in which the vector linguistic evaluation has the statistic properties, to deal with the multigranularity decision-making problems. In the MGVLS, we extend the PVLT into the GPVLT.(3)
We study several forms of the probabilistic vector linguistic units (PVLUs), which are different kinds of combinations of the GPVLTs. The nondirectional and the directional potentials of PVLUs are proposed.(4)
We apply the potentials of the PVLUs into the physician-patient communication for COPD to illustrate their effectiveness and practicality. Some comparisons are illustrated to show the advantages of the new proposed methods, where some drawbacks of them are also indicated.The rest of this paper is organized as follows: Section2 briefly reviews some related concepts of the PVLT. Section 3 defines the individual vector LESs, the MGVLS and the GPVLT. We also present the PVLU and discuss the different forms of it in this section. Section 4 introduces two kinds of potentials for different forms of the PVLUs, i.e., the nondirectional and the directional potentials. The two potentials are applied to deal with the physician-patient communication for COPD in Section 5, which manifests the effectiveness and the applicability of the potentials. Section 6 first compares the new proposed methods with the other related methods and then discusses the drawbacks and the advantages of them. Finally, Section 7 draws some conclusions of the paper and indicates the relevant further studies.


## 2. Preliminaries

In this section, we review some basic operations of the PVLTs.Definition 1 (see [38]).
LetSkk=1N=Skk=1,2,…,N;N∈N+ be a set of LESs. When a group of experts choose the linguistic terms from the LES Sk to evaluate the objects, the following steps can be used to transform all the selected linguistic evaluations into the normalized vector linguistic terms:
Letsα∈sατ,sατ+1 be a linguistic term [42] of Sk, where sατ and sατ+1 are the τ−th and τ+1−th linguistic terms of Sk, respectively. Here, we take sα as an example to show the transformation steps:(1)
Normalizing the linguistic termsα to the normalized linguistic term sα↔ by the equation α↔=α−minsβsβ∈Sk/maxsβsβ∈Sk−minsβsβ∈Sk.(2)
Calculating the relative change ratiorrsα↔ of the normalized linguistic term sα↔ by rrsα↔=α−ατ/ατ+1−α−1.(3)
Obtaining the absolute change ratioarsα↔ of sα↔ by arsα↔=∑k=1τrrsα↔k+α−ατ/ατ+1−α−1.(4)
Transforming the normalized linguistic termsα↔ into a vector. In the plane rectangular coordinate system sα↔−O−arsα↔, the abscissa axis indicates the linguistic evaluation sα↔ and the vertical axis indicates the absolute change rate of sα↔. Let i→ be the unit vector of the abscissa axis and j→ be the unit vector of the vertical axis, then the linguistic evaluation sα↔ can be transformed into the normalized vector linguistic term s→α↔=sα↔i→+arsα↔j→.
Additionally, if we consider the statistic property ofs→α↔ (denoted as ps→α↔), then the PVLT can be obtained, which is s→α↔,ps→α↔=sα↔i→+arsα↔j→,ps→α↔.
Note that, in the above normalization and transformation processes of the linguistic termsα, all the calculations are applied for the subscript α, where the corresponding linguistic word is unchanged.


## 3. The GPVLT and the Forms of the PVLUs in the MGVLS

In this section, we first define the individual linguistic system to describe the individual characteristic relations among the domain of discussion, the membership functions, the LESs, and the interval0,1. Due to that different people may have different expertise and knowledge levels, in the first section, we also propose the multigranularity linguistic space to simultaneously analyze the experts’ linguistic evaluations based on the individual LESs. Through the transformation steps of Definition 1, each linguistic evaluation in the multigranularity linguistic space can be transformed into the normalized vector based on the individual LES. Thus, the MGVLS is proposed in the second section, based on which the concept of GPVLT is presented in the third section. Last but not least, we study the different forms of the PVLUs.


### 3.1. The Individual Linguistic System and the Multigranularity Linguistic Space

#### 3.1.1. The Individual Linguistic System

Linguistic evaluation, a popular way to describe the evaluations and the opinions of the experts, is hard to be quantified for its fuzziness of the words. To handle this qualitative information in the decision-making problems, the LES is utilized to link the linguistic evaluations and the real numbers. LetX=x be the domain of discourse and LEs be the linguistic evaluations. Based on a set of membership functions μx:X⟶LEs, we can evaluate each x∈X by a set of linguistic evaluations with the membership degrees. Then the normalized LES translates the linguistic evaluations into the values in the interval 0,1 by the membership functions μS:0,1⟶LEs. Therefore, the normalized LES can be regarded as the distribution of the linguistic evaluations on 0,1.For example, for an issue “selecting the appropriate retirement age to alleviate the employment pressure,” we choose the interval0,100 as the domain of discourse X and assign L = {l1 = “young,” l2 = “appropriate,” l3 = “old”} as the set of the linguistic evaluations. As shown in Figure 1, the relation between X and L can be described by μx:X⟶L in the plane rectangular coordinate system X−O−L, which corresponds to people’s linguistic evaluation procedure.Figure 1
An example of the individual linguistic systemΩX,μx,L,μS,S.
(a)
(b)The right plane rectangular coordinate systemS−O−L in Figure 1 describes the relations among the linguistic evaluations, the normalized LES S, and the interval 0,1, which corresponds to people’s numerical evaluation criteria. For instance, a person illustrated by Figure 1 thinks value 0 is equal to “young,” 0.5 is equal to “appropriate” and 1 is equal to “old.” Therefore, the membership levels of 0 to “young,” 0.5 to “appropriate,” and 1 to “old” are all 1, where the normalized LES S1 = {s0 = “young,” s0.5 = “appropriate,” s1 = “old”} expresses this strong correspondences. In addition, the membership levels of the relations between all other values of 0,1 and the linguistic evaluations are less than 1, which can be seen as the weak correspondences. If we denote 0,1 as the interval s0,s1, then all the strong and weak correspondences between L and 0,1 can be interpreted by μS:s0,s1⟶L.Definition 2.
LetX be the domain of discourse, L=lkk=1,2,…,N;N∈N+ be a set of the linguistic evaluations, S=sατα∈0,1;τ=1,2,…,N,N∈N+ be the normalized LES with respect to L, μx:X⟶L, and μS:s0,s1⟶L be two sets of the membership functions, then we call the system built by the above components as an individual linguistic system, denoted by ΩX,μx,L,μS,S.
For the individual linguistic systemΩX,μx,L,μS,S, there are two plane rectangular coordinate systems X−O−L and S−O−L, which correspond to people’s linguistic evaluation procedures and numerical evaluation criteria, respectively.
By Figure1, we find that (i) X is divided by the linguistic evaluations L based on the membership functions μtx:X⟶Lt=1,2,…,N1;N1∈N+ and (ii) the mapping between L and S is one to one; (iii) the interval s0,s1 is divided by the normalized LES S based on the membership function set utS:s0,s1⟶Lt=1,2,…,N2;N∈N+. Let a positive integer N be the number of subregions of X divided by μx:X⟶L, then we can deduce that the numbers of L, S, and the segmentations of s0,s1 are equal to N. This reflects the uniformity of the individual linguistic system ΩX,μx,L,μS,S with respect to N, which is demonstrated in Figure 2.Figure 2
The relations amongX, L, S, and the interval s0,s1.


#### 3.1.2. The Multigranularity Linguistic Space

Let us continue with the problem of “selecting the appropriate retirement age for alleviating the employment pressure.” Another person may assign age 60 as the “appropriate” with the membership level 1, and his/her normalized LES may beS2 = {s0 = “young,” s0.3 = “slightly young,” s0.5 = “appropriate,” s0.75 = “slightly old,” s1 = “old”}. Thus, from the normalized LESs S1 and S2, it is obvious that people’s evaluation thinking and judgement criteria are different. Denoting the individual linguistic systems provided by the two persons as Ω1=ΩX,μ1x,L1,μ1S,S1, and Ω2=ΩX,μ2x,L2,μ2S,S2, respectively, and gathering the two individual linguistic systems together, we can obtain an information space (as shown in Figure 3).Figure 3
An example of an information space constructed withΩ1 and Ω2. (a) Ω1 with S1 = {s0 = young, s0.5 = appropriate, s1 = old}. (b) Ω2 with S2 = {s0 = young, s0.3 = slightly young, s0.5 = appropriate, s0.75 = slightly old, s1 = old}.
(a)
(b)In Figure3, the individual linguistic systems Ω1 and Ω2 are two different linguistic evaluation sources, where the numbers of S1 and S2 are 3 and 5, respectively. For any given linguistic evaluation, it comes from Ω1 or Ω2. Then the status-parallel components Ω1 and Ω2 develop a greater information space.Definition 3.
LetΩτ=ΩX,μτx,Lτ,μτS,Sτ, τ=1,2,…,N be N individual linguistic systems for the same domain of discourse X, we call Ωττ=1N=ΩX,μτx,Lτ,μτS,Sττ=1,2,…,N a multigranularity linguistic space constructed by the individual linguistic systems Ω1, Ω2, …, and ΩN.
The multigranularity linguistic spaceΩττ=1N is the largest collection of the linguistic evaluations got from different individual linguistic systems Ω1, Ω2, …, and ΩN. Let sα be a linguistic term of the multigranularity linguistic space. Adding the statistic property of it into sα, we get the probabilistic linguistic term sα,psα. When we use any sα,psα from the multigranularity linguistic space to make a decision, the fundamental work is to analyze the meanings of sα. Taking the multigranularity linguistic space illustrated by Figure 3 as an example, we deeply discuss the meanings of s0.4. It is associated with “appropriate” with the membership level 0.65 in Ω1, while it is associated with “appropriate” with the membership level 0.45 in Ω2. Therefore, when several people give a symbol like s0.4 to express their evaluations, it is hard to distinguish the specific meaning of each people.


### 3.2. The MGVLS

By Definition 1, we transform the above LESsS1 and S2 into the normalized vector forms S→1 and S→2, where S→1 = {s0i→−j→ = “young,” s0.5i→−j→ = “appropriate,” s1i→+∞j→ = “old”} and S→2 = {s0i→−j→ = “young,” s0.3i→−0.5j→ = “slightly young,” s0.5i→−0.7j→ = “appropriate,” s0.75i→−0.7j→ = “slightly old,” s1i→+∞j→ = “old”}. Based on S→1, s0.4 is transformed into s0.4i→+2j→. Similarly, based on S→2, s0.4 is transformed into s0.4i→−0.5j→. By this way, it is easy to distinguish the meanings of s0.4 that comes from Ω1 and Ω2, respectively.Definition 4.
LetΩX,μx,L,μS,S be any individual linguistic system. If any probabilistic linguistic term sα,psα of Ω is transformed into the PVLT s→α,ps→α, then Ω can be converted into the individual vector linguistic system Ω→X,μx,L,μS→,S→.
Based on Definitions 3 and 4, we naturally get the concept of the MGVLS as follows:Definition 5.
LetΩ→τ=Ω→X,μτx,Lτ,μτS→,S→ττ=1,2,…,N be N individual vector linguistic systems for the same domain of discourse X, then the individual vector linguistic systems Ω→1, Ω→2, …, and Ω→N construct a MGVLS, denoted by Ω→ττ=1N=Ω→X,μτx,Lτ,μτS→,S→ττ=1,2,…,N.Example 1.
LetΩ→1 and Ω→2 be the corresponding individual vector linguistic systems of S→1 = {s0i→−j→ = “poor,” s0.5i→ = “neutral,” s1i→+∞j→ = “good”} and S→2 = {s0i→−j→ = “poor,” s0.3i→+0.5j→ = “slightly poor,” s0.5i→−0.2j→ = “neutral,” s0.75i→ = “slightly good,” s1i→+∞j→ = “good”}. Then the part of people’s numerical evaluation criteria of the MGVLS Ω→ττ=12 is the infinite plane area s0≤sα≤s1,−∞≤rsαi→≤+∞ depicted in Figure 4.
Comparing the expression of people’s numerical evaluation criteria in the multigranularity linguistic space (refer to Figure3) and the MGVLS in Figure 4, we get that the linguistic evaluations without probabilities in the multigranularity linguistic space are unidimensional, whereas the ones in the MGVLS are dimensional. Therefore, the MGVLS is a stretched form of the multigranularity linguistic space. In the MGVLS, the linguistic evaluation can be distinguished subtler.Figure 4
The part of people’s numerical evaluation criteria of MGVLSΩ→ττ=12 derived by Ω→1 and Ω→2.


