Processing math: 32%
Research article

Effectiveness of the genus Riccia (Marchantiophyta: Ricciaceae) as a biofilter for particulate matter adsorption from air pollution

  • Received: 03 September 2022 Revised: 06 December 2022 Accepted: 13 December 2022 Published: 17 January 2023
  • The study of plants as a biofilter is highly relevant in the field of air pollution science to ecological restoration in urban, which is connected to the ecosystem and human health. The aim of this present study was designed to evaluate the use of Riccia as a biofilter for particulate matter. The treatment box was designed using the Computational Fluid Dynamic (CFD) model. The alignment of the biofilter plant was designed and performed in three different arrangements blocking, zigzag, and parallel panels. The particulate matter was generated by simulated B7 diesel fuel combustion smoke using a smoke generator and loaded into the chamber with air velocities of 0.5, 1.0, 1.5, and 2.0 m/s via a Laser dust sensor for both inlet and outlet air. The adsorption efficiency of the PM adsorbed on the biofilter plant was calculated. The physical properties, physiological, and biochemical parameters of the study plant such as Air pollution tolerance index (APTI), Dust capturing potential were investigated. Moreover, the micromorphological details of the plant, the volatile organic compounds (VOCs), polyaromatic hydrocarbons (PAHs), and adsorbed metal were analyzed. The study revealed adsorption efficiency was in the range of 2.3%–49.6 %. The highest efficiency values for PM1, PM2.5, and PM10 were 31.4, 40.1, and 49.6, respectively, which belonged to the horizontal panel with a velocity of 2.0 m/s. The alignment of the panel and air velocities affects the efficiency. HS-GC-MS revealed that Riccia can be adsorbed the particulate matter and the quantity of Cd, Pb, and Na were 0.0044 ± 0.0069 mg/gDW, 0.0208 ± 0.0278 mg/gDW, and 0.9395 ± 0.1009 mg/gDW, respectively. The morphological study exhibited a rough surface to enhance the efficiency of the trapped particle matter. The results showed that Riccia was suitable for adsorbing the particulate matter with a diameter of 1–4 μm.

    Citation: Winai Meesang, Erawan Baothong, Aphichat Srichat, Sawai Mattapha, Wiwat Kaensa, Pathomsorn Juthakanok, Wipaporn Kitisriworaphan, Kanda Saosoong. Effectiveness of the genus Riccia (Marchantiophyta: Ricciaceae) as a biofilter for particulate matter adsorption from air pollution[J]. AIMS Environmental Science, 2023, 10(1): 157-177. doi: 10.3934/environsci.2023009

    Related Papers:

    [1] Chunfeng Suo, Yan Wang, Dan Mou . The new construction of knowledge measure on intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets. AIMS Mathematics, 2023, 8(11): 27113-27127. doi: 10.3934/math.20231387
    [2] Li Li, Xin Wang . Hamming distance-based knowledge measure and entropy for interval-valued Pythagorean fuzzy sets. AIMS Mathematics, 2025, 10(4): 8707-8720. doi: 10.3934/math.2025399
    [3] Muhammad Riaz, Maryam Saba, Muhammad Abdullah Khokhar, Muhammad Aslam . Novel concepts of m-polar spherical fuzzy sets and new correlation measures with application to pattern recognition and medical diagnosis. AIMS Mathematics, 2021, 6(10): 11346-11379. doi: 10.3934/math.2021659
    [4] Li Li, Mengjing Hao . Interval-valued Pythagorean fuzzy entropy and its application to multi-criterion group decision-making. AIMS Mathematics, 2024, 9(5): 12511-12528. doi: 10.3934/math.2024612
    [5] T. M. Athira, Sunil Jacob John, Harish Garg . A novel entropy measure of Pythagorean fuzzy soft sets. AIMS Mathematics, 2020, 5(2): 1050-1061. doi: 10.3934/math.2020073
    [6] Atiqe Ur Rahman, Muhammad Saeed, Hamiden Abd El-Wahed Khalifa, Walaa Abdullah Afifi . Decision making algorithmic techniques based on aggregation operations and similarity measures of possibility intuitionistic fuzzy hypersoft sets. AIMS Mathematics, 2022, 7(3): 3866-3895. doi: 10.3934/math.2022214
    [7] Hu Wang . A novel bidirectional projection measures of circular intuitionistic fuzzy sets and its application to multiple attribute group decision-making problems. AIMS Mathematics, 2025, 10(5): 10283-10307. doi: 10.3934/math.2025468
    [8] Changlin Xu, Yaqing Wen . New measure of circular intuitionistic fuzzy sets and its application in decision making. AIMS Mathematics, 2023, 8(10): 24053-24074. doi: 10.3934/math.20231226
    [9] Shichao Li, Zeeshan Ali, Peide Liu . Prioritized Hamy mean operators based on Dombi t-norm and t-conorm for the complex interval-valued Atanassov-Intuitionistic fuzzy sets and their applications in strategic decision-making problems. AIMS Mathematics, 2025, 10(3): 6589-6635. doi: 10.3934/math.2025302
    [10] Muhammad Arshad, Muhammad Saeed, Khuram Ali Khan, Nehad Ali Shah, Wajaree Weera, Jae Dong Chung . A robust MADM-approach to recruitment-based pattern recognition by using similarity measures of interval-valued fuzzy hypersoft set. AIMS Mathematics, 2023, 8(5): 12321-12341. doi: 10.3934/math.2023620
  • The study of plants as a biofilter is highly relevant in the field of air pollution science to ecological restoration in urban, which is connected to the ecosystem and human health. The aim of this present study was designed to evaluate the use of Riccia as a biofilter for particulate matter. The treatment box was designed using the Computational Fluid Dynamic (CFD) model. The alignment of the biofilter plant was designed and performed in three different arrangements blocking, zigzag, and parallel panels. The particulate matter was generated by simulated B7 diesel fuel combustion smoke using a smoke generator and loaded into the chamber with air velocities of 0.5, 1.0, 1.5, and 2.0 m/s via a Laser dust sensor for both inlet and outlet air. The adsorption efficiency of the PM adsorbed on the biofilter plant was calculated. The physical properties, physiological, and biochemical parameters of the study plant such as Air pollution tolerance index (APTI), Dust capturing potential were investigated. Moreover, the micromorphological details of the plant, the volatile organic compounds (VOCs), polyaromatic hydrocarbons (PAHs), and adsorbed metal were analyzed. The study revealed adsorption efficiency was in the range of 2.3%–49.6 %. The highest efficiency values for PM1, PM2.5, and PM10 were 31.4, 40.1, and 49.6, respectively, which belonged to the horizontal panel with a velocity of 2.0 m/s. The alignment of the panel and air velocities affects the efficiency. HS-GC-MS revealed that Riccia can be adsorbed the particulate matter and the quantity of Cd, Pb, and Na were 0.0044 ± 0.0069 mg/gDW, 0.0208 ± 0.0278 mg/gDW, and 0.9395 ± 0.1009 mg/gDW, respectively. The morphological study exhibited a rough surface to enhance the efficiency of the trapped particle matter. The results showed that Riccia was suitable for adsorbing the particulate matter with a diameter of 1–4 μm.



    Since Zadeh introduced fuzzy set theory, extensive research and extension related to fuzzy sets have been pursued. Among the various generalized forms that have been proposed, Atanassov [1] introduced the definition of intuitionistic fuzzy sets (IFSs). This was a broader concept of fuzzy sets designed to address uncertain information. It offered a more precise characterization of such information through the triad of degree of membership, degree of nonmembership, and degree of hesitation. As an extension of IFS, Atanassov [2] presented an alternative definition of interval-valued intuitionistic fuzzy sets (IVIFSs), wherein the degree of membership, nonmembership, and hesitation are represented by subintervals within the range of [0,1]. As a result, it can precisely capture the dynamic features. Furthermore, Xu [3] examined certain attributes and average operators of IVIFSs. Currently, IVIFSs have been extensively utilized in diverse areas, including quality evaluation, venture capital, and medical diagnosis [4,5,6,7], owing to their remarkable flexibility in managing uncertainty or ambiguity.