### 3.3. The GPVLT

Lets→α,ps→α be any PVLT of the MGVLS Ω→ττ=12 as shown in Figure 4. Noticing the ordinate rsαi→ of s→α, we find that it monotonically increases while sα moves from sατ to sατ+1, where sατ and sατ+1 are the τ−th and τ+1−th linguistic terms of the normalized LES S, respectively. The monotonicity of rsαi→ accords with the left side membership function μsα. In such a case, rsαi→ can be regarded as the rough expression of μsα. Due to the complexity in the membership function determination process, the PVLT is definitely a sensible choice to deal with the probabilistic vector linguistic evaluations in the MGVLS. However, these exist some problems in the simulations with the PVLT: (i) rsαi→ cannot express the right side of μsα and (ii) the values of rsαi→ are all located in −∞,+∞, but the values of the membership function μsα are limited in 0,1. If we regard rsαi→ as the simulation of μsα, the ranges of rsαi→ and μsα are different. Furthermore, ∞ is not able to be handled in the specific computational process. For example, the infinity ∞ can change the convergence and divergence of the integral. To solve these two issues, we present the concept of the GPVLT below.Definition 6.
LetΩ=ΩX,μx,L,μS,S be any individual linguistic system, where S=sατsατ∈s0,s1,τ=1,2,…,N,sα1=s0,sαN=s1,N∈N+ sα1=s0,sαN=s1,N∈N+ is a normalized LES and sατ is the τ−th linguistic term of the normalized LES. For any probabilistic linguistic term sα,psα of Ω, if we transform it into s→α,ps→α, where s→α=sαi→+rsαj→, in which rsα:s0,s1⟶R is a function that reflects the membership functions μS, then s→α,ps→α is the GPVLT with respect to sα,psα.Remark 1.
Although Definition 6 does not restrict the specific characteristic ofrsα, we usually select the functions for rsα according to the following criteria.
For anysατ∈S, which is the target linguistic term, then,(i)
rsα monotonically increases, if sατ−1<sα<sατ, τ=2,3,…,N(ii)
rsα monotonically decreases, if sατ<sα<sατ+1, τ=1,2,…,N−1
The PVLT presented by Definition 1 is a particular case of the GPVLT.
For the second normalized LESS2 in Figure 3, according to Definition 6 and Remark 1, we can assign(1)r1sα=α−ατ−1ατ−ατ−1,ατ−1<α<ατ,τ−1≥0;1,α=ατ;ατ+1−αατ+1−ατ,ατ<α<ατ+1,τ+1≤5,orr2sα=α−ατ−1⋅4/π⋅arctan1/ατ−α−1ατ−ατ−1,ατ−1<α<ατ,τ−1≥0;1,α=ατ;ατ+1−α⋅4/π⋅arctan1/ατ−α−1ατ+1−ατ,ατ<α<ατ+1,τ+1≤5,as the coefficient function of j→ within s→α=sαi→+rsαj→,ps→α, where τ=1,2,3,4,5, sα1=s0 and sα5=s1. Functions r1sα and r2sα can be seen in Figure 5.
As shown in Figure5, the monotonicity and ranges of r1sα and r2sα conform to the membership functions μ2S. Therefore, r1sα or r2sα is more reasonable and precise to be the coefficient function of j→ compared with the one obtained by Definition 1.
As stated in Section3.2, the part of people’s numerical evaluation criteria of any MGVLS can be expressed by a plane area A:s0≤sα≤s1,a≤rsα≤b in a plane rectangular coordinate system S−O−rsα, where a,b is the range of rsα. Each point D in the area A corresponds to a vector s→α=sαi→+rsαj→. If more than one person provides the point D to express the linguistic evaluations, then the statistic property of the point D can be presented as ps→α=pD, i.e., GPVLT D,pD. In the practical linguistic evaluation process, the experts only need to provide the LESs and the linguistic evaluations, which is as easy as the existing linguistic decision-making methods. By calculating the frequency of the occurrence of the linguistic terms, their statistics properties can be obtained.
The GPVLTs are the fundamental components of the MGVLSΩ→ττ=1N. Considering that the corresponding strength of the linguistic evaluations and the values in s0,s1, we give the following definition to classify the GPVLTs:Figure 5
Two examples ofrsα within Definition 6.
(a)
(b)Definition 7.
LetΩ→ττ=1N=Ω→X,μτx,Lτ,μτS→,S→ττ=1,2,…,N be a MGVLS and s→α,ps→α be any GPVLT in Ω→ττ=1N. If s→α is an element of S→τ, then we call s→α,ps→α the explicit GPVLT. Otherwise, we call it the implicit GPVLT.
According to Definition 7, for any explicit GPVLTs→α,ps→α, the ordinate coefficient of s→α is the maximum of rsα, which means that the linguistic evaluation (described by s→α) strongly corresponds to the value α. For example, in the MGVLS of Example 1, s0.75i→,ps0.75i→ is an explicit GPVLT, whereas s0.76i→,ps0.76i→ is an implicit PVLT. Then the explicit-implicit property of the GPVLTs is the location property in the plane area A:s0≤sα≤s1,a≤rsα≤b. Based on the understandings with respect to s→α, rsα, and ps→α, we can compare them by a score function:Definition 8.
The score function of a GPVLTsαi→+rsαj→,ps→α expressed as SCs→α,ps→α=α⋅rsα⋅ps→α was proposed to rank any two GPVLTs sα1i→+rsα1j→,ps→α1 and sα2i→+rsα2j→,ps→α2 in a MGVLS.(i)
IfSCs→α1,ps→α1>SCs→α2,ps→α2, then sα1i→+rsα1j→,ps→α1≻sα2i→+rsα2j→,ps→α2(ii)
IfSCs→α1,ps→α1<SCs→α2,ps→α2, then sα1i→+rsα1j→,ps→α1≺sα2i→+rsα2j→,ps→α2(iii)
IfSCs→α1,ps→α1=SCs→α2,ps→α2, then sα1i→+rsα1j→,ps→α1=sα2i→+rsα2j→,ps→α2
where “≻” indicates “superior to” and “≺” indicates “inferior to.”
The score function of GPVLT is a real mapping froms0,s1×minrsα,maxrsα×0,1 to the interval 0,1. Besides, it is an increasing function of sα, rsα, and ps→α, respectively.Example 2.
LetΩττ=12=ΩX,μτx,Lτ,μτS,Sττ=1,2 be a multigranularity linguistic space, where S1 = {s0 = “none,” s0.3 = “slightly low,” s0.5 = “neutral,” s0.75 = “slightly high,” s1 = “high”} and S2 = {s0 = “none,” s0.28 = “very low,” s0.42 = “low,” s0.5 = “neutral,” s0.58 = “high,” s0.72 = “very high,” s1 = “perfect”}.
For the given linguistic evaluations0.65, we analyze the meanings of it in Ω1 and Ω2, respectively. By using rsα, r1sα, and r2sα shown in Definition 1 and equation (1), we can translate s0.65 and the elements in S1, S2 into the GPVLTs. Assume that there is no repetition of the GPVLT, and the probability of each GPVLT is equal to 1.
All the transformed results listed in Table1 are the implicit GPVLTs of Ωττ=12; hence, we use the nearby principle to select the linguistic evaluation for s0.65. According to the comparisons of the abscissa of s→α, all transformed results are between “neutral” and “slightly high” in Ω1. But from the ordinate of s→α, we find that s0.65i→+3/5j→,1 is the highest. By using the Euclidean measure of the plane rectangular coordinate system, we can calculate all the distances between the transformed GPVLTs and their adjacent explicit GPVLTs. For the given r1sα, the distance between s0.65i→+3/5j→,1 and “slightly high” is less than the distance between s0.65i→+2/5j→,1 and “neutral”; then we should translate s0.65 into “slightly high” for Ω1 when the ordinate coefficient function of s→α is r1sα. In the same manner, we can get other translated results. Although we get the different distances between the transformed GPVLTs and their adjacent explicit GPVLTs through r1sα and r2sα, respectively, the selections are same. Moreover, if we transform s0.65 by the rsα presented in Definition 1, we cannot select the better one between “high” and “very high” for s0.65 in Ω2.
Compared to the above nearby principle, we can use the score function of GPVLT to easily make a choice between two adjacent explicit GPVLTs. Fors0.65i→+2/5j→,1 and s0.65i→+3/5j→,1 in Ω1, which are transformed based on r1sα, the difference between them is the ordinates of j→. Considering that the ordinates of j→ in the adjacent explicit GPVLTs are 1 (maxr1sα=1) and the score function is the increasing function of r1sα, we should translate s0.65 into “slightly high” for Ω1. It is obvious that we can get the same selections if we use the score function of GPVLT to compare the transformed GPVLTs in Table 1.Table 1
The transformed results ofs0.65.
The coefficient function ofj→
Source
The transformed GPVLTs
The adjacent explicit GPVLTs
The score of each transformed GPVLT
The Euclidean distance between the transformed LVET and the adjacent explicit GPVLTs
The meanings ofs0.65
r
Ω
1
s
0.65
i
→
−
1
/
5
j
→
,
1
s
0.5
i
→
+
j
→
,
1 neutral
0.130
1.209
Not select
r
1
s
0.65
i
→
+
2
/
5
j
→
,
1
0.260
0.618
Not select
r
2
s
0
.
65
i
→
+
81
/
2
5
0
j
→
,
1
0.211
0.692
Not select
r
s
0.65
i
→
−
1
/
5
j
→
,
1
s
0.75
i
→
+
j
→
,
1 slightly high
0.130
1.204
Select
r
1
s
0
.
65
i
→
+
3
/
5
j
→
,
1
0.390
0.412
Select
r
2
s
0.65
i
→
+
131
/
250
j
→
,
1
0.341
0.486
Select
r
Ω
2
s
0.65
i
→
+
37
/
195
j
→
,
1
s
0.58
i
→
+
j
→
,
1 high
0.123
0.813
—
r
1
s
0.65
i
→
+
13
/
25
j
→
,
1
0.338
0.485
Select
r
2
s
0.65
i
→
+
129
/
250
j
→
,
1
0.335
0.489
Select
r
s
0.65
i
→
+
37
/
195
j
→
,
1
s
0.72
i
→
+
j
→
,
1 very high
0.123
0.813
—
r
1
s
0
.
65
i
→
+
12
/
2
5
j
→
,
1
0.312
0.525
Not select
r
2
s
0.65
i
→
+
109
/
250
j
→
,
1
0.283
0.568
Not select


### 3.4. The Forms of the PVLU

Case 1.
As the pointe1 in Figure 3, for x∈X, there exists a discrete single-point s0.4 to express the evaluation of it. Whether we transform s0.4 into s0.4i→+2j→,1 based on Definition 1 or translate s0.4 into s0.4i→+0.8j→,1 by r1sα given in equation (1), the results are both the discretely single GPVLTs. In this case, we call the GPVLT the discrete single probabilistic vector linguistic unit (DSPVLU) and denote it by sαi→+rsαi→j→,psαi→+rsαi→j→.Case 2.
As the individual linguistic system in Figure6, the element x∈X can be evaluated by “neutral” with the membership level 0.65 and “slightly good” with the membership level 0.25 simultaneously.Translating “neutral” and “slightly good” into the intervals0,s1, we can obtain the linguistic terms sα1 and sα2. The set sα1,sα2 expresses the linguistic evaluation with respect to x. Since sα1 and sα2 are two discrete points, s→α1,ps→α1,s→α2,ps→α2 is a set with two discrete transformed GPVLTs. Generalizing the number of the elements of s→α1,ps→α1,s→α2,ps→α2 to N, we get a discrete multiple probabilistic vector linguistic unit (DMPVLU) and denote it by s→αk,ps→αkk=1N=s→αk,ps→αkk=1,2,…,N;N∈N+;∑k=1Nps→αk≤1.Figure 6
An example of the individual linguistic system.S = {s0 = poor, s0.3 = slightly poor, s0.5 = neutral, s0.75 = slightly good, s1 = good}.Remark 2.
The DMPVLU can be regarded as a collection of the DSPVLUs. But the DSPVLU and the DMPVLU have different meanings for people’s judgement thinking, where the DSPVLU indicates no hesitance and the DMPVLU indicates the hesitance in the judgements. Therefore, if we take any DSPVLU for a unit of the probabilistic vector linguistic evaluation, then the probability ins→α,ps→α reduces to 1. Moreover, if we take any DMPVLU s→αk,ps→αkk=1N for a unit of the probabilistic vector linguistic evaluation, then the probabilities in it should be normalized by p↔s→αk=ps→αk/∑k=1Nps→αk. Generally, the probability of any DSPVLU used below defaults to 1, and all probabilities in any DMPVLU used below are normalized.Case 3.
In Figure6, “s0.5 = neutral” and “s0.75 = slightly good” are two consecutive elements in the normalized LES S. The interval sα1,sα2⊂s0.5,s0.75 interprets people’s fuzziness between the judgements with “neutral” and “slightly good” [42]. If we transform all values in sα1,sα2 into the GPVLTs, then we can obtain a continuous single-interval probabilistic vector linguistic unit s→α1,s→α2,ps→α, where sα1≤sα≤sα2 and ps→α is the probability function of s→α. In such a situation, we call s→αa,s→αb,ps→α the continuous single-interval probabilistic vector linguistic unit (CSIPVLU), where sαa≤sα≤sαb, s→αa,s→αb⊆sατ,sατ+1, and sατ and sατ+1 are two consecutive elements in the normalized LES S, ps→α:s→αa,s→αb⟶0,1 is the probability function of s→α.Remark 3.
Firstly, lets→αa,s→αb,ps→α be any CSIPVLU, then the vector interval s→αa,s→αb is originated from sαa,sαb. Considering that the process of transforming the linguistic term sα into the normalized vector linguistic term s→α does not change the probability of sα, we know that the probabilistic distribution in sαa,sαb is the same as s→αa,s→αb, i.e., psα:sαa,sαb⟶0,1 is identical to ps→α: s→αa,s→αb⟶0,1. Secondly, the numbers of the elements in the DSPVLU and the DMPVLU are both positive integers, whereas the number of the elements in the CSIPVLU is uncountable. Hence, the CSIPVLU can be regarded as the extended form of the DSPVLU and the DMPVLU. Thirdly, any CSIPVLU s→αa,s→αb,ps→α can be normalized by p↔s→α=ps→α/∫s→αas→αbps→α, where ∫s→αas→αbps→α is the definite integral of ps→α on the interval s→αa,s→αb. In general, all CSIPVLUs used below are normalized.Case 4.
For the CSIPVLUs→αa,s→αb,ps→α depicted in Case 3, the condition s→αa,s→αb⊆sατ,sατ+1 and sατ and sατ+1 are two consecutive linguistic terms in the normalized LES” is somewhat rigorous. In the practical situations, it is difficult to reach it for the flexibility of the linguistic expressions and the fuzziness of the judgements. When the interval sαa,sαb falls on sατ,sατ+k⊆s0,s1, where sατ is the τ−th element of the LES S and k≥2, we can divide sαa,sαb into a sequence of the CSIPVLUs s→αa,s→ατ+1,psα,s→ατ+1,s→ατ+2,psα,…,s→ατ+k−1,s→αb,psα. The divided sequence of the CSIPVLUs s→αk,s→αk+1,pks→αk=1N is called as the continuous multi-intervals probabilistic vector linguistic unit (CMIPVLU), where pks→α is a probabilistic distribution on s→αk,s→αk+1.Remark 4.
The probabilities in any CMIPVLUs→αk,s→αk+1,pks→αk=1N can be normalized by p↔ks→α=pks→α/∑k=1Npks→α. Generally, all CMIPVLUs used below are normalized.


## 3.1. The Individual Linguistic System and the Multigranularity Linguistic Space

### 3.1.1. The Individual Linguistic System

Linguistic evaluation, a popular way to describe the evaluations and the opinions of the experts, is hard to be quantified for its fuzziness of the words. To handle this qualitative information in the decision-making problems, the LES is utilized to link the linguistic evaluations and the real numbers. LetX=x be the domain of discourse and LEs be the linguistic evaluations. Based on a set of membership functions μx:X⟶LEs, we can evaluate each x∈X by a set of linguistic evaluations with the membership degrees. Then the normalized LES translates the linguistic evaluations into the values in the interval 0,1 by the membership functions μS:0,1⟶LEs. Therefore, the normalized LES can be regarded as the distribution of the linguistic evaluations on 0,1.For example, for an issue “selecting the appropriate retirement age to alleviate the employment pressure,” we choose the interval0,100 as the domain of discourse X and assign L = {l1 = “young,” l2 = “appropriate,” l3 = “old”} as the set of the linguistic evaluations. As shown in Figure 1, the relation between X and L can be described by μx:X⟶L in the plane rectangular coordinate system X−O−L, which corresponds to people’s linguistic evaluation procedure.Figure 1
An example of the individual linguistic systemΩX,μx,L,μS,S.
(a)
(b)The right plane rectangular coordinate systemS−O−L in Figure 1 describes the relations among the linguistic evaluations, the normalized LES S, and the interval 0,1, which corresponds to people’s numerical evaluation criteria. For instance, a person illustrated by Figure 1 thinks value 0 is equal to “young,” 0.5 is equal to “appropriate” and 1 is equal to “old.” Therefore, the membership levels of 0 to “young,” 0.5 to “appropriate,” and 1 to “old” are all 1, where the normalized LES S1 = {s0 = “young,” s0.5 = “appropriate,” s1 = “old”} expresses this strong correspondences. In addition, the membership levels of the relations between all other values of 0,1 and the linguistic evaluations are less than 1, which can be seen as the weak correspondences. If we denote 0,1 as the interval s0,s1, then all the strong and weak correspondences between L and 0,1 can be interpreted by μS:s0,s1⟶L.Definition 2.
LetX be the domain of discourse, L=lkk=1,2,…,N;N∈N+ be a set of the linguistic evaluations, S=sατα∈0,1;τ=1,2,…,N,N∈N+ be the normalized LES with respect to L, μx:X⟶L, and μS:s0,s1⟶L be two sets of the membership functions, then we call the system built by the above components as an individual linguistic system, denoted by ΩX,μx,L,μS,S.
For the individual linguistic systemΩX,μx,L,μS,S, there are two plane rectangular coordinate systems X−O−L and S−O−L, which correspond to people’s linguistic evaluation procedures and numerical evaluation criteria, respectively.
By Figure1, we find that (i) X is divided by the linguistic evaluations L based on the membership functions μtx:X⟶Lt=1,2,…,N1;N1∈N+ and (ii) the mapping between L and S is one to one; (iii) the interval s0,s1 is divided by the normalized LES S based on the membership function set utS:s0,s1⟶Lt=1,2,…,N2;N∈N+. Let a positive integer N be the number of subregions of X divided by μx:X⟶L, then we can deduce that the numbers of L, S, and the segmentations of s0,s1 are equal to N. This reflects the uniformity of the individual linguistic system ΩX,μx,L,μS,S with respect to N, which is demonstrated in Figure 2.Figure 2
The relations amongX, L, S, and the interval s0,s1.


### 3.1.2. The Multigranularity Linguistic Space

Let us continue with the problem of “selecting the appropriate retirement age for alleviating the employment pressure.” Another person may assign age 60 as the “appropriate” with the membership level 1, and his/her normalized LES may beS2 = {s0 = “young,” s0.3 = “slightly young,” s0.5 = “appropriate,” s0.75 = “slightly old,” s1 = “old”}. Thus, from the normalized LESs S1 and S2, it is obvious that people’s evaluation thinking and judgement criteria are different. Denoting the individual linguistic systems provided by the two persons as Ω1=ΩX,μ1x,L1,μ1S,S1, and Ω2=ΩX,μ2x,L2,μ2S,S2, respectively, and gathering the two individual linguistic systems together, we can obtain an information space (as shown in Figure 3).Figure 3
An example of an information space constructed withΩ1 and Ω2. (a) Ω1 with S1 = {s0 = young, s0.5 = appropriate, s1 = old}. (b) Ω2 with S2 = {s0 = young, s0.3 = slightly young, s0.5 = appropriate, s0.75 = slightly old, s1 = old}.
(a)
(b)In Figure3, the individual linguistic systems Ω1 and Ω2 are two different linguistic evaluation sources, where the numbers of S1 and S2 are 3 and 5, respectively. For any given linguistic evaluation, it comes from Ω1 or Ω2. Then the status-parallel components Ω1 and Ω2 develop a greater information space.Definition 3.
LetΩτ=ΩX,μτx,Lτ,μτS,Sτ, τ=1,2,…,N be N individual linguistic systems for the same domain of discourse X, we call Ωττ=1N=ΩX,μτx,Lτ,μτS,Sττ=1,2,…,N a multigranularity linguistic space constructed by the individual linguistic systems Ω1, Ω2, …, and ΩN.
The multigranularity linguistic spaceΩττ=1N is the largest collection of the linguistic evaluations got from different individual linguistic systems Ω1, Ω2, …, and ΩN. Let sα be a linguistic term of the multigranularity linguistic space. Adding the statistic property of it into sα, we get the probabilistic linguistic term sα,psα. When we use any sα,psα from the multigranularity linguistic space to make a decision, the fundamental work is to analyze the meanings of sα. Taking the multigranularity linguistic space illustrated by Figure 3 as an example, we deeply discuss the meanings of s0.4. It is associated with “appropriate” with the membership level 0.65 in Ω1, while it is associated with “appropriate” with the membership level 0.45 in Ω2. Therefore, when several people give a symbol like s0.4 to express their evaluations, it is hard to distinguish the specific meaning of each people.


## 3.1.1. The Individual Linguistic System

Linguistic evaluation, a popular way to describe the evaluations and the opinions of the experts, is hard to be quantified for its fuzziness of the words. To handle this qualitative information in the decision-making problems, the LES is utilized to link the linguistic evaluations and the real numbers. LetX=x be the domain of discourse and LEs be the linguistic evaluations. Based on a set of membership functions μx:X⟶LEs, we can evaluate each x∈X by a set of linguistic evaluations with the membership degrees. Then the normalized LES translates the linguistic evaluations into the values in the interval 0,1 by the membership functions μS:0,1⟶LEs. Therefore, the normalized LES can be regarded as the distribution of the linguistic evaluations on 0,1.For example, for an issue “selecting the appropriate retirement age to alleviate the employment pressure,” we choose the interval0,100 as the domain of discourse X and assign L = {l1 = “young,” l2 = “appropriate,” l3 = “old”} as the set of the linguistic evaluations. As shown in Figure 1, the relation between X and L can be described by μx:X⟶L in the plane rectangular coordinate system X−O−L, which corresponds to people’s linguistic evaluation procedure.Figure 1
An example of the individual linguistic systemΩX,μx,L,μS,S.
(a)
(b)The right plane rectangular coordinate systemS−O−L in Figure 1 describes the relations among the linguistic evaluations, the normalized LES S, and the interval 0,1, which corresponds to people’s numerical evaluation criteria. For instance, a person illustrated by Figure 1 thinks value 0 is equal to “young,” 0.5 is equal to “appropriate” and 1 is equal to “old.” Therefore, the membership levels of 0 to “young,” 0.5 to “appropriate,” and 1 to “old” are all 1, where the normalized LES S1 = {s0 = “young,” s0.5 = “appropriate,” s1 = “old”} expresses this strong correspondences. In addition, the membership levels of the relations between all other values of 0,1 and the linguistic evaluations are less than 1, which can be seen as the weak correspondences. If we denote 0,1 as the interval s0,s1, then all the strong and weak correspondences between L and 0,1 can be interpreted by μS:s0,s1⟶L.Definition 2.
LetX be the domain of discourse, L=lkk=1,2,…,N;N∈N+ be a set of the linguistic evaluations, S=sατα∈0,1;τ=1,2,…,N,N∈N+ be the normalized LES with respect to L, μx:X⟶L, and μS:s0,s1⟶L be two sets of the membership functions, then we call the system built by the above components as an individual linguistic system, denoted by ΩX,μx,L,μS,S.
For the individual linguistic systemΩX,μx,L,μS,S, there are two plane rectangular coordinate systems X−O−L and S−O−L, which correspond to people’s linguistic evaluation procedures and numerical evaluation criteria, respectively.
By Figure1, we find that (i) X is divided by the linguistic evaluations L based on the membership functions μtx:X⟶Lt=1,2,…,N1;N1∈N+ and (ii) the mapping between L and S is one to one; (iii) the interval s0,s1 is divided by the normalized LES S based on the membership function set utS:s0,s1⟶Lt=1,2,…,N2;N∈N+. Let a positive integer N be the number of subregions of X divided by μx:X⟶L, then we can deduce that the numbers of L, S, and the segmentations of s0,s1 are equal to N. This reflects the uniformity of the individual linguistic system ΩX,μx,L,μS,S with respect to N, which is demonstrated in Figure 2.Figure 2
The relations amongX, L, S, and the interval s0,s1.


## 3.1.2. The Multigranularity Linguistic Space

Let us continue with the problem of “selecting the appropriate retirement age for alleviating the employment pressure.” Another person may assign age 60 as the “appropriate” with the membership level 1, and his/her normalized LES may beS2 = {s0 = “young,” s0.3 = “slightly young,” s0.5 = “appropriate,” s0.75 = “slightly old,” s1 = “old”}. Thus, from the normalized LESs S1 and S2, it is obvious that people’s evaluation thinking and judgement criteria are different. Denoting the individual linguistic systems provided by the two persons as Ω1=ΩX,μ1x,L1,μ1S,S1, and Ω2=ΩX,μ2x,L2,μ2S,S2, respectively, and gathering the two individual linguistic systems together, we can obtain an information space (as shown in Figure 3).Figure 3
An example of an information space constructed withΩ1 and Ω2. (a) Ω1 with S1 = {s0 = young, s0.5 = appropriate, s1 = old}. (b) Ω2 with S2 = {s0 = young, s0.3 = slightly young, s0.5 = appropriate, s0.75 = slightly old, s1 = old}.
(a)
(b)In Figure3, the individual linguistic systems Ω1 and Ω2 are two different linguistic evaluation sources, where the numbers of S1 and S2 are 3 and 5, respectively. For any given linguistic evaluation, it comes from Ω1 or Ω2. Then the status-parallel components Ω1 and Ω2 develop a greater information space.Definition 3.
LetΩτ=ΩX,μτx,Lτ,μτS,Sτ, τ=1,2,…,N be N individual linguistic systems for the same domain of discourse X, we call Ωττ=1N=ΩX,μτx,Lτ,μτS,Sττ=1,2,…,N a multigranularity linguistic space constructed by the individual linguistic systems Ω1, Ω2, …, and ΩN.
The multigranularity linguistic spaceΩττ=1N is the largest collection of the linguistic evaluations got from different individual linguistic systems Ω1, Ω2, …, and ΩN. Let sα be a linguistic term of the multigranularity linguistic space. Adding the statistic property of it into sα, we get the probabilistic linguistic term sα,psα. When we use any sα,psα from the multigranularity linguistic space to make a decision, the fundamental work is to analyze the meanings of sα. Taking the multigranularity linguistic space illustrated by Figure 3 as an example, we deeply discuss the meanings of s0.4. It is associated with “appropriate” with the membership level 0.65 in Ω1, while it is associated with “appropriate” with the membership level 0.45 in Ω2. Therefore, when several people give a symbol like s0.4 to express their evaluations, it is hard to distinguish the specific meaning of each people.


## 3.2. The MGVLS

By Definition 1, we transform the above LESsS1 and S2 into the normalized vector forms S→1 and S→2, where S→1 = {s0i→−j→ = “young,” s0.5i→−j→ = “appropriate,” s1i→+∞j→ = “old”} and S→2 = {s0i→−j→ = “young,” s0.3i→−0.5j→ = “slightly young,” s0.5i→−0.7j→ = “appropriate,” s0.75i→−0.7j→ = “slightly old,” s1i→+∞j→ = “old”}. Based on S→1, s0.4 is transformed into s0.4i→+2j→. Similarly, based on S→2, s0.4 is transformed into s0.4i→−0.5j→. By this way, it is easy to distinguish the meanings of s0.4 that comes from Ω1 and Ω2, respectively.Definition 4.
LetΩX,μx,L,μS,S be any individual linguistic system. If any probabilistic linguistic term sα,psα of Ω is transformed into the PVLT s→α,ps→α, then Ω can be converted into the individual vector linguistic system Ω→X,μx,L,μS→,S→.
Based on Definitions 3 and 4, we naturally get the concept of the MGVLS as follows:Definition 5.
LetΩ→τ=Ω→X,μτx,Lτ,μτS→,S→ττ=1,2,…,N be N individual vector linguistic systems for the same domain of discourse X, then the individual vector linguistic systems Ω→1, Ω→2, …, and Ω→N construct a MGVLS, denoted by Ω→ττ=1N=Ω→X,μτx,Lτ,μτS→,S→ττ=1,2,…,N.Example 1.
LetΩ→1 and Ω→2 be the corresponding individual vector linguistic systems of S→1 = {s0i→−j→ = “poor,” s0.5i→ = “neutral,” s1i→+∞j→ = “good”} and S→2 = {s0i→−j→ = “poor,” s0.3i→+0.5j→ = “slightly poor,” s0.5i→−0.2j→ = “neutral,” s0.75i→ = “slightly good,” s1i→+∞j→ = “good”}. Then the part of people’s numerical evaluation criteria of the MGVLS Ω→ττ=12 is the infinite plane area s0≤sα≤s1,−∞≤rsαi→≤+∞ depicted in Figure 4.
Comparing the expression of people’s numerical evaluation criteria in the multigranularity linguistic space (refer to Figure3) and the MGVLS in Figure 4, we get that the linguistic evaluations without probabilities in the multigranularity linguistic space are unidimensional, whereas the ones in the MGVLS are dimensional. Therefore, the MGVLS is a stretched form of the multigranularity linguistic space. In the MGVLS, the linguistic evaluation can be distinguished subtler.Figure 4
The part of people’s numerical evaluation criteria of MGVLSΩ→ττ=12 derived by Ω→1 and Ω→2.


## 3.3. The GPVLT

Lets→α,ps→α be any PVLT of the MGVLS Ω→ττ=12 as shown in Figure 4. Noticing the ordinate rsαi→ of s→α, we find that it monotonically increases while sα moves from sατ to sατ+1, where sατ and sατ+1 are the τ−th and τ+1−th linguistic terms of the normalized LES S, respectively. The monotonicity of rsαi→ accords with the left side membership function μsα. In such a case, rsαi→ can be regarded as the rough expression of μsα. Due to the complexity in the membership function determination process, the PVLT is definitely a sensible choice to deal with the probabilistic vector linguistic evaluations in the MGVLS. However, these exist some problems in the simulations with the PVLT: (i) rsαi→ cannot express the right side of μsα and (ii) the values of rsαi→ are all located in −∞,+∞, but the values of the membership function μsα are limited in 0,1. If we regard rsαi→ as the simulation of μsα, the ranges of rsαi→ and μsα are different. Furthermore, ∞ is not able to be handled in the specific computational process. For example, the infinity ∞ can change the convergence and divergence of the integral. To solve these two issues, we present the concept of the GPVLT below.Definition 6.
LetΩ=ΩX,μx,L,μS,S be any individual linguistic system, where S=sατsατ∈s0,s1,τ=1,2,…,N,sα1=s0,sαN=s1,N∈N+ sα1=s0,sαN=s1,N∈N+ is a normalized LES and sατ is the τ−th linguistic term of the normalized LES. For any probabilistic linguistic term sα,psα of Ω, if we transform it into s→α,ps→α, where s→α=sαi→+rsαj→, in which rsα:s0,s1⟶R is a function that reflects the membership functions μS, then s→α,ps→α is the GPVLT with respect to sα,psα.Remark 1.
Although Definition 6 does not restrict the specific characteristic ofrsα, we usually select the functions for rsα according to the following criteria.
For anysατ∈S, which is the target linguistic term, then,(i)
rsα monotonically increases, if sατ−1<sα<sατ, τ=2,3,…,N(ii)
rsα monotonically decreases, if sατ<sα<sατ+1, τ=1,2,…,N−1
The PVLT presented by Definition 1 is a particular case of the GPVLT.
For the second normalized LESS2 in Figure 3, according to Definition 6 and Remark 1, we can assign(1)r1sα=α−ατ−1ατ−ατ−1,ατ−1<α<ατ,τ−1≥0;1,α=ατ;ατ+1−αατ+1−ατ,ατ<α<ατ+1,τ+1≤5,orr2sα=α−ατ−1⋅4/π⋅arctan1/ατ−α−1ατ−ατ−1,ατ−1<α<ατ,τ−1≥0;1,α=ατ;ατ+1−α⋅4/π⋅arctan1/ατ−α−1ατ+1−ατ,ατ<α<ατ+1,τ+1≤5,as the coefficient function of j→ within s→α=sαi→+rsαj→,ps→α, where τ=1,2,3,4,5, sα1=s0 and sα5=s1. Functions r1sα and r2sα can be seen in Figure 5.
As shown in Figure5, the monotonicity and ranges of r1sα and r2sα conform to the membership functions μ2S. Therefore, r1sα or r2sα is more reasonable and precise to be the coefficient function of j→ compared with the one obtained by Definition 1.
As stated in Section3.2, the part of people’s numerical evaluation criteria of any MGVLS can be expressed by a plane area A:s0≤sα≤s1,a≤rsα≤b in a plane rectangular coordinate system S−O−rsα, where a,b is the range of rsα. Each point D in the area A corresponds to a vector s→α=sαi→+rsαj→. If more than one person provides the point D to express the linguistic evaluations, then the statistic property of the point D can be presented as ps→α=pD, i.e., GPVLT D,pD. In the practical linguistic evaluation process, the experts only need to provide the LESs and the linguistic evaluations, which is as easy as the existing linguistic decision-making methods. By calculating the frequency of the occurrence of the linguistic terms, their statistics properties can be obtained.
The GPVLTs are the fundamental components of the MGVLSΩ→ττ=1N. Considering that the corresponding strength of the linguistic evaluations and the values in s0,s1, we give the following definition to classify the GPVLTs:Figure 5
Two examples ofrsα within Definition 6.
(a)
(b)Definition 7.
LetΩ→ττ=1N=Ω→X,μτx,Lτ,μτS→,S→ττ=1,2,…,N be a MGVLS and s→α,ps→α be any GPVLT in Ω→ττ=1N. If s→α is an element of S→τ, then we call s→α,ps→α the explicit GPVLT. Otherwise, we call it the implicit GPVLT.
According to Definition 7, for any explicit GPVLTs→α,ps→α, the ordinate coefficient of s→α is the maximum of rsα, which means that the linguistic evaluation (described by s→α) strongly corresponds to the value α. For example, in the MGVLS of Example 1, s0.75i→,ps0.75i→ is an explicit GPVLT, whereas s0.76i→,ps0.76i→ is an implicit PVLT. Then the explicit-implicit property of the GPVLTs is the location property in the plane area A:s0≤sα≤s1,a≤rsα≤b. Based on the understandings with respect to s→α, rsα, and ps→α, we can compare them by a score function:Definition 8.
The score function of a GPVLTsαi→+rsαj→,ps→α expressed as SCs→α,ps→α=α⋅rsα⋅ps→α was proposed to rank any two GPVLTs sα1i→+rsα1j→,ps→α1 and sα2i→+rsα2j→,ps→α2 in a MGVLS.(i)
IfSCs→α1,ps→α1>SCs→α2,ps→α2, then sα1i→+rsα1j→,ps→α1≻sα2i→+rsα2j→,ps→α2(ii)
IfSCs→α1,ps→α1<SCs→α2,ps→α2, then sα1i→+rsα1j→,ps→α1≺sα2i→+rsα2j→,ps→α2(iii)
IfSCs→α1,ps→α1=SCs→α2,ps→α2, then sα1i→+rsα1j→,ps→α1=sα2i→+rsα2j→,ps→α2
where “≻” indicates “superior to” and “≺” indicates “inferior to.”
The score function of GPVLT is a real mapping froms0,s1×minrsα,maxrsα×0,1 to the interval 0,1. Besides, it is an increasing function of sα, rsα, and ps→α, respectively.Example 2.
LetΩττ=12=ΩX,μτx,Lτ,μτS,Sττ=1,2 be a multigranularity linguistic space, where S1 = {s0 = “none,” s0.3 = “slightly low,” s0.5 = “neutral,” s0.75 = “slightly high,” s1 = “high”} and S2 = {s0 = “none,” s0.28 = “very low,” s0.42 = “low,” s0.5 = “neutral,” s0.58 = “high,” s0.72 = “very high,” s1 = “perfect”}.
For the given linguistic evaluations0.65, we analyze the meanings of it in Ω1 and Ω2, respectively. By using rsα, r1sα, and r2sα shown in Definition 1 and equation (1), we can translate s0.65 and the elements in S1, S2 into the GPVLTs. Assume that there is no repetition of the GPVLT, and the probability of each GPVLT is equal to 1.
All the transformed results listed in Table1 are the implicit GPVLTs of Ωττ=12; hence, we use the nearby principle to select the linguistic evaluation for s0.65. According to the comparisons of the abscissa of s→α, all transformed results are between “neutral” and “slightly high” in Ω1. But from the ordinate of s→α, we find that s0.65i→+3/5j→,1 is the highest. By using the Euclidean measure of the plane rectangular coordinate system, we can calculate all the distances between the transformed GPVLTs and their adjacent explicit GPVLTs. For the given r1sα, the distance between s0.65i→+3/5j→,1 and “slightly high” is less than the distance between s0.65i→+2/5j→,1 and “neutral”; then we should translate s0.65 into “slightly high” for Ω1 when the ordinate coefficient function of s→α is r1sα. In the same manner, we can get other translated results. Although we get the different distances between the transformed GPVLTs and their adjacent explicit GPVLTs through r1sα and r2sα, respectively, the selections are same. Moreover, if we transform s0.65 by the rsα presented in Definition 1, we cannot select the better one between “high” and “very high” for s0.65 in Ω2.
Compared to the above nearby principle, we can use the score function of GPVLT to easily make a choice between two adjacent explicit GPVLTs. Fors0.65i→+2/5j→,1 and s0.65i→+3/5j→,1 in Ω1, which are transformed based on r1sα, the difference between them is the ordinates of j→. Considering that the ordinates of j→ in the adjacent explicit GPVLTs are 1 (maxr1sα=1) and the score function is the increasing function of r1sα, we should translate s0.65 into “slightly high” for Ω1. It is obvious that we can get the same selections if we use the score function of GPVLT to compare the transformed GPVLTs in Table 1.Table 1
The transformed results ofs0.65.
The coefficient function ofj→
Source
The transformed GPVLTs
The adjacent explicit GPVLTs
The score of each transformed GPVLT
The Euclidean distance between the transformed LVET and the adjacent explicit GPVLTs
The meanings ofs0.65
r
Ω
1
s
0.65
i
→
−
1
/
5
j
→
,
1
s
0.5
i
→
+
j
→
,
1 neutral
0.130
1.209
Not select
r
1
s
0.65
i
→
+
2
/
5
j
→
,
1
0.260
0.618
Not select
r
2
s
0
.
65
i
→
+
81
/
2
5
0
j
→
,
1
0.211
0.692
Not select
r
s
0.65
i
→
−
1
/
5
j
→
,
1
s
0.75
i
→
+
j
→
,
1 slightly high
0.130
1.204
Select
r
1
s
0
.
65
i
→
+
3
/
5
j
→
,
1
0.390
0.412
Select
r
2
s
0.65
i
→
+
131
/
250
j
→
,
1
0.341
0.486
Select
r
Ω
2
s
0.65
i
→
+
37
/
195
j
→
,
1
s
0.58
i
→
+
j
→
,
1 high
0.123
0.813
—
r
1
s
0.65
i
→
+
13
/
25
j
→
,
1
0.338
0.485
Select
r
2
s
0.65
i
→
+
129
/
250
j
→
,
1
0.335
0.489
Select
r
s
0.65
i
→
+
37
/
195
j
→
,
1
s
0.72
i
→
+
j
→
,
1 very high
0.123
0.813
—
r
1
s
0
.
65
i
→
+
12
/
2
5
j
→
,
1
0.312
0.525
Not select
r
2
s
0.65
i
→
+
109
/
250
j
→
,
1
0.283
0.568
Not select