    Information measures pertaining to IFSs and IVIFSs have garnered escalating attention due to their exceptional performance in addressing practical applications. Interested readers can to [8,9,10]. Axiomatic definitions of distance measures on IFSs and IVIFSs were respectively provided in [11,12]. Furthermore, a novel distance measure for IFSs was introduced through the utilization of line integrals [13]. Drawing from the relationship between distance measure and similarity measure, similarity measure has also been paid attention to by a considerable number of researchers, and numerous scholars investigated the similarity measure for IVIFSs on the basis of distance measures [14,15,16,29,30,31,32]. Fuzzy entropy was initially introduced by Zadeh, and in recent years, there has been a growing interest among scholars in exploring this concept. Interested readers are encouraged to consult references [17,18,19] for further details. An entropy on IVIFSs constructed based on a distance measure was proposed by Zhang et al. [20], and the relationship between similarity measure and entropy was investigated in the paper. Che et al. [21] presented a method for constructing an entropy on IVIFSs in terms of the distance function. The cross-entropy approach, originating from the information theory presented by Shannon [22], was later applied to decision-making problems by Ye [23] who developed the fuzzy cross-entropy of IVIFSs using an analogy with the intuitionistic fuzzy cross-entropy. In recent years, there has been a notable surge in the interest surrounding knowledge measures within the frameworks of IFSs and IVIFSs, as clearly demonstrated by the references [24,25,26]. Das et al. [27] employed the knowledge measures under IFSs and IVIFSs to compute the criterion weights in the decision problem, and Guo [28] considered both aspects of knowledge related to IVIFSs, i.e., information content and information clarity to construct a model of knowledge measure and apply it to a decision-making problem under unknown weights. In addition, some scholars have also extended IVIFS [33,34].

    The structure of this paper is as follows: Section 2 delves into several definitions and axioms that will be utilized in the subsequent research outlined in this paper. Section 3 centers on the construction of similarity measures, knowledge measures, and entropy, all grounded in the axiomatic definition of the closest crisp set and the distance measure. Furthermore, it explores the intricate transformation relationships among these three constructs. Subsequently, we present a range of formulas for information measures, each utilizing distinct functions. In Section 4, we demonstrate the efficacy and versatility of our newly introduced methodologies by applying the proposed knowledge measure to tackle a decision-making problem and the similarity measure to address a pattern recognition problem. Lastly, in Section 5, we offer some concluding remarks on the contents of this paper.

    In this section, we will briefly introduce some of fuzzy set theories and concepts which should be used in this paper. The X in the following text is represented as: X=(x1,x2,,xn). Let N denote the set of natural numbers.

    Definition 2.1. ( [2]) An IVIFS ˜A on a finite set X can be defined as the following form:

    ˜A={<x,μ˜A(x),ν˜A(x)>|xX},

    where μ˜A(x)=[μL˜A(x),μU˜A(x)][0,1] and ν˜A(x)=[νL˜A(x),νU˜A(x)][0,1], which satisfies 0μU˜A(x)+νU˜A(x)1, for any xX. For the interval hesitation margin π˜A(x)=[πL˜A(x),πU˜A(x)][0,1], we have πL˜A(x)=1μU˜A(x)νU˜A(x) and πU˜A(x)=1μL˜A(x)νL˜A(x).

    Specifically, if μL˜A(x)=μU˜A(x) and νL˜A(x)=νU˜A(x), then the IVIFS is reduced to an intuitionistic fuzzy set.

    From the interval representation of membership degree and nonmembership degree of IVIFSs, it is not difficult to see that the geometric meaning of IVIFSs at a certain point is a rectangle, as shown in Figure 1. The three vertices of the triangle are denoted as (0,0),(1,0),(0,1), and the two sides of the triangle represent the degree of membership μ˜A(x) and nonmembership ν˜A(x), so that (0,0) expresses the case if π˜A(x)=1. (1,0) and (0,1) mean the two cases of the crisp sets. For the hypotenuse of the triangle, that is, the segment from (1,0) to (0,1), it corresponds to ν˜A(x)=1μ˜A(x). Moreover, there is 0μ˜A(x)+ν˜A(x)1 for an IVIFS, which also means that none of the individual vertices of the rectangle in Figure 1 can extend beyond the hypotenuse of the triangle.