## 3.4. The Forms of the PVLU

Case 1.
As the pointe1 in Figure 3, for x∈X, there exists a discrete single-point s0.4 to express the evaluation of it. Whether we transform s0.4 into s0.4i→+2j→,1 based on Definition 1 or translate s0.4 into s0.4i→+0.8j→,1 by r1sα given in equation (1), the results are both the discretely single GPVLTs. In this case, we call the GPVLT the discrete single probabilistic vector linguistic unit (DSPVLU) and denote it by sαi→+rsαi→j→,psαi→+rsαi→j→.Case 2.
As the individual linguistic system in Figure6, the element x∈X can be evaluated by “neutral” with the membership level 0.65 and “slightly good” with the membership level 0.25 simultaneously.Translating “neutral” and “slightly good” into the intervals0,s1, we can obtain the linguistic terms sα1 and sα2. The set sα1,sα2 expresses the linguistic evaluation with respect to x. Since sα1 and sα2 are two discrete points, s→α1,ps→α1,s→α2,ps→α2 is a set with two discrete transformed GPVLTs. Generalizing the number of the elements of s→α1,ps→α1,s→α2,ps→α2 to N, we get a discrete multiple probabilistic vector linguistic unit (DMPVLU) and denote it by s→αk,ps→αkk=1N=s→αk,ps→αkk=1,2,…,N;N∈N+;∑k=1Nps→αk≤1.Figure 6
An example of the individual linguistic system.S = {s0 = poor, s0.3 = slightly poor, s0.5 = neutral, s0.75 = slightly good, s1 = good}.Remark 2.
The DMPVLU can be regarded as a collection of the DSPVLUs. But the DSPVLU and the DMPVLU have different meanings for people’s judgement thinking, where the DSPVLU indicates no hesitance and the DMPVLU indicates the hesitance in the judgements. Therefore, if we take any DSPVLU for a unit of the probabilistic vector linguistic evaluation, then the probability ins→α,ps→α reduces to 1. Moreover, if we take any DMPVLU s→αk,ps→αkk=1N for a unit of the probabilistic vector linguistic evaluation, then the probabilities in it should be normalized by p↔s→αk=ps→αk/∑k=1Nps→αk. Generally, the probability of any DSPVLU used below defaults to 1, and all probabilities in any DMPVLU used below are normalized.Case 3.
In Figure6, “s0.5 = neutral” and “s0.75 = slightly good” are two consecutive elements in the normalized LES S. The interval sα1,sα2⊂s0.5,s0.75 interprets people’s fuzziness between the judgements with “neutral” and “slightly good” [42]. If we transform all values in sα1,sα2 into the GPVLTs, then we can obtain a continuous single-interval probabilistic vector linguistic unit s→α1,s→α2,ps→α, where sα1≤sα≤sα2 and ps→α is the probability function of s→α. In such a situation, we call s→αa,s→αb,ps→α the continuous single-interval probabilistic vector linguistic unit (CSIPVLU), where sαa≤sα≤sαb, s→αa,s→αb⊆sατ,sατ+1, and sατ and sατ+1 are two consecutive elements in the normalized LES S, ps→α:s→αa,s→αb⟶0,1 is the probability function of s→α.Remark 3.
Firstly, lets→αa,s→αb,ps→α be any CSIPVLU, then the vector interval s→αa,s→αb is originated from sαa,sαb. Considering that the process of transforming the linguistic term sα into the normalized vector linguistic term s→α does not change the probability of sα, we know that the probabilistic distribution in sαa,sαb is the same as s→αa,s→αb, i.e., psα:sαa,sαb⟶0,1 is identical to ps→α: s→αa,s→αb⟶0,1. Secondly, the numbers of the elements in the DSPVLU and the DMPVLU are both positive integers, whereas the number of the elements in the CSIPVLU is uncountable. Hence, the CSIPVLU can be regarded as the extended form of the DSPVLU and the DMPVLU. Thirdly, any CSIPVLU s→αa,s→αb,ps→α can be normalized by p↔s→α=ps→α/∫s→αas→αbps→α, where ∫s→αas→αbps→α is the definite integral of ps→α on the interval s→αa,s→αb. In general, all CSIPVLUs used below are normalized.Case 4.
For the CSIPVLUs→αa,s→αb,ps→α depicted in Case 3, the condition s→αa,s→αb⊆sατ,sατ+1 and sατ and sατ+1 are two consecutive linguistic terms in the normalized LES” is somewhat rigorous. In the practical situations, it is difficult to reach it for the flexibility of the linguistic expressions and the fuzziness of the judgements. When the interval sαa,sαb falls on sατ,sατ+k⊆s0,s1, where sατ is the τ−th element of the LES S and k≥2, we can divide sαa,sαb into a sequence of the CSIPVLUs s→αa,s→ατ+1,psα,s→ατ+1,s→ατ+2,psα,…,s→ατ+k−1,s→αb,psα. The divided sequence of the CSIPVLUs s→αk,s→αk+1,pks→αk=1N is called as the continuous multi-intervals probabilistic vector linguistic unit (CMIPVLU), where pks→α is a probabilistic distribution on s→αk,s→αk+1.Remark 4.
The probabilities in any CMIPVLUs→αk,s→αk+1,pks→αk=1N can be normalized by p↔ks→α=pks→α/∑k=1Npks→α. Generally, all CMIPVLUs used below are normalized.


## 4. The Potentials for the PVLUs

According to the different forms of PVLUs discussed in Section3.4, in this section, we introduce two types of the potentials for PVLUs.


### 4.1. Nondirectional Potentials of the PVLUs

#### 4.1.1. Nondirectional Potential of the DSPVLU

Definition 9.
Lets→α,ps→α=sαi→+rsαj→,ps→α be a DSPVLU, then its isolated point potential can be obtained by(2)IPPs→α,ps→α=ps→α⋅sαi→+rsαj→=ps→α⋅α2+r2sα.
The isolated point potential of the DSPVLU, which is a real increasing function ofsα, rsα, and ps→α, is derived from the norm of vector in mathematical theory. Usually, the ps→α in equation (2) reduces to 1.


#### 4.1.2. Nondirectional Potential of the DMPVLU

As stated in Remark 2, the DMPVLU can be regarded as a series of DSPVLUs. Based on equation (2), we define the multipoints potential of the DMPVLU as follows:Definition 10.
Lets→αk,ps→αkk=1N=s→αk,ps→αk,k=1,2,…,N;N∈N+;∑k=1Nps→αk≤1 be a DMPVLU, then its multiple points potential can be given by(3)MPPs→αk,ps→αkk=1N=∑k=1Nps→αk⋅s→αk=∑k=1Nps→αk⋅sαki→+rsαkj→=∑k=1Nps→αk⋅αk2+r2sαk.
Similar to Definition 9, the multiple points potential is a real increasing function ofsαk, rsαk, and ps→αk.


#### 4.1.3. Nondirectional Potential of the CSIPVLU

Lets→a,s→b,ps→ be any CSIPVLU of the MGVLS. If D0=s→a and Dn=s→b, then s→a,s→b,ps→ is equal to D0,Dn,ps→. As described in Section 3.3, when we transform every value of sa,sb to s→a,s→b, a continuous finite curve C:sαα=α,rsα=rα,α∈a,b is obtained.In Figure7, the finite curve C can be regarded as the locus of the point D=s→α=sαi→+rsαj→ moving from D0 to Dn, where ps→:D0,Dn⟶0,1 is the probability function of D. Taking ps→ as the density function of the finite curve C, we can calculate the weight of C by the following four steps:Step 1 (subdividing curve C into a set of curve segments).
Insert a set of dotsDk,pDkk=0N to subdivide the curve C into n curve segments Ckk=1N=Ckk=1,2,⋯,n;n∈N+, and denote the length of the k−th curve segment by ΔaDk−1Dk for k=1,2,…,n.Step 2 (taking dots from every curve segment).
Take any point from each curve segmentCk, denoted as ξk,ηk, and then obtain a set of dots ξk,ηkk=1,2,…,n;n∈N+.Step 3 (taking approximations).
(1)
Approximate the arc-length of eachCk. Replace Ck by the line segment Lk, where the dots Dk−1 and Dk are the endpoints of Lk. In the plane rectangular coordinate system S−O−rsα, the length of Lk can be described by Δlk=Δsαk2+Δrsαk2, where Δsαk=αk−αk−1 and Δrsαk=rsαk−rsαk. So, the arc-length of Ck can be approximated by Δlk, i.e., Δak=ΔaDk−1Dk≈Δsαk2+Δr2sαk.(2)
Approximate the weight ofCk. Utilize pξki→+ηkj→ to represent the density of Ck, then the weight of Ck, denoted as ΔWk, can be approximated by pξki→+ηkj→⋅Δlk.(3)
Approximate the weight of the curveC. Adding all the approximations of ΔWk together, we get the approximation of the weight of the curve C as follows:(4)W=∑k=1NΔWk=∑k=1Nps→⋅Δak≈∑k=1Npξki→+ηkj→⋅Δlk.Step 4 (calculating the limit of∑k=1Npξki→+ηkj→⋅Δlk).
Leta be the maximum value of Δak, where Δak is the arc-length of the curve segment Ck, then we can subdivide C infinitely when a approaches to 0. The limit value of ∑k=1Npξki→+ηkj→⋅Δlk with a⟶0 is the accurate value of W.Figure 7
The visual representation of the CSIPVLUD0,Dn,ps→.Definition 11.
If the limit of∑k=1Npξki→+ηkj→⋅Δlk exists when a approaches to 0, then we call it the nondirectional curve potential of the CSIPVLU s→a,s→b,ps→ and denote it by ∫Cps→⋅da, where C:sαα=α,rsα=rα,α∈a,b is the integral curve, ps→:D0,Dn⟶0,1 is the integrand function, da is the arc-length integral infinitesimal.Obviously,NDCPs→a,s→b,ps→=∫Cps→da=∫abps→⋅1+r′α2dα, which is the first curvilinear integral of ps→:D0,Dn⟶0,1 on the curve C. The nondirectional curve potential is a quantity property of the CSIPVLU, and its physical significance is the weight of C.Remark 5.
In Definition 11, the segmentation process of the curveC and the way of selecting dots from Ckk=1N are arbitrary. Particularly, in NDCPs→a,s→b,ps→=∫abps→⋅1+r′α2dα, to guarantee the meaningfulness of the arc-length, the lower integral limit a must be less than the upper integral limit b.Until now, it can be easy to deduce that the values of the nondirectional curve potentials will be different if the probability distributionps→:D0,Dn⟶0,1 changes, even though the integral curve C remains the same, which manifests the significance of the statistical properties of the vector linguistic evaluation.


#### 4.1.4. Nondirectional Potential of the CMIPVLU

For any CMIPVLUs→αk,s→αk+1,ps→αk=1N, we can get its nondirectional curve potential based on the properties of the first curvilinear integral and Definition 11.Definition 12.
Lets→αk,s→αk+1,ps→αk=1N be a CMIPVLU, then its nondirectional curve potential can be obtained by(5)NDCPs→αk,s→αk+1,ps→αk=1N=∑k=1NNDCPs→αk,s→αk+1,ps→α.


### 4.2. Directional Potentials of the PVLU

To enhance the practicability and effectiveness of the tools to investigate the directional linguistic evaluation in physician-patient communication, this subsection introduces the directional potentials for the PVLUs.


#### 4.2.1. Directional Potential of the DSPVLU

Definition 13.
LetDk,pDk=s→αki→+rαkj→,ps→αk be a DSPVLU, then its directional potential is(6)DPDk,pDk=ps→αk⋅s→αki→+rαkj→=ps→αk⋅s→αki→+ps→αk⋅rαkj→.
This equation obtains a directional vector which contains the potential of the DSPVLU.


#### 4.2.2. Directional Potential of the DMPVLU

Furthermore, Definition 13 can be generalized to the DMPVLU that contains more than one point.Definition 14.
LetDk,ps→αkk=1N be a DMPVLU, then its directional potential can be defined by(7)DPDk,ps→αkk=1N=∑k=2NpDk⋅αk⋅i→+pDk⋅rsαkj→.


#### 4.2.3. Directional Potential of the CSIPVLU

For a CSIPVLUs→a,s→b,ps→=D0,Dn,ps→, there exist two trajectories that vary from D0 to Dn and Dn to D0, whose paths are denoted as s→a⟶s→b,ps→=D0⟶Dn,ps→ and s→b⟶s→a,ps→ =Dn⟶D0,ps→, respectively. As shown in Figure 8, for the path D0⟶Dn,ps→, we can get its corresponding directional curve C→:sαα=α,rsα=rα,α:a⟶b, where D0=sαai→+rsαaj→ is the starting point and Dn=sαbi→+rsαbj→ is the terminal point. Based on the probability function ps→:s→b⟶s→a⟶0,1, we can establish a directional function as F→s→=ps→⋅αi→+ps→⋅rαj→=Fsi→+Frj→, where α∈a,b. If we regard it as the variable force on C→, then the work of F→s→ along with C→ can be calculated by the following four steps:Figure 8
The visual representation of the work ofF→s→ along with C→.Step 1 (subdividingC→ into an ordered set of directional curve segments).
Insert a sequence of ordered dotsDk,pDkk=0N into the directional curve, then C→ can be subdivided into a sequence of ordered directional curve segments C→kk=1N=C→kk=1,2,⋯,n;n∈N+. For the k‐th directional curve segment, the dot Dk−1 is the starting point and Dk is the terminal point.Step 2 (taking a sequence of ordered dots).
Take any point in each directional curve segmentC→k, and denote it as sξk,ηk, then the dots of sξk,ηkk=1,2,⋯,n;n∈N+ successively fall on the directional curve C→.Step 3.
(approximating the work ofF→s→ along with the curve segment C→k)(1)
Approximate the directional curveC→k. Replace C→k by the directional line segment L→k, where the dots Dk−1 and Dk are the starting point and the terminal point, respectively. In the plane rectangular coordinate system S−O−rsα, L→k can be described by the vector ΔL→k=Δsαki→+Δrsαkj→, which means ΔC→k≈Δsαki→+Δrsαkj→, where Δsαk=αk−αk−1 and Δrsαk=rsαk−rsαk−1.(2)
Approximate the work ofF→s→ on C→k. Let ΔW→k be the work of F→s→ along with C→k. For each point of sξk,ηkk=1,2,…,n;n∈N+, we compute the value of F→s→ on it and denote the result by F→sξki→+ηkj→=psξki→+ηkj→⋅ξki→+psξki→+ηkj→⋅rsξkj→=Fssξki→+ηkj→i→+Frsξki→+ηkj→j→. Replace the value of F→s→ at every point in C→k by F→sξki→+ηkj→, then the work of C→k, denoted as ΔW→k, can be approximated by Fssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk.(3)
Approximate the work ofF→s→ on C→. Let W→ be the work of F→s→ on C→. Sum all ΔW→k, then we can get the approximation of W→:(8)
W
→
=
∑
k
=
1
N
Δ
W
→
k
≈
∑
k
=
1
N
F
s
s
ξ
k
i
→
+
η
k
j
→
⋅
Δ
s
α
k
+
F
r
s
ξ
k
i
→
+
η
k
j
→
⋅
Δ
r
s
α
k
.Step 4 (calculate the limit of equation (8)). Let a→ be the maximum value of Δa→k, where Δa→k is the arc-length of the directional curve C→k, then we can subdivide the directional curve C→ infinitely when a→ approaches to 0. The limit value of ∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk with a→⟶0 is an accurate value of W→.Definition 15.
If the limit of∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk exists with a→⟶0, then the directional curve potential of the CSIPVLU s→a⟶s→b,ps→=D0⟶Dn,ps→ can be defined as(9)DCPs→a⟶s→b,ps→=∫C→F→s→⋅da→=lima→⟶0∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk=∫C→Fssξki→+ηkj→⋅dsα+Frsξki→+ηkj→⋅drsα=∫C→Fssξki→+ηkj→⋅dsα+∫C→Frsξki→+ηkj→⋅drsα.
In this equation,∫C→Fssξki→+ηkj→⋅dsα and ∫C→Frsξki→+ηkj→⋅drsα are the components of W→ falling on the abscissa and vertical axes, respectively.Remark 6.
The segmentation process of the directional curveC→ and the way of selecting dots from C→kk=1N are arbitrary. In addition, the directional curve potential of the CSIPVLU s→a⟶s→b,ps→ can be calculated by(10)∫C→F→s→⋅da→=∫C→Fssξki→+ηkj→⋅dsα+∫C→Frsξki→+ηkj→⋅drsα=∫abpsξki→+ηkj→⋅αdα+∫abpsξki→+ηkj→⋅rαdrα.
Since the integral infinitesimalΔa→ of ∫C→F→s→⋅da→ is directional, the integral lower limit a is not required to be less than the integral upper limit b when we calculate ∫C→F→s→⋅da→.


#### 4.2.4. Directional Potential of the CMIPVLU

For a CMIPVLUs→αk,s→αk+1,ps→αk=1N, we can get its directional curve potential based on the properties of the second curvilinear integral and Definition 15.Definition 16.
Lets→αk⟶s→αk+1,ps→αk=1N be a directional CMIPVLU, then its directional curve potential can be obtained by(11)DCPs→αk⟶s→αk+1,ps→αk=1N=∑k=1NDCPs→αk⟶s→αk+1,ps→α.


## 4.1. Nondirectional Potentials of the PVLUs

### 4.1.1. Nondirectional Potential of the DSPVLU

Definition 9.
Lets→α,ps→α=sαi→+rsαj→,ps→α be a DSPVLU, then its isolated point potential can be obtained by(2)IPPs→α,ps→α=ps→α⋅sαi→+rsαj→=ps→α⋅α2+r2sα.
The isolated point potential of the DSPVLU, which is a real increasing function ofsα, rsα, and ps→α, is derived from the norm of vector in mathematical theory. Usually, the ps→α in equation (2) reduces to 1.


### 4.1.2. Nondirectional Potential of the DMPVLU

As stated in Remark 2, the DMPVLU can be regarded as a series of DSPVLUs. Based on equation (2), we define the multipoints potential of the DMPVLU as follows:Definition 10.
Lets→αk,ps→αkk=1N=s→αk,ps→αk,k=1,2,…,N;N∈N+;∑k=1Nps→αk≤1 be a DMPVLU, then its multiple points potential can be given by(3)MPPs→αk,ps→αkk=1N=∑k=1Nps→αk⋅s→αk=∑k=1Nps→αk⋅sαki→+rsαkj→=∑k=1Nps→αk⋅αk2+r2sαk.
Similar to Definition 9, the multiple points potential is a real increasing function ofsαk, rsαk, and ps→αk.


### 4.1.3. Nondirectional Potential of the CSIPVLU

Lets→a,s→b,ps→ be any CSIPVLU of the MGVLS. If D0=s→a and Dn=s→b, then s→a,s→b,ps→ is equal to D0,Dn,ps→. As described in Section 3.3, when we transform every value of sa,sb to s→a,s→b, a continuous finite curve C:sαα=α,rsα=rα,α∈a,b is obtained.In Figure7, the finite curve C can be regarded as the locus of the point D=s→α=sαi→+rsαj→ moving from D0 to Dn, where ps→:D0,Dn⟶0,1 is the probability function of D. Taking ps→ as the density function of the finite curve C, we can calculate the weight of C by the following four steps:Step 1 (subdividing curve C into a set of curve segments).
Insert a set of dotsDk,pDkk=0N to subdivide the curve C into n curve segments Ckk=1N=Ckk=1,2,⋯,n;n∈N+, and denote the length of the k−th curve segment by ΔaDk−1Dk for k=1,2,…,n.Step 2 (taking dots from every curve segment).
Take any point from each curve segmentCk, denoted as ξk,ηk, and then obtain a set of dots ξk,ηkk=1,2,…,n;n∈N+.Step 3 (taking approximations).
(1)
Approximate the arc-length of eachCk. Replace Ck by the line segment Lk, where the dots Dk−1 and Dk are the endpoints of Lk. In the plane rectangular coordinate system S−O−rsα, the length of Lk can be described by Δlk=Δsαk2+Δrsαk2, where Δsαk=αk−αk−1 and Δrsαk=rsαk−rsαk. So, the arc-length of Ck can be approximated by Δlk, i.e., Δak=ΔaDk−1Dk≈Δsαk2+Δr2sαk.(2)
Approximate the weight ofCk. Utilize pξki→+ηkj→ to represent the density of Ck, then the weight of Ck, denoted as ΔWk, can be approximated by pξki→+ηkj→⋅Δlk.(3)
Approximate the weight of the curveC. Adding all the approximations of ΔWk together, we get the approximation of the weight of the curve C as follows:(4)W=∑k=1NΔWk=∑k=1Nps→⋅Δak≈∑k=1Npξki→+ηkj→⋅Δlk.Step 4 (calculating the limit of∑k=1Npξki→+ηkj→⋅Δlk).
Leta be the maximum value of Δak, where Δak is the arc-length of the curve segment Ck, then we can subdivide C infinitely when a approaches to 0. The limit value of ∑k=1Npξki→+ηkj→⋅Δlk with a⟶0 is the accurate value of W.Figure 7
The visual representation of the CSIPVLUD0,Dn,ps→.Definition 11.
If the limit of∑k=1Npξki→+ηkj→⋅Δlk exists when a approaches to 0, then we call it the nondirectional curve potential of the CSIPVLU s→a,s→b,ps→ and denote it by ∫Cps→⋅da, where C:sαα=α,rsα=rα,α∈a,b is the integral curve, ps→:D0,Dn⟶0,1 is the integrand function, da is the arc-length integral infinitesimal.Obviously,NDCPs→a,s→b,ps→=∫Cps→da=∫abps→⋅1+r′α2dα, which is the first curvilinear integral of ps→:D0,Dn⟶0,1 on the curve C. The nondirectional curve potential is a quantity property of the CSIPVLU, and its physical significance is the weight of C.Remark 5.
In Definition 11, the segmentation process of the curveC and the way of selecting dots from Ckk=1N are arbitrary. Particularly, in NDCPs→a,s→b,ps→=∫abps→⋅1+r′α2dα, to guarantee the meaningfulness of the arc-length, the lower integral limit a must be less than the upper integral limit b.Until now, it can be easy to deduce that the values of the nondirectional curve potentials will be different if the probability distributionps→:D0,Dn⟶0,1 changes, even though the integral curve C remains the same, which manifests the significance of the statistical properties of the vector linguistic evaluation.


### 4.1.4. Nondirectional Potential of the CMIPVLU

For any CMIPVLUs→αk,s→αk+1,ps→αk=1N, we can get its nondirectional curve potential based on the properties of the first curvilinear integral and Definition 11.Definition 12.
Lets→αk,s→αk+1,ps→αk=1N be a CMIPVLU, then its nondirectional curve potential can be obtained by(5)NDCPs→αk,s→αk+1,ps→αk=1N=∑k=1NNDCPs→αk,s→αk+1,ps→α.


## 4.1.1. Nondirectional Potential of the DSPVLU

Definition 9.
Lets→α,ps→α=sαi→+rsαj→,ps→α be a DSPVLU, then its isolated point potential can be obtained by(2)IPPs→α,ps→α=ps→α⋅sαi→+rsαj→=ps→α⋅α2+r2sα.
The isolated point potential of the DSPVLU, which is a real increasing function ofsα, rsα, and ps→α, is derived from the norm of vector in mathematical theory. Usually, the ps→α in equation (2) reduces to 1.


## 4.1.2. Nondirectional Potential of the DMPVLU

As stated in Remark 2, the DMPVLU can be regarded as a series of DSPVLUs. Based on equation (2), we define the multipoints potential of the DMPVLU as follows:Definition 10.
Lets→αk,ps→αkk=1N=s→αk,ps→αk,k=1,2,…,N;N∈N+;∑k=1Nps→αk≤1 be a DMPVLU, then its multiple points potential can be given by(3)MPPs→αk,ps→αkk=1N=∑k=1Nps→αk⋅s→αk=∑k=1Nps→αk⋅sαki→+rsαkj→=∑k=1Nps→αk⋅αk2+r2sαk.
Similar to Definition 9, the multiple points potential is a real increasing function ofsαk, rsαk, and ps→αk.


## 4.1.3. Nondirectional Potential of the CSIPVLU

Lets→a,s→b,ps→ be any CSIPVLU of the MGVLS. If D0=s→a and Dn=s→b, then s→a,s→b,ps→ is equal to D0,Dn,ps→. As described in Section 3.3, when we transform every value of sa,sb to s→a,s→b, a continuous finite curve C:sαα=α,rsα=rα,α∈a,b is obtained.In Figure7, the finite curve C can be regarded as the locus of the point D=s→α=sαi→+rsαj→ moving from D0 to Dn, where ps→:D0,Dn⟶0,1 is the probability function of D. Taking ps→ as the density function of the finite curve C, we can calculate the weight of C by the following four steps:Step 1 (subdividing curve C into a set of curve segments).
Insert a set of dotsDk,pDkk=0N to subdivide the curve C into n curve segments Ckk=1N=Ckk=1,2,⋯,n;n∈N+, and denote the length of the k−th curve segment by ΔaDk−1Dk for k=1,2,…,n.Step 2 (taking dots from every curve segment).
Take any point from each curve segmentCk, denoted as ξk,ηk, and then obtain a set of dots ξk,ηkk=1,2,…,n;n∈N+.Step 3 (taking approximations).
(1)
Approximate the arc-length of eachCk. Replace Ck by the line segment Lk, where the dots Dk−1 and Dk are the endpoints of Lk. In the plane rectangular coordinate system S−O−rsα, the length of Lk can be described by Δlk=Δsαk2+Δrsαk2, where Δsαk=αk−αk−1 and Δrsαk=rsαk−rsαk. So, the arc-length of Ck can be approximated by Δlk, i.e., Δak=ΔaDk−1Dk≈Δsαk2+Δr2sαk.(2)
Approximate the weight ofCk. Utilize pξki→+ηkj→ to represent the density of Ck, then the weight of Ck, denoted as ΔWk, can be approximated by pξki→+ηkj→⋅Δlk.(3)
Approximate the weight of the curveC. Adding all the approximations of ΔWk together, we get the approximation of the weight of the curve C as follows:(4)W=∑k=1NΔWk=∑k=1Nps→⋅Δak≈∑k=1Npξki→+ηkj→⋅Δlk.Step 4 (calculating the limit of∑k=1Npξki→+ηkj→⋅Δlk).
Leta be the maximum value of Δak, where Δak is the arc-length of the curve segment Ck, then we can subdivide C infinitely when a approaches to 0. The limit value of ∑k=1Npξki→+ηkj→⋅Δlk with a⟶0 is the accurate value of W.Figure 7
The visual representation of the CSIPVLUD0,Dn,ps→.Definition 11.
If the limit of∑k=1Npξki→+ηkj→⋅Δlk exists when a approaches to 0, then we call it the nondirectional curve potential of the CSIPVLU s→a,s→b,ps→ and denote it by ∫Cps→⋅da, where C:sαα=α,rsα=rα,α∈a,b is the integral curve, ps→:D0,Dn⟶0,1 is the integrand function, da is the arc-length integral infinitesimal.Obviously,NDCPs→a,s→b,ps→=∫Cps→da=∫abps→⋅1+r′α2dα, which is the first curvilinear integral of ps→:D0,Dn⟶0,1 on the curve C. The nondirectional curve potential is a quantity property of the CSIPVLU, and its physical significance is the weight of C.Remark 5.
In Definition 11, the segmentation process of the curveC and the way of selecting dots from Ckk=1N are arbitrary. Particularly, in NDCPs→a,s→b,ps→=∫abps→⋅1+r′α2dα, to guarantee the meaningfulness of the arc-length, the lower integral limit a must be less than the upper integral limit b.Until now, it can be easy to deduce that the values of the nondirectional curve potentials will be different if the probability distributionps→:D0,Dn⟶0,1 changes, even though the integral curve C remains the same, which manifests the significance of the statistical properties of the vector linguistic evaluation.


## 4.1.4. Nondirectional Potential of the CMIPVLU

For any CMIPVLUs→αk,s→αk+1,ps→αk=1N, we can get its nondirectional curve potential based on the properties of the first curvilinear integral and Definition 11.Definition 12.
Lets→αk,s→αk+1,ps→αk=1N be a CMIPVLU, then its nondirectional curve potential can be obtained by(5)NDCPs→αk,s→αk+1,ps→αk=1N=∑k=1NNDCPs→αk,s→αk+1,ps→α.


## 4.2. Directional Potentials of the PVLU

To enhance the practicability and effectiveness of the tools to investigate the directional linguistic evaluation in physician-patient communication, this subsection introduces the directional potentials for the PVLUs.


### 4.2.1. Directional Potential of the DSPVLU

Definition 13.
LetDk,pDk=s→αki→+rαkj→,ps→αk be a DSPVLU, then its directional potential is(6)DPDk,pDk=ps→αk⋅s→αki→+rαkj→=ps→αk⋅s→αki→+ps→αk⋅rαkj→.
This equation obtains a directional vector which contains the potential of the DSPVLU.


### 4.2.2. Directional Potential of the DMPVLU

Furthermore, Definition 13 can be generalized to the DMPVLU that contains more than one point.Definition 14.
LetDk,ps→αkk=1N be a DMPVLU, then its directional potential can be defined by(7)DPDk,ps→αkk=1N=∑k=2NpDk⋅αk⋅i→+pDk⋅rsαkj→.


### 4.2.3. Directional Potential of the CSIPVLU

For a CSIPVLUs→a,s→b,ps→=D0,Dn,ps→, there exist two trajectories that vary from D0 to Dn and Dn to D0, whose paths are denoted as s→a⟶s→b,ps→=D0⟶Dn,ps→ and s→b⟶s→a,ps→ =Dn⟶D0,ps→, respectively. As shown in Figure 8, for the path D0⟶Dn,ps→, we can get its corresponding directional curve C→:sαα=α,rsα=rα,α:a⟶b, where D0=sαai→+rsαaj→ is the starting point and Dn=sαbi→+rsαbj→ is the terminal point. Based on the probability function ps→:s→b⟶s→a⟶0,1, we can establish a directional function as F→s→=ps→⋅αi→+ps→⋅rαj→=Fsi→+Frj→, where α∈a,b. If we regard it as the variable force on C→, then the work of F→s→ along with C→ can be calculated by the following four steps:Figure 8
The visual representation of the work ofF→s→ along with C→.Step 1 (subdividingC→ into an ordered set of directional curve segments).
Insert a sequence of ordered dotsDk,pDkk=0N into the directional curve, then C→ can be subdivided into a sequence of ordered directional curve segments C→kk=1N=C→kk=1,2,⋯,n;n∈N+. For the k‐th directional curve segment, the dot Dk−1 is the starting point and Dk is the terminal point.Step 2 (taking a sequence of ordered dots).
Take any point in each directional curve segmentC→k, and denote it as sξk,ηk, then the dots of sξk,ηkk=1,2,⋯,n;n∈N+ successively fall on the directional curve C→.Step 3.
(approximating the work ofF→s→ along with the curve segment C→k)(1)
Approximate the directional curveC→k. Replace C→k by the directional line segment L→k, where the dots Dk−1 and Dk are the starting point and the terminal point, respectively. In the plane rectangular coordinate system S−O−rsα, L→k can be described by the vector ΔL→k=Δsαki→+Δrsαkj→, which means ΔC→k≈Δsαki→+Δrsαkj→, where Δsαk=αk−αk−1 and Δrsαk=rsαk−rsαk−1.(2)
Approximate the work ofF→s→ on C→k. Let ΔW→k be the work of F→s→ along with C→k. For each point of sξk,ηkk=1,2,…,n;n∈N+, we compute the value of F→s→ on it and denote the result by F→sξki→+ηkj→=psξki→+ηkj→⋅ξki→+psξki→+ηkj→⋅rsξkj→=Fssξki→+ηkj→i→+Frsξki→+ηkj→j→. Replace the value of F→s→ at every point in C→k by F→sξki→+ηkj→, then the work of C→k, denoted as ΔW→k, can be approximated by Fssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk.(3)
Approximate the work ofF→s→ on C→. Let W→ be the work of F→s→ on C→. Sum all ΔW→k, then we can get the approximation of W→:(8)
W
→
=
∑
k
=
1
N
Δ
W
→
k
≈
∑
k
=
1
N
F
s
s
ξ
k
i
→
+
η
k
j
→
⋅
Δ
s
α
k
+
F
r
s
ξ
k
i
→
+
η
k
j
→
⋅
Δ
r
s
α
k
.Step 4 (calculate the limit of equation (8)). Let a→ be the maximum value of Δa→k, where Δa→k is the arc-length of the directional curve C→k, then we can subdivide the directional curve C→ infinitely when a→ approaches to 0. The limit value of ∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk with a→⟶0 is an accurate value of W→.Definition 15.
If the limit of∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk exists with a→⟶0, then the directional curve potential of the CSIPVLU s→a⟶s→b,ps→=D0⟶Dn,ps→ can be defined as(9)DCPs→a⟶s→b,ps→=∫C→F→s→⋅da→=lima→⟶0∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk=∫C→Fssξki→+ηkj→⋅dsα+Frsξki→+ηkj→⋅drsα=∫C→Fssξki→+ηkj→⋅dsα+∫C→Frsξki→+ηkj→⋅drsα.
In this equation,∫C→Fssξki→+ηkj→⋅dsα and ∫C→Frsξki→+ηkj→⋅drsα are the components of W→ falling on the abscissa and vertical axes, respectively.Remark 6.
The segmentation process of the directional curveC→ and the way of selecting dots from C→kk=1N are arbitrary. In addition, the directional curve potential of the CSIPVLU s→a⟶s→b,ps→ can be calculated by(10)∫C→F→s→⋅da→=∫C→Fssξki→+ηkj→⋅dsα+∫C→Frsξki→+ηkj→⋅drsα=∫abpsξki→+ηkj→⋅αdα+∫abpsξki→+ηkj→⋅rαdrα.
Since the integral infinitesimalΔa→ of ∫C→F→s→⋅da→ is directional, the integral lower limit a is not required to be less than the integral upper limit b when we calculate ∫C→F→s→⋅da→.