    Figure 1.  Graphicalization of IVIFS.

    For convenience, let IVIFS(X) express the set of IVIFSs on X. Besides, we use P(X) to denote the set of all crisp sets defined on X.

    Definition 2.2. ( [3]) For any ˜A,˜BIVIFS(X), the following relations can be defined:

    (1) ˜A˜B if μL˜A(x)μL˜B(x), μU˜A(x)μU˜B(x) and νL˜A(x)νL˜B(x), νU˜A(x)νU˜B(x) for any xX;

    (2) ˜A=˜B if ˜A˜B and ˜A˜B;

    (3) The complement set ˜Ac={x,ν˜A(x),μ˜A(x)|xX}.

    Definition 2.3. ( [12]) For any ˜A,˜B,˜CIVIFS(X), a mapping D:IVIFS(X)×IVIFS(X)[0,1] is called a distance measure on IVIFSs, and it satisfies the following properties:

    (D1)0D(˜A,˜B)1;

    (D2)D(˜A,˜B)=0 if ˜A=˜B;

    (D3)D(˜A,˜B)=D(˜B,˜A);

    (D4)D(˜A,˜C)D(˜A,˜B)+D(˜B,˜C);

    (D5)If˜A˜B˜C, then D(˜A,˜C)max{D(˜A,˜B),D(˜B,˜C)}.

    Consider ˜A,˜BIVIFS(X), where ˜A={x,[μL˜A(x),μU˜A(x)],[νL˜A(x),νU˜A(x)]|xX}, ˜B={x,[μL˜B(x),μU˜B(x)],[νL˜B(x),νU˜B(x)]|xX}.

    For convenience, we denote:

    |μL˜A(xi)μL˜B(xi)|=αi,|μU˜A(xi)μU˜B(xi)|=βi,|νL˜A(xi)νL˜B(xi)|=γi,|νU˜A(xi)νU˜B(xi)|=δi.

    For arbitrary p, the distance measure between ˜A and ˜B is presented in the following form:

    Dp(˜A,˜B)=(1qnni=1(αpiβpiγpiδpi))1p, (2.1)

    where the parameter q is determined by the two operations "" and "".

    When p assumes different values, various distance measures can be obtained, as demonstrated in Table 1.

    Table 1.  Different distance measures.
    =+
    =+
    Minkovsky q=4 p=1 DH(˜A,˜B)=(14nni=1(αi+βi+γi+δi))
    p=2 DE(˜A,˜B)=(14nni=1(α2i+β2i+γ2i+δ2i))12
    p=3 DM(˜A,˜B)=(14nni=1(α3i+β3i+γ3i+δ3i))13
    =,=+

    =,=
    p=1 q=2 Hamming-Hausdorff DHH(˜A,˜B)=(12nni=1(αiβi+γiδi))
    q=1 Hausdorff Dh(˜A,˜B)=(1nni=1(αiβiγiδi))

     | Show Table
    DownLoad: CSV

    Definition 2.4. ( [28]) Let ˜A,˜BIVIFS(X), and a mapping K:IVIFS(X)[0,1] is called a knowledge measure on IVIFS(X) if K satisfies the following conditions:

    (K1)K(˜A)=1 if ˜AP(X);

    (K2)K(˜A)=0 if π˜A(x)=[1,1] for any xX;

    (K3)K(˜A)K(˜B) if ˜A is less fuzzy than ˜B, i.e., ˜A˜B for μL˜B(x)νL˜B(x) and μU˜B(x)νU˜B(x) or ˜A˜B for μL˜B(x)νL˜B(x) and μU˜B(x)νU˜B(x), for any xX;

    (K4)K(˜A)=K(˜Ac).

    Definition 2.5. ( [18]) Assume that the entropy is a function E:IVIFS(X)[0,1], which satisfies the nether four conditions:

    (E1)E(˜A)=0 if ˜AP(X);

    (E2)E(˜A)=1 if μ˜A(x)=ν˜A(x) for any xX;

    (E3)E(˜A)E(˜B) if ˜A is less fuzzy than ˜B, i.e., μ˜A(x)μ˜B(x) and ν˜A(x)ν˜B(x) for μ˜B(x)ν˜B(x) or μ˜A(x)μ˜B(x) and ν˜A(x)ν˜B(x) for μ˜B(x)ν˜B(x);

    (E4)E(˜A)=E(˜Ac).

    Furthermore, this paper introduces the definition of the closest crisp set within the realm of IVIFSs and derives several equations pertaining to similarity measure, knowledge measure, and entropy, all grounded in the distance measure. Additionally, it delves into the methodology for converting between these measures.

    The extent of resemblance between two comparable entities is frequently characterized or elucidated through the utilization of similarity measure. Within this segment, we primarily introduce a novel formulation of similarity measure pertaining to IVIFSs, which relies on distance measure and the proximate crisp set denoted as C˜A.

    To this end, inspired by the concept of the closest crisp set considering intuitionistic fuzzy sets in [17], the notion of closest crisp set for IVIFSs is introduced.

    Definition 3.1. Let ˜AIVIFS(X), and the closest crisp set to ˜A, which is denoted by C˜A, satisfies the following condition: If μ˜A(x)ν˜A(x), then xC˜A; if μ˜A(x)ν˜A(x), then xC˜A.

    Since any crisp set is an IVIFS with zero interval hesitation index, the closest crisp set of ˜A can be expressed as:

    μLC˜A(x)=μUC˜A(x)={1ifμL˜A(x)νL˜A(x),μU˜A(x)νU˜A(x),0otherwise, (3.1)
    νLC˜A(x)=νUC˜A(x)={0ifμL˜A(x)νL˜A(x),μU˜A(x)νU˜A(x),1otherwise. (3.2)

    From Figure 2, it can be seen that in the left panel, μL˜A(x)>νL˜A(x),μU˜A(x)>νU˜A(x), so that there is xC˜A, or μLC˜A(x)=μUC˜A(x)=1, νLC˜A(x)=νUC˜A(x)=0. The opposite is true for the right panel, μL˜A(x)<νL˜A(x),μU˜A(x)<νU˜A(x); thus, there is xC˜A, or μLC˜A(x)=μUC˜A(x)=0, νLC˜A(x)=νUC˜A(x)=1.

    Figure 2.  Examples of the closest crisp sets to IVIFSs ˜A and ˜B.

    Next, a corollary is given below that presents practical properties of the closest crisp set.

    Corollary 3.2. Assume that ˜AIVIFS(X), and let C˜A be its closest crisp set. It has the following properties:

    (1) ˜AP(X) if, and only if, ˜A=C˜A.

    (2) If μL˜A(x)νL˜A(x),μU˜A(x)νU˜A(x), then Cc˜A(x)=C˜Ac(x) for any xX, where C˜Ac expresses the closest crisp set to Ac.

    The preceding Corollary 3.2(2) can be elucidated more intuitively by scrutinizing Figure 3, which illustrates that the rectangular positions designated by ˜A(x) and ~Ac(x) are precisely as depicted in the image when the conditions μL˜A(x)νL˜A(x) and μU˜A(x)νU˜A(x) are satisfied. Furthermore, it becomes evident that C˜A(x) and C˜Ac(x) occupy opposing corners, thereby establishing the equality Cc˜A(x)=C˜Ac(x). Note here that the conditional requirement of Corollary 3.2(2) is μL˜A(x)νL˜A(x),μU˜A(x)νU˜A(x), the reason for which is that if μL˜A(x)=νL˜A(x),μU˜A(x)=νU˜A(x), as shown in Figure 4, then ˜A(x) and ˜Ac(x) are coincident and their closest crisp sets are the same.

    Figure 3.  Graphical representation of Corollary 3.2(2).
    Figure 4.  Graphical representation of μL˜A(x)=νL˜A(x),μU˜A(x)=νU˜A(x).

    Based on the proposed the closest crisp set, information measure functions are explored.

    Definition 3.3. A function f:[0,1]2[0,1] is called a binary aggregation function, for any x,y[0,1], it satisfies the following conditions:

    (1) f is component-wise increasing;

    (2) f(0,0)=0;

    (3) f(1,1)=1;

    (4) f(x,y)=f(y,x).

    Definition 3.4. ( [9]) Let ˜A,˜BIVIFS(X), and a mapping S: IVIFS(X)×IVIFS(X)[0,1] is called a similarity measure on IVIFSs if S satisfies the following properties:

    (S1) Boundary: 0S(˜A,˜B)1;

    (S2) Symmetry: S(˜A,˜B)=S(˜B,˜A);

    (S3) Reflexivity: S(˜A,˜B)=1 iff ˜A=˜B;

    (S4) Complementarity: S(˜A,˜Ac)=0 iff ˜AP(X).

    Theorem 3.5. Let f:[0,1]×[0,1][0,1] be a binary aggregation function and D be an aforementioned distance in Definition 2.3 for IVIFSs. Then, for any ˜A,˜BIVIFS(X),

    S(˜A,˜B)=f(D(˜A,C˜A),D(˜B,C˜B))

    is a similarity measure for IVIFSs.

    Proof. In the following, we will prove the proposed formula satisfies four conditions in Definition 3.4.

    (S1): It is clearly established.

    (S2): By the definition of function f, it can be clearly deduced that

    S(˜A,˜B)=f(D(˜A,C˜A),D(˜B,C˜B))=f(D(˜B,C˜B),D(˜A,C˜A))=S(˜B,˜A).

    (S3): If S(˜A,˜B)=1, according to the definition of f and the distance measure, we can get

    f(D(˜A,C˜A),D(˜B,C˜B))=1D(˜A,C˜A)=D(˜B,C˜B)=1,

    thus, ˜A=˜B.

    (S4): If S(˜A,˜Ac)=0, based on the definition of f and the distance measure,

    f(D(˜A,C˜A),D(˜Ac,C˜Ac))=0D(˜A,C˜A)=D(˜Ac,C˜Ac)=0

    can be deduced. Then, ˜AP(X).

    With regards to Theorem 3.5, it is feasible to devise diverse formulations for calculating similarity measures on IVIFSs, leveraging various binary aggregation functions f:[0,1]×[0,1][0,1]. By incorporating distinct functions f and distance measures D, a range of similarity measures can be derived, as exemplified in Table 2.

    Table 2.  Different similarity measures.
    Distance measures Conversion functions Similarity measures
    DH(˜A,˜B) f1(x,y)=x+y2 SH1(˜A,˜B)=DH(˜A,C˜A)+DH(˜B,C˜B)2
    f2(x,y)=sin(x+y)π4 SH2(˜A,˜B)=sin(DH(˜A,C˜A)+DH(˜B,C˜B))π4
    f3(x,y)=kx+y2(k>1) SH3(˜A,˜B)=kDH(˜A,C˜A)+DH(˜B,C˜B)2(k>1)
    f4(x,y)=logx+y+13 SH4(˜A,˜B)=log(DH(˜A,C˜A)+DH(˜B,C˜B))3
    f5(x,y)=2xy1 SH5(˜A,˜B)=2DH(˜A,C˜A)DH(˜B,C˜B)1
    DE(˜A,˜B) f1(x,y)=x+y2 SE1(˜A,˜B)=DE(˜A,C˜A)+DE(˜B,C˜B)2
    f2(x,y)=sin(x+y)π4 SE2(˜A,˜B)=sin(DE(˜A,C˜A)+DE(˜B,C˜B))π4
    f3(x,y)=kx+y2(k>1) SE3(˜A,˜B)=kDE(˜A,C˜A)+DE(˜B,C˜B)2(k>1)
    f4(x,y)=logx+y+13 SE4(˜A,˜B)=log(DE(˜A,C˜A)+DE(˜B,C˜B))3
    f5(x,y)=2xy1 SE5(˜