### 4.2.4. Directional Potential of the CMIPVLU

For a CMIPVLUs→αk,s→αk+1,ps→αk=1N, we can get its directional curve potential based on the properties of the second curvilinear integral and Definition 15.Definition 16.
Lets→αk⟶s→αk+1,ps→αk=1N be a directional CMIPVLU, then its directional curve potential can be obtained by(11)DCPs→αk⟶s→αk+1,ps→αk=1N=∑k=1NDCPs→αk⟶s→αk+1,ps→α.


## 4.2.1. Directional Potential of the DSPVLU

Definition 13.
LetDk,pDk=s→αki→+rαkj→,ps→αk be a DSPVLU, then its directional potential is(6)DPDk,pDk=ps→αk⋅s→αki→+rαkj→=ps→αk⋅s→αki→+ps→αk⋅rαkj→.
This equation obtains a directional vector which contains the potential of the DSPVLU.


## 4.2.2. Directional Potential of the DMPVLU

Furthermore, Definition 13 can be generalized to the DMPVLU that contains more than one point.Definition 14.
LetDk,ps→αkk=1N be a DMPVLU, then its directional potential can be defined by(7)DPDk,ps→αkk=1N=∑k=2NpDk⋅αk⋅i→+pDk⋅rsαkj→.


## 4.2.3. Directional Potential of the CSIPVLU

For a CSIPVLUs→a,s→b,ps→=D0,Dn,ps→, there exist two trajectories that vary from D0 to Dn and Dn to D0, whose paths are denoted as s→a⟶s→b,ps→=D0⟶Dn,ps→ and s→b⟶s→a,ps→ =Dn⟶D0,ps→, respectively. As shown in Figure 8, for the path D0⟶Dn,ps→, we can get its corresponding directional curve C→:sαα=α,rsα=rα,α:a⟶b, where D0=sαai→+rsαaj→ is the starting point and Dn=sαbi→+rsαbj→ is the terminal point. Based on the probability function ps→:s→b⟶s→a⟶0,1, we can establish a directional function as F→s→=ps→⋅αi→+ps→⋅rαj→=Fsi→+Frj→, where α∈a,b. If we regard it as the variable force on C→, then the work of F→s→ along with C→ can be calculated by the following four steps:Figure 8
The visual representation of the work ofF→s→ along with C→.Step 1 (subdividingC→ into an ordered set of directional curve segments).
Insert a sequence of ordered dotsDk,pDkk=0N into the directional curve, then C→ can be subdivided into a sequence of ordered directional curve segments C→kk=1N=C→kk=1,2,⋯,n;n∈N+. For the k‐th directional curve segment, the dot Dk−1 is the starting point and Dk is the terminal point.Step 2 (taking a sequence of ordered dots).
Take any point in each directional curve segmentC→k, and denote it as sξk,ηk, then the dots of sξk,ηkk=1,2,⋯,n;n∈N+ successively fall on the directional curve C→.Step 3.
(approximating the work ofF→s→ along with the curve segment C→k)(1)
Approximate the directional curveC→k. Replace C→k by the directional line segment L→k, where the dots Dk−1 and Dk are the starting point and the terminal point, respectively. In the plane rectangular coordinate system S−O−rsα, L→k can be described by the vector ΔL→k=Δsαki→+Δrsαkj→, which means ΔC→k≈Δsαki→+Δrsαkj→, where Δsαk=αk−αk−1 and Δrsαk=rsαk−rsαk−1.(2)
Approximate the work ofF→s→ on C→k. Let ΔW→k be the work of F→s→ along with C→k. For each point of sξk,ηkk=1,2,…,n;n∈N+, we compute the value of F→s→ on it and denote the result by F→sξki→+ηkj→=psξki→+ηkj→⋅ξki→+psξki→+ηkj→⋅rsξkj→=Fssξki→+ηkj→i→+Frsξki→+ηkj→j→. Replace the value of F→s→ at every point in C→k by F→sξki→+ηkj→, then the work of C→k, denoted as ΔW→k, can be approximated by Fssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk.(3)
Approximate the work ofF→s→ on C→. Let W→ be the work of F→s→ on C→. Sum all ΔW→k, then we can get the approximation of W→:(8)
W
→
=
∑
k
=
1
N
Δ
W
→
k
≈
∑
k
=
1
N
F
s
s
ξ
k
i
→
+
η
k
j
→
⋅
Δ
s
α
k
+
F
r
s
ξ
k
i
→
+
η
k
j
→
⋅
Δ
r
s
α
k
.Step 4 (calculate the limit of equation (8)). Let a→ be the maximum value of Δa→k, where Δa→k is the arc-length of the directional curve C→k, then we can subdivide the directional curve C→ infinitely when a→ approaches to 0. The limit value of ∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk with a→⟶0 is an accurate value of W→.Definition 15.
If the limit of∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk exists with a→⟶0, then the directional curve potential of the CSIPVLU s→a⟶s→b,ps→=D0⟶Dn,ps→ can be defined as(9)DCPs→a⟶s→b,ps→=∫C→F→s→⋅da→=lima→⟶0∑k=1NFssξki→+ηkj→⋅Δsαk+Frsξki→+ηkj→⋅Δrsαk=∫C→Fssξki→+ηkj→⋅dsα+Frsξki→+ηkj→⋅drsα=∫C→Fssξki→+ηkj→⋅dsα+∫C→Frsξki→+ηkj→⋅drsα.
In this equation,∫C→Fssξki→+ηkj→⋅dsα and ∫C→Frsξki→+ηkj→⋅drsα are the components of W→ falling on the abscissa and vertical axes, respectively.Remark 6.
The segmentation process of the directional curveC→ and the way of selecting dots from C→kk=1N are arbitrary. In addition, the directional curve potential of the CSIPVLU s→a⟶s→b,ps→ can be calculated by(10)∫C→F→s→⋅da→=∫C→Fssξki→+ηkj→⋅dsα+∫C→Frsξki→+ηkj→⋅drsα=∫abpsξki→+ηkj→⋅αdα+∫abpsξki→+ηkj→⋅rαdrα.
Since the integral infinitesimalΔa→ of ∫C→F→s→⋅da→ is directional, the integral lower limit a is not required to be less than the integral upper limit b when we calculate ∫C→F→s→⋅da→.


## 4.2.4. Directional Potential of the CMIPVLU

For a CMIPVLUs→αk,s→αk+1,ps→αk=1N, we can get its directional curve potential based on the properties of the second curvilinear integral and Definition 15.Definition 16.
Lets→αk⟶s→αk+1,ps→αk=1N be a directional CMIPVLU, then its directional curve potential can be obtained by(11)DCPs→αk⟶s→αk+1,ps→αk=1N=∑k=1NDCPs→αk⟶s→αk+1,ps→α.


## 5. The Application Examples of the Potentials

In this section, we illustrate two numerical examples to show how to use the potentials of PVLUs to address the linguistic evaluation problems in the physician-patient communication for COPD.


### 5.1. The Applications of the Nondirectional Curve Potential in Measuring the Fuzziness of the CSIPVLU

Example 3.
Lets0.8,s0.85⊂s0,s1 be an interval linguistic evaluation, S1 and S2 be two normalized LESs illustrated in Example 2, r1sα and r2sα be the ordinate coefficient functions given in equation (1). We transform s0.8,s0.85 into the CSIPVLUs by using r1sα and r2sα based on S1 and S2, respectively, and calculate the nondirectional curve potentials of the transformed units; the results are listed in Table 2:
By Table2, U1 and U2 are transformed from S1 with respect to “s0.75i→+j→=slightly high” by r1sα and r2sα, respectively. Referring to Figure 5, we know that r2sα is twisted more than r1sα, which demonstrates that the judgement thinking reflected by r2sα is fuzzier than the one reflected by r1sα. The nondirectional curve potentials of U1 and U2 present the fuzzy comparisons between U1 and U2. The same explanations can be conducted to the fuzzy comparisons between U3 and U4.
In addition,U1 and U3 are both transformed by r1sα, while U1 is from S1 and U3 is from S2. Comparing S1 with S2, we know that S2 is subdivided more. Therefore, the fuzziness of U3 should be less than U1. The nondirectional curve potentials of U1 and U3 show the difference of fuzziness between U1 and U3. The comparative results can also be summarized for U2 and U4.Table 2
The nondirectional potentials of the CSIPVLUs.
CSIPVLUs
Source
NDCP
U
1
=
s
0.8
i
→
+
0.80
j
→
,
s
0.85
i
→
+
0.60
j
→
,
20
FromS1 to “s0.75i→+j→=slightly high” based on r1sα
4.12
U
2
=
s
0.8
i
→
+
0.75
j
→
,
s
0.85
i
→
+
0.52
j
→
,
20
FromS1 to “s0.75i→+j→=slightly high” based on r2sα
4.62
U
3
=
s
0.8
i
→
+
0.72
j
→
,
s
0.85
i
→
+
0.54
j
→
,
20
FromS2 to “s0.72i→+j→=very high” based on r1sα
3.74
U
4
=
s
0.8
i
→
+
0.65
j
→
,
s
0.85
i
→
+
0.45
j
→
,
20
FromS2 to “s0.72i→+j→=very high” based on r2sα
4.05


### 5.2. The Applications of the Potentials in Dealing with the Linguistic Evaluation in Physician-Patient Communication for COPD

Nowadays, many major hospitals pay more attention to patients’ evaluations and satisfactions to improve the physician-patient relation. On the website of the West China Hospital, there are questionnaires to track and collect the social evaluation opinions and patients’ satisfactions for medicine service. In the evaluation systems, the indexes about the quality of physician-patient communications are necessary and important. The ways to obtain patients’ evaluations are various, such as questionnaire survey, visiting patients by mails or telephones, and so on.Suppose that the physicianE is a specialist on COPD. His performance is regularly evaluated according to the rules of the hospital. By mail and telephone following up, we get the same linguistic evaluation from patients H1, H2 H3, and H4 for E, which is s0.43,s0.54, where s0.43 is the earlier information and s0.54 is the later information. The values s0.43 and s0.54 are the any given patients’ evaluation information based on the multigranularity linguistic space Ωττ=12 illustrated in Example 2, which are used to illustrate the usage of the directional potentials with respect to the CMIPVLUs. Based on the LESs S1, S2, and the functions r1sα, and r2sα mentioned in equation (1), we can transform s0.43,s0.54 into four CMIPVLUs. By equation (11), we can calculate the abscissa component, the vertical component, and the total value of the directional curve potential for each CMIPVLU, respectively. The results are shown in Table 3.Table 3
The directional potentials of the CMIPVLUs.
CMIPVLUs
Source
The work in the left side ofs0.5i→+j→=neutral
The work in the right side ofs0.5i→+j→=neutral
The work of CMIPVLU
The abscissa work
The vertical work
The abscissa work
The vertical work
The abscissa work
The vertical work
U
1
=
s
0.43
i
→
+
0.67
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.83
j
→
,
120
/
13
FromS1 based on r1sα
0.29
2.56
0.2
−1.41
0.49
1.15
2.85
−1.21
1.64
U
2
=
s
0.43
i
→
+
0.61
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.79
j
→
,
120
/
13
FromS1 based on r2sα
0.29
2.90
0.2
−1.74
0.49
1.16
3.19
−1.54
1.65
U
3
=
s
0.43
i
→
+
0.2
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.5
j
→
,
120
/
13
FromS2 based on r1sα
0.29
4.43
0.2
−3.46
0.49
0.97
4.72
−3.26
1.46
U
4
=
s
0.43
i
→
+
0.18
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.47
j
→
,
120
/
13
FromS2 based on r2sα
0.29
4.46
0.2
−3.58
0.49
0.88
4.75
−3.38
1.37In the above table,U1 and U2 are both got from S1 with respect to “s0.5i→+j→=neutral,” and obtained by r1sα and r2sα, respectively. Making comparisons between the works of U1 and U2, we can find that (i) their abscissa works are always same since the ranges of sα are s0.43,s0.54, either U1 or U2 and (ii) their vertical works are different due to the integral curves of ∫C→F→s→⋅da→ are different with r1sα and r2sα.ForU1, in the left side of “s0.5i→+j→=neutral,” the abscissa work and the vertical work are both positive, which indicates that either evaluations or satisfactions develop in the positive direction. However, in the right side of “s0.5i→+j→=neutral,” the abscissa work is positive but the vertical work is negative, which means that the evaluations develop in the positive direction but the satisfactions develop in the negative direction. Moreover, the total work of U1 is positive, which expresses that the overall work of evaluations and satisfactions is positively effective.Comparing the works amongU1, U2, U3, and U4, we get the individual meaning of s0.43,s0.54 for each patient. When they give “s0.5i→+j→=neutral” to express the evaluations for the current physician-patient communication, the first patient’s positive development is greater than others, and the fourth patient requires more attentions and improvements in the medical treatment process.


## 5.1. The Applications of the Nondirectional Curve Potential in Measuring the Fuzziness of the CSIPVLU

Example 3.
Lets0.8,s0.85⊂s0,s1 be an interval linguistic evaluation, S1 and S2 be two normalized LESs illustrated in Example 2, r1sα and r2sα be the ordinate coefficient functions given in equation (1). We transform s0.8,s0.85 into the CSIPVLUs by using r1sα and r2sα based on S1 and S2, respectively, and calculate the nondirectional curve potentials of the transformed units; the results are listed in Table 2:
By Table2, U1 and U2 are transformed from S1 with respect to “s0.75i→+j→=slightly high” by r1sα and r2sα, respectively. Referring to Figure 5, we know that r2sα is twisted more than r1sα, which demonstrates that the judgement thinking reflected by r2sα is fuzzier than the one reflected by r1sα. The nondirectional curve potentials of U1 and U2 present the fuzzy comparisons between U1 and U2. The same explanations can be conducted to the fuzzy comparisons between U3 and U4.
In addition,U1 and U3 are both transformed by r1sα, while U1 is from S1 and U3 is from S2. Comparing S1 with S2, we know that S2 is subdivided more. Therefore, the fuzziness of U3 should be less than U1. The nondirectional curve potentials of U1 and U3 show the difference of fuzziness between U1 and U3. The comparative results can also be summarized for U2 and U4.Table 2
The nondirectional potentials of the CSIPVLUs.
CSIPVLUs
Source
NDCP
U
1
=
s
0.8
i
→
+
0.80
j
→
,
s
0.85
i
→
+
0.60
j
→
,
20
FromS1 to “s0.75i→+j→=slightly high” based on r1sα
4.12
U
2
=
s
0.8
i
→
+
0.75
j
→
,
s
0.85
i
→
+
0.52
j
→
,
20
FromS1 to “s0.75i→+j→=slightly high” based on r2sα
4.62
U
3
=
s
0.8
i
→
+
0.72
j
→
,
s
0.85
i
→
+
0.54
j
→
,
20
FromS2 to “s0.72i→+j→=very high” based on r1sα
3.74
U
4
=
s
0.8
i
→
+
0.65
j
→
,
s
0.85
i
→
+
0.45
j
→
,
20
FromS2 to “s0.72i→+j→=very high” based on r2sα
4.05


## 5.2. The Applications of the Potentials in Dealing with the Linguistic Evaluation in Physician-Patient Communication for COPD

Nowadays, many major hospitals pay more attention to patients’ evaluations and satisfactions to improve the physician-patient relation. On the website of the West China Hospital, there are questionnaires to track and collect the social evaluation opinions and patients’ satisfactions for medicine service. In the evaluation systems, the indexes about the quality of physician-patient communications are necessary and important. The ways to obtain patients’ evaluations are various, such as questionnaire survey, visiting patients by mails or telephones, and so on.Suppose that the physicianE is a specialist on COPD. His performance is regularly evaluated according to the rules of the hospital. By mail and telephone following up, we get the same linguistic evaluation from patients H1, H2 H3, and H4 for E, which is s0.43,s0.54, where s0.43 is the earlier information and s0.54 is the later information. The values s0.43 and s0.54 are the any given patients’ evaluation information based on the multigranularity linguistic space Ωττ=12 illustrated in Example 2, which are used to illustrate the usage of the directional potentials with respect to the CMIPVLUs. Based on the LESs S1, S2, and the functions r1sα, and r2sα mentioned in equation (1), we can transform s0.43,s0.54 into four CMIPVLUs. By equation (11), we can calculate the abscissa component, the vertical component, and the total value of the directional curve potential for each CMIPVLU, respectively. The results are shown in Table 3.Table 3
The directional potentials of the CMIPVLUs.
CMIPVLUs
Source
The work in the left side ofs0.5i→+j→=neutral
The work in the right side ofs0.5i→+j→=neutral
The work of CMIPVLU
The abscissa work
The vertical work
The abscissa work
The vertical work
The abscissa work
The vertical work
U
1
=
s
0.43
i
→
+
0.67
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.83
j
→
,
120
/
13
FromS1 based on r1sα
0.29
2.56
0.2
−1.41
0.49
1.15
2.85
−1.21
1.64
U
2
=
s
0.43
i
→
+
0.61
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.79
j
→
,
120
/
13
FromS1 based on r2sα
0.29
2.90
0.2
−1.74
0.49
1.16
3.19
−1.54
1.65
U
3
=
s
0.43
i
→
+
0.2
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.5
j
→
,
120
/
13
FromS2 based on r1sα
0.29
4.43
0.2
−3.46
0.49
0.97
4.72
−3.26
1.46
U
4
=
s
0.43
i
→
+
0.18
j
→
⟶
s
0.5
i
→
+
j
→
,
120
/
13
,
s
0.5
i
→
+
j
→
⟶
s
0.54
i
→
+
0.47
j
→
,
120
/
13
FromS2 based on r2sα
0.29
4.46
0.2
−3.58
0.49
0.88
4.75
−3.38
1.37In the above table,U1 and U2 are both got from S1 with respect to “s0.5i→+j→=neutral,” and obtained by r1sα and r2sα, respectively. Making comparisons between the works of U1 and U2, we can find that (i) their abscissa works are always same since the ranges of sα are s0.43,s0.54, either U1 or U2 and (ii) their vertical works are different due to the integral curves of ∫C→F→s→⋅da→ are different with r1sα and r2sα.ForU1, in the left side of “s0.5i→+j→=neutral,” the abscissa work and the vertical work are both positive, which indicates that either evaluations or satisfactions develop in the positive direction. However, in the right side of “s0.5i→+j→=neutral,” the abscissa work is positive but the vertical work is negative, which means that the evaluations develop in the positive direction but the satisfactions develop in the negative direction. Moreover, the total work of U1 is positive, which expresses that the overall work of evaluations and satisfactions is positively effective.Comparing the works amongU1, U2, U3, and U4, we get the individual meaning of s0.43,s0.54 for each patient. When they give “s0.5i→+j→=neutral” to express the evaluations for the current physician-patient communication, the first patient’s positive development is greater than others, and the fourth patient requires more attentions and improvements in the medical treatment process.


## 6. Comparisons and Discussions

### 6.1. Comparisons among Different PVLUs

The nondirectional potentials and the directional potentials of the PVLUs introduced in Section4 can be summarized in the following table.In Table4, the sign “√” expresses “existence,” which indicates that the corresponding values of the directional potentials of the PVLUs can be obtained. For example, for the CSIPVLU, there are two “√” existing in the columns of the nondirectional curve potential and the directional curve potential, respectively. It means that the nondirectional curve potential and the directional curve potential of the CSIPVLU can be calculated. Additionally, the sign “√v” interprets “existence and vector,” which manifests that the results of the directional potentials of the PVLUs are vectors. Moreover, different forms of the PVLUs can be compared by the potentials that fall in the same colored area.Table 4
The summaries of the potentials of the PVLUs.
Potentials
PVLUs
The nondirectional potentials
The directional potentials
IPP
MPP
NDCP
DP
DCP
DSPVLU
√
√v
DMPVLU
√
√v
CSIPVLU
√
√
CMIPVLU
√
√
IPP = isolated point potential; MPP = multiple point potential; NDCP=nondirectional curve potential; DP = directional potential; DCP = directional curve potential.Example 4.
For several DSPVLUs and DMPVLUs listed in Table5, their nondirectional potentials and directional potentials can be calculated. The results are shown in Table 5.
In Table5, we calculate the IPPs of the DSPVLUs, the MPPs of the DMPVLUs, and the DPs of the DSPVLUs and the DMPVLUs. For the nondirectional potentials of the IPP and the MPP, we get an order that U1≻U3≻U5≻U4≻U2. Additionally, for the single vectors DPs, by assigning the probability of them as 1, the order U3≻U5≻U1≻U4≻U2 can be obtained. Obviously, the orders based on the nondirectional and directional potentials are different. The reason is that the IPP and the MPP are the holistic reflection of sα, rsα, and psαi→+rsαj→, while the DP is the component reflection of the sα with psαi→+rsαj→ and the rsα with psαi→+rsαj→. Thus, in practice, we should choose a desirable potential to cope with the probabilistic vector linguistic evaluation according to the different reflections about the properties of the nondirectional and directional potentials.Table 5
The potentials of several DSPVLUs and DMPVLUs.
Potentials
PVLUs
IPP
MPP
The order based on IPP and MPP
DP
The order based on DP
U
1
=
s
0.5
i
→
+
j
→
,
1
1.12
—
1
s
0.5
i
→
+
j
→
3
U
2
=
s
0.43
i
→
+
0.2
j
→
,
1
0.48
—
5
s
0.43
i
→
+
0.2
j
→
5
U
3
=
s
0.54
i
→
+
0.5
j
→
,
1
0.74
—
2
s
0.54
i
→
+
0.5
j
→
1
U
4
=
s
0.43
i
→
+
0.2
j
→
,
1
/
2
,
s
0.54
i
→
+
0.5
j
→
,
1
/
2
—
0.61
4
s
0.49
i
→
+
0.35
j
→
4
U
5
=
s
0.43
i
→
+
0.2
j
→
,
1
/
5
,
s
0.54
i
→
+
0.5
j
→
,
4
/
5
—
0.69
3
s
0.52
i
→
+
0.44
j
→
2


### 6.2. Comparisons between the New Proposed Method and the Other Related Methods

Example 5.
This example is to apply the GPVLT in personal hospital selection-recommender system, which has been illustrated with PVLT in Section5 of reference [38]. A COPD patient consults three experts groups (denoted by e1, e2, and e3) to choose a hospital from two alternatives H1 and H2 for the follow-up treatment. Because the experts can use different LESs to give the linguistic evaluations, this problem is a multigranularity linguistic decision-making. Let Se1, Se2 and Se3 be the experts’ LESs, respectively, where Se1 = {s−4 = “extremely low,” s−3 = “very low,” s−2 = “low,” s−1 = “few low,” s0 = “moderate,” s1 = “few high,” s2 = “high,” s3 = “very high,” s4 = “extremely high”}, Se2 = {s0 = “extremely low,” s0.25 = “few low,” s0.5 = “moderate,” s0.75 = “high,” s1 = “extremely high”} and Se3 = {s0 = “extremely low,” s0.28 = “very low,” s0.42 = “low,” s0.5 = “moderate,” s0.58 = “high,” s0.72 = “very high,” s1 = “extremely high”}.
Replacing the vector in each PVLT of Tables6 and 7 in reference [38] by the GPVLT based on r2sα in equation (1), we can get the DMPVLUs matrices provided by the experts on two hospitals over four attributes (denoted by A1, A2, A3, and A4) as
Denote each element of the above table ass→,pmn=s→,pσmnmn, where σmn=1,2,…,#s→,pmn and #s→,pmn is the granularity of the GPVLTs in the DMPVLU s→,pmn. Normalizing the probabilities in each element of Table 6 by ∑σmn=1#s→,pmnpσmn=1 and taking pσmn as the weight of s→σmn, we can aggregate s→,pmn into a GPVLT by the directional potential of DMPVLU defined in Definition 14. Thus, Table 6 can be rewritten into Table 7 as follows:
Note that the probability of each GPVLT in the above table defaults to 1. Based on the Euclidean distance, we can get the distance between each GPVLT in Table7 and the adjacent linguistic terms based on each individual LES, denoted by demn,α→mn, where emn is the element of row m, column n in the above table and sα→mn is the most adjacent linguistic term from semn in the LES Sen. According to the idea of the value function defined by equation (4) in Step 3 of Algorithm 4.1 in reference [38], we use EPemn=θkλi+wkυt−demn,α→mn+1 as the value function can get the selection sequences (11, 34, 32, 13) for H1 and (14, 11, 23, 22) for H2. For each vector in Table 7, take the subscript of the most adjacent linguistic term from semn in the LES Sen as the evaluation value to aggregate them by Step 5 of Algorithm 4.1 in reference [38], we get 1.92 as the final evaluation value of H1 and get 1.50 as the final evaluation value of H2. Obviously, we should recommend the first hospital to the patient, which is the same as the result in reference [38].
Comparing the above computing process to the one in reference [38], it is clear that each GPVLT in Table 7 can be explained into a linguistic term based on each individual LES. For example, for the vector 0.52i→+0.90j→ in the first row and the first column in Table 7, by the Euclidean distance and the nearby principle, it can be explained by the linguistic term “moderate” in the LES Se1. But in reference [38], the aggregated results of the experts on two hospitals over four attributes are real values with fuzzy degrees, which is hard to explain as a linguistic term in the LES. This is because that the ordinate of the vector in the GPVLT is finite but the one of the vector in the PVLT defined in reference [38] may be infinite. Thus, the result with the GPVLT in this paper helps the decision maker understand computed result more easily than the one based on the PVLT in reference [38], which is one of the advantages of GPVLT over the PVLT introduced in reference [38].
Moreover, as shown in Example 5, all DMPVLUs given by the experts for all attributes can be aggregated by the nondirectional potential and directional potential defined in Definition 10 (the aggregated result is an accurate real number) and Definition 14 (the aggregated result is an accurate vector), respectively. But in reference [38], because of the infinity ordinate of vector in the PVLT, we have to aggregate the experts’ linguistic evaluations by the curve fitting method, which is an approximation method with error. In this sense, the GPVLT proposed in this paper improves the computational performance of the ordinate of the PVLT defined in reference [38].Table 6
The decision matrix of the experts on two hospitals over four attributes.
A
1
0.33
A
2
0.27
A
3
0.25
A
4
0.15
H
1
e
1
0.45
0.38
i
→
+
j
→
,
0.2
,
0.5
i
→
+
j
→
,
0.38
,
0.75
i
→
+
j
→
,
0.42
0.38
i
→
+
j
→
,
0.15
,
0.42
i
→
+
0.63
j
→
,
0.36
,
0.46
i
→
+
0.63
j
→
,
0.29
,
0.75
i
→
+
j
→
,
0.2
0.46
i
→
+
0.63
j
→
,
0.07
,
0.5
i
→
+
j
→
,
0.31
,
0.72
i
→
+
0.72
j
→
,
0.21
,
0.85
i
→
+
0.74
j
→
,
0.42
0.38
i
→
+
j
→
,
0.31
,
0.42
i
→
+
0.63
j
→
,
0.16
,
0.83
i
→
+
j
→
,
0.53
e
2
0.35
0.28
i
→
+
0.85
j
→
,
0.15
,
0.75
i
→
+
j
→
,
0.49
,
0.9
i
→
+
0.52
j
→
,
0.36
0.42
i
→
+
0.61
j
→
,
0.4
,
0.5
i
→
+
j
→
,
0.13
,
0.75
i
→
+
j
→
,
0.47
0.13
i
→
+
0.44
j
→
,
0.34
,
0.42
i
→
+
0.61
j
→
,
0.45
,
0.95
i
→
+
0.75
j
→
,
0.21
0.33
i
→
+
0.61
j
→
,
0.2
,
0.42
i
→
+
0.61
j
→
,
0.19
,
0.5
i
→
+
j
→
,
0.33
,
0.58
i
→
+
0.61
j
→
,
0.29
e
3
0.20
0.42
i
→
+
j
→
,
0.26
,
0.5
i
→
+
j
→
,
0.21
,
0.58
i
→
+
j
→
,
0.53
0.42
i
→
+
j
→
,
0.2
,
0.54
i
→
+
0.47
j
→
,
0.35
,
0.5
i
→
+
j
→
,
0.27
,
0.75
i
→
+
j
→
,
0.18
0.5
i
→
+
j
→
,
0.43
,
0.58
i
→
+
j
→
,
0.17
,
0.75
i
→
+
j
→
,
0.39
0.42
i
→
+
j
→
,
0.3
,
0.5
i
→
+
j
→
,
0.21
,
0.58
i
→
+
j
→
,
0.49
H
2
e
1
0.45
0.28
i
→
+
0.74
j
→
,
0.25
,
0.45
i
→
+
0.55
j
→
,
0.4
,
0.75
i
→
+
j
→
,
0.35
0.35
i
→
+
0.74
j
→
,
0.17
,
0.47
i
→
+
0.72
j
→
,
0.22
,
0.6
i
→
+
0.74
j
→
,
0.4
,
0.75
i
→
+
j
→
,
0.2
0.38
i
→
+
j
→
,
0.06
,
0.47
i
→
+
0.72
j
→
,
0.33
,
0.57
i
→
+
0.5
j
→
,
0.23
,
0.65
i
→
+
j
→
,
0.38
0.38
i
→
+
j
→
,
0.13
,
0.45
i
→
+
0.55
j
→
,
0.38
,
0.93
i
→
+
0.55
j
→
,
0.5
e
2
0.35
0.35
i
→
+
0.52
j
→
,
0.14
,
0.7
i
→
+
0.75
j
→
,
0.43
,
0.8
i
→
+
0.75
j
→
,
0.43
0.39
i
→
+
0.48
j
→
,
0.26
,
0.5
i
→
+
j
→
,
0.39
,
0.75
i
→
+
j
→
,
0.36
0.13
i
→
+
j
→
,
0.38
,
0.66
i
→
+
j
→
,
0.41
,
0.9
i
→
+
j
→
,
0.21
0.45
i
→
+
0.75
j
→
,
0.18
,
0.47
i
→
+
0.85
j
→
,
0.25
,
0.5
i
→
+
j
→
,
0.3
,
0.78
i
→
+
0.85
j
→
,
0.26
e
3
0.20
0.4
i
→
+
0.84
j
→
,
0.26
,
0.5
i
→
+
j
→
,
0.21
,
0.68
i
→
+
0.68
j
→
,
0.53
0.4
i
→
+
0.84
j
→
,
0.23
,
0.56
i
→
+
0.73
j
→
,
0.31
,
0.7
i
→
+
0.84
j
→
,
0.28
,
0.9
i
→
+
0.56
j
→
,
0.19
0.5
i
→
+
j
→
,
0.27
,
0.7
i
→
+
0.84
j
→
,
0.41
,
0.75
i
→
+
0.86
j
→
,
0.32
0.49
i
→
+
0.86
j
→
,
0.35
,
0.66
i
→
+
0.53
j
→
,
0.19
,
0.98
i
→
+
0.9
j
→
,
0.47Table 7
The aggregated results of the experts on two hospitals over four attributes.
A
1
0.33
A
2
0.27
A
3
0.25
A
4
0.15
H
1
e
1
0.45
0.52
i
→
+
0.90
j
→
0.44
i
→
+
0.68
j
→
0.50
i
→
+
0.58
j
→
0.32
i
→
+
0.48
j
→
e
2
0.35
0.53
i
→
+
0.58
j
→
0.31
i
→
+
0.45
j
→
0.31
i
→
+
0.41
j
→
0.38
i
→
+
0.59
j
→
e
3
0.20
0.40
i
→
−
0.76
j
→
0.54
i
→
+
0.81
j
→
0.42
i
→
+
0.69
j
→
0.38
i
→
+
0.73
j
→
H
2
e
1
0.45
0.56
i
→
+
0.83
j
→
0.50
i
→
+
0.70
j
→
0.44
i
→
+
0.63
j
→
0.32
i
→
+
0.29
j
→
e
2
0.35
0.49
i
→
+
0.50
j
→
0.39
i
→
+
0.61
j
→
0.37
i
→
+
0.73
j
→
0.48
i
→
+
0.76
j
→
e
3
0.20
0.43
i
→
+
0.60
j
→
0.61
i
→
+
0.73
j
→
0.56
i
→
+
0.76
j
→
0.64
i
→
+
0.70
j
→


### 6.3. Discussions

In this subsection, we point out some drawbacks and advantages of the new proposed methods.Drawbacks:(1)
In practical decision-making problems, it is hard to determine the probability functions of the CSIPVLU and the CMIPVLU.(2)
The nondirectional potentials cannot be utilized to compare all forms of the PVLUs together since various potentials have different meanings. For example, the multiple point potential of the DMPVLU and the nondirectional curve potential of the CSIPVLU mean diversely.Advantages:(1)
The GPVLT is more appropriate and comprehensive than the PVLT in portraying people’s judgements.(2)
The nondirectional potentials and the directional potentials possess the strong abilities to compare not only the discrete PVLUs, but also the continuous PVLUs.(3)
For an object, we can grasp its directional changes from two angles, i.e., the evaluations and the satisfactions, by the directional curve potential of the continuous PVLUs. It provides the flexibility and effectiveness to let people choose the component works or the aggregated work of the continuous PVLUs according to real problems.


## 6.1. Comparisons among Different PVLUs

The nondirectional potentials and the directional potentials of the PVLUs introduced in Section4 can be summarized in the following table.In Table4, the sign “√” expresses “existence,” which indicates that the corresponding values of the directional potentials of the PVLUs can be obtained. For example, for the CSIPVLU, there are two “√” existing in the columns of the nondirectional curve potential and the directional curve potential, respectively. It means that the nondirectional curve potential and the directional curve potential of the CSIPVLU can be calculated. Additionally, the sign “√v” interprets “existence and vector,” which manifests that the results of the directional potentials of the PVLUs are vectors. Moreover, different forms of the PVLUs can be compared by the potentials that fall in the same colored area.Table 4
The summaries of the potentials of the PVLUs.
Potentials
PVLUs
The nondirectional potentials
The directional potentials
IPP
MPP
NDCP
DP
DCP
DSPVLU
√
√v
DMPVLU
√
√v
CSIPVLU
√
√
CMIPVLU
√
√
IPP = isolated point potential; MPP = multiple point potential; NDCP=nondirectional curve potential; DP = directional potential; DCP = directional curve potential.Example 4.
For several DSPVLUs and DMPVLUs listed in Table5, their nondirectional potentials and directional potentials can be calculated. The results are shown in Table 5.
In Table5, we calculate the IPPs of the DSPVLUs, the MPPs of the DMPVLUs, and the DPs of the DSPVLUs and the DMPVLUs. For the nondirectional potentials of the IPP and the MPP, we get an order that U1≻U3≻U5≻U4≻U2. Additionally, for the single vectors DPs, by assigning the probability of them as 1, the order U3≻U5≻U1≻U4≻U2 can be obtained. Obviously, the orders based on the nondirectional and directional potentials are different. The reason is that the IPP and the MPP are the holistic reflection of sα, rsα, and psαi→+rsαj→, while the DP is the component reflection of the sα with psαi→+rsαj→ and the rsα with psαi→+rsαj→. Thus, in practice, we should choose a desirable potential to cope with the probabilistic vector linguistic evaluation according to the different reflections about the properties of the nondirectional and directional potentials.Table 5
The potentials of several DSPVLUs and DMPVLUs.
Potentials
PVLUs
IPP
MPP
The order based on IPP and MPP
DP
The order based on DP
U
1
=
s
0.5
i
→
+
j
→
,
1
1.12
—
1
s
0.5
i
→
+
j
→
3
U
2
=
s
0.43
i
→
+
0.2
j
→
,
1
0.48
—
5
s
0.43
i
→
+
0.2
j
→
5
U
3
=
s
0.54
i
→
+
0.5
j
→
,
1
0.74
—
2
s
0.54
i
→
+
0.5
j
→
1
U
4
=
s
0.43
i
→
+
0.2
j
→
,
1
/
2
,
s
0.54
i
→
+
0.5
j
→
,
1
/
2
—
0.61
4
s
0.49
i
→
+
0.35
j
→
4
U
5
=
s
0.43
i
→
+
0.2
j
→
,
1
/
5
,
s
0.54
i
→
+
0.5
j
→
,
4
/
5
—
0.69
3
s
0.52
i
→
+
0.44
j
→
2


## 6.2. Comparisons between the New Proposed Method and the Other Related Methods

Example 5.
This example is to apply the GPVLT in personal hospital selection-recommender system, which has been illustrated with PVLT in Section5 of reference [38]. A COPD patient consults three experts groups (denoted by e1, e2, and e3) to choose a hospital from two alternatives H1 and H2 for the follow-up treatment. Because the experts can use different LESs to give the linguistic evaluations, this problem is a multigranularity linguistic decision-making. Let Se1, Se2 and Se3 be the experts’ LESs, respectively, where Se1 = {s−4 = “extremely low,” s−3 = “very low,” s−2 = “low,” s−1 = “few low,” s0 = “moderate,” s1 = “few high,” s2 = “high,” s3 = “very high,” s4 = “extremely high”}, Se2 = {s0 = “extremely low,” s0.25 = “few low,” s0.5 = “moderate,” s0.75 = “high,” s1 = “extremely high”} and Se3 = {s0 = “extremely low,” s0.28 = “very low,” s0.42 = “low,” s0.5 = “moderate,” s0.58 = “high,” s0.72 = “very high,” s1 = “extremely high”}.
Replacing the vector in each PVLT of Tables6 and 7 in reference [38] by the GPVLT based on r2sα in equation (1), we can get the DMPVLUs matrices provided by the experts on two hospitals over four attributes (denoted by A1, A2, A3, and A4) as
Denote each element of the above table ass→,pmn=s→,pσmnmn, where σmn=1,2,…,#s→,pmn and #s→,pmn is the granularity of the GPVLTs in the DMPVLU s→,pmn. Normalizing the probabilities in each element of Table 6 by ∑σmn=1#s→,pmnpσmn=1 and taking pσmn as the weight of s→σmn, we can aggregate s→,pmn into a GPVLT by the directional potential of DMPVLU defined in Definition 14. Thus, Table 6 can be rewritten into Table 7 as follows:
Note that the probability of each GPVLT in the above table defaults to 1. Based on the Euclidean distance, we can get the distance between each GPVLT in Table7 and the adjacent linguistic terms based on each individual LES, denoted by demn,α→mn, where emn is the element of row m, column n in the above table and sα→mn is the most adjacent linguistic term from semn in the LES Sen. According to the idea of the value function defined by equation (4) in Step 3 of Algorithm 4.1 in reference [38], we use EPemn=θkλi+wkυt−demn,α→mn+1 as the value function can get the selection sequences (11, 34, 32, 13) for H1 and (14, 11, 23, 22) for H2. For each vector in Table 7, take the subscript of the most adjacent linguistic term from semn in the LES Sen as the evaluation value to aggregate them by Step 5 of Algorithm 4.1 in reference [38], we get 1.92 as the final evaluation value of H1 and get 1.50 as the final evaluation value of H2. Obviously, we should recommend the first hospital to the patient, which is the same as the result in reference [38].
Comparing the above computing process to the one in reference [38], it is clear that each GPVLT in Table 7 can be explained into a linguistic term based on each individual LES. For example, for the vector 0.52i→+0.90j→ in the first row and the first column in Table 7, by the Euclidean distance and the nearby principle, it can be explained by the linguistic term “moderate” in the LES Se1. But in reference [38], the aggregated results of the experts on two hospitals over four attributes are real values with fuzzy degrees, which is hard to explain as a linguistic term in the LES. This is because that the ordinate of the vector in the GPVLT is finite but the one of the vector in the PVLT defined in reference [38] may be infinite. Thus, the result with the GPVLT in this paper helps the decision maker understand computed result more easily than the one based on the PVLT in reference [38], which is one of the advantages of GPVLT over the PVLT introduced in reference [38].
Moreover, as shown in Example 5, all DMPVLUs given by the experts for all attributes can be aggregated by the nondirectional potential and directional potential defined in Definition 10 (the aggregated result is an accurate real number) and Definition 14 (the aggregated result is an accurate vector), respectively. But in reference [38], because of the infinity ordinate of vector in the PVLT, we have to aggregate the experts’ linguistic evaluations by the curve fitting method, which is an approximation method with error. In this sense, the GPVLT proposed in this paper improves the computational performance of the ordinate of the PVLT defined in reference [38].Table 6
The decision matrix of the experts on two hospitals over four attributes.
A
1
0.33
A
2
0.27
A
3
0.25
A
4
0.15
H
1
e
1
0.45
0.38
i
→
+
j
→
,
0.2
,
0.5
i
→
+
j
→
,
0.38
,
0.75
i
→
+
j
→
,
0.42
0.38
i
→
+
j
→
,
0.15
,
0.42
i
→
+
0.63
j
→
,
0.36
,
0.46
i
→
+
0.63
j
→
,
0.29
,
0.75
i
→
+
j
→
,
0.2
0.46
i
→
+
0.63
j
→
,
0.07
,
0.5
i
→
+
j
→
,
0.31
,
0.72
i
→
+
0.72
j
→
,
0.21
,
0.85
i
→
+
0.74
j
→
,
0.42
0.38
i
→
+
j
→
,
0.31
,
0.42
i
→
+
0.63
j
→
,
0.16
,
0.83
i
→
+
j
→
,
0.53
e
2
0.35
0.28
i
→
+
0.85
j
→
,
0.15
,
0.75
i
→
+
j
→
,
0.49
,
0.9
i
→
+
0.52
j
→
,
0.36
0.42
i
→
+
0.61
j
→
,
0.4
,
0.5
i
→
+
j
→
,
0.13
,
0.75
i
→
+
j
→
,
0.47
0.13
i
→
+
0.44
j
→
,
0.34
,
0.42
i
→
+
0.61
j
→
,
0.45
,
0.95
i
→
+
0.75
j
→
,
0.21
0.33
i
→
+
0.61
j
→
,
0.2
,
0.42
i
→
+
0.61
j
→
,
0.19
,
0.5
i
→
+
j
→
,
0.33
,
0.58
i
→
+
0.61
j
→
,
0.29
e
3
0.20
0.42
i
→
+
j
→
,
0.26
,
0.5
i
→
+
j
→
,
0.21
,
0.58
i
→
+
j
→
,
0.53
0.42
i
→
+
j
→
,
0.2
,
0.54
i
→
+
0.47
j
→
,
0.35
,
0.5
i
→
+
j
→
,
0.27
,
0.75
i
→
+
j
→
,
0.18
0.5
i
→
+
j
→
,
0.43
,
0.58
i
→
+
j
→
,
0.17
,
0.75
i
→
+
j
→
,
0.39
0.42
i
→
+
j
→
,
0.3
,
0.5
i
→
+
j
→
,
0.21
,
0.58
i
→
+
j
→
,
0.49
H
2
e
1
0.45
0.28
i
→
+
0.74
j
→
,
0.25
,
0.45
i
→
+
0.55
j
→
,
0.4
,
0.75
i
→
+
j
→
,
0.35
0.35
i
→
+
0.74
j
→
,
0.17
,
0.47
i
→
+
0.72
j
→
,
0.22
,
0.6
i
→
+
0.74
j
→
,
0.4
,
0.75
i
→
+
j
→
,
0.2
0.38
i
→
+
j
→
,
0.06
,
0.47
i
→
+
0.72
j
→
,
0.33
,
0.57
i
→
+
0.5
j
→
,
0.23
,
0.65
i
→
+
j
→
,
0.38
0.38
i
→
+
j
→
,
0.13
,
0.45
i
→
+
0.55
j
→
,
0.38
,
0.93
i
→
+
0.55
j
→
,
0.5
e
2
0.35
0.35
i
→
+
0.52
j
→
,
0.14
,
0.7
i
→
+
0.75
j
→
,
0.43
,
0.8
i
→
+
0.75
j
→
,
0.43
0.39
i
→
+
0.48
j
→
,
0.26
,
0.5
i
→
+
j
→
,
0.39
,
0.75
i
→
+
j
→
,
0.36
0.13
i
→
+
j
→
,
0.38
,
0.66
i
→
+
j
→
,
0.41
,
0.9
i
→
+
j
→
,
0.21
0.45
i
→
+
0.75
j
→
,
0.18
,
0.47
i
→
+
0.85
j
→
,
0.25
,
0.5
i
→
+
j
→
,
0.3
,
0.78
i
→
+
0.85
j
→
,
0.26
e
3
0.20
0.4
i
→
+
0.84
j
→
,
0.26
,
0.5
i
→
+
j
→
,
0.21
,
0.68
i
→
+
0.68
j
→
,
0.53
0.4
i
→
+
0.84
j
→
,
0.23
,
0.56
i
→
+
0.73
j
→
,
0.31
,
0.7
i
→
+
0.84
j
→
,
0.28
,
0.9
i
→
+
0.56
j
→
,
0.19
0.5
i
→
+
j
→
,
0.27
,
0.7
i
→
+
0.84
j
→
,
0.41
,
0.75
i
→
+
0.86
j
→
,
0.32
0.49
i
→
+
0.86
j
→
,
0.35
,
0.66
i
→
+
0.53
j
→
,
0.19
,
0.98
i
→
+
0.9
j
→
,
0.47Table 7
The aggregated results of the experts on two hospitals over four attributes.
A
1
0.33
A
2
0.27
A
3
0.25
A
4
0.15
H
1
e
1
0.45
0.52
i
→
+
0.90
j
→
0.44
i
→
+
0.68
j
→
0.50
i
→
+
0.58
j
→
0.32
i
→
+
0.48
j
→
e
2
0.35
0.53
i
→
+
0.58
j
→
0.31
i
→
+
0.45
j
→
0.31
i
→
+
0.41
j
→
0.38
i
→
+
0.59
j
→
e
3
0.20
0.40
i
→
−
0.76
j
→
0.54
i
→
+
0.81
j
→
0.42
i
→
+
0.69
j
→
0.38
i
→
+
0.73
j
→
H
2
e
1
0.45
0.56
i
→
+
0.83
j
→
0.50
i
→
+
0.70
j
→
0.44
i
→
+
0.63
j
→
0.32
i
→
+
0.29
j
→
e
2
0.35
0.49
i
→
+
0.50
j
→
0.39
i
→
+
0.61
j
→
0.37
i
→
+
0.73
j
→
0.48
i
→
+
0.76
j
→
e
3
0.20
0.43
i
→
+
0.60
j
→
0.61
i
→
+
0.73
j
→
0.56
i
→
+
0.76
j
→
0.64
i
→
+
0.70
j
→


## 6.3. Discussions

In this subsection, we point out some drawbacks and advantages of the new proposed methods.Drawbacks:(1)
In practical decision-making problems, it is hard to determine the probability functions of the CSIPVLU and the CMIPVLU.(2)
The nondirectional potentials cannot be utilized to compare all forms of the PVLUs together since various potentials have different meanings. For example, the multiple point potential of the DMPVLU and the nondirectional curve potential of the CSIPVLU mean diversely.Advantages:(1)
The GPVLT is more appropriate and comprehensive than the PVLT in portraying people’s judgements.(2)
The nondirectional potentials and the directional potentials possess the strong abilities to compare not only the discrete PVLUs, but also the continuous PVLUs.(3)
For an object, we can grasp its directional changes from two angles, i.e., the evaluations and the satisfactions, by the directional curve potential of the continuous PVLUs. It provides the flexibility and effectiveness to let people choose the component works or the aggregated work of the continuous PVLUs according to real problems.


## 7. Conclusions

In this paper, we have extended the PVLT into the GPVLT to improve the computational performance of the ordinate of the vector in PVLT. Based on GPVLT, we have studied the forms of the PVLUs, i.e., DSPVLUs, DMPVLUs, CSIPVLUs, and CMIPVLUs and proposed the nondirectional potentials and the directional potentials for them. Because the GPVLT can distinguish the directional linguistic evaluation, the new proposed potentials have enriched the theories of the PVLTs to open a new prospect of analyzing and utilizing the linguistic evaluation. Later on, the cases about the physician-patient communication for COPD have been illustrated to demonstrate the effectiveness and practicability of the potentials of PVLUs. Furthermore, we have compared the new proposed methods with other related methods and listed the drawbacks and advantages of the GPVLT and the potentials of the PVLUs.Moreover, there are some potential directions for the further investigation. For example, we have developed several basic ways to study the linguistic evaluations through the analytical properties of functions, such as the directional potentials of the CSIPVLUs and the CMIPVLUs. Thus, we can develop more tools on these results to promote the application of GPVLT in analyzing the linguistic evaluations for the physician-patient communication of COPD. Secondly, considering the advantages of GPVLT in describing the same linguistic evaluation from different sources in the multigranularity linguistic decision-making, we will continue to study the practical applications of applying the GPVLT in the multigranularity linguistic decision-making methods.



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*Source: 2574075-2019-11-21.xml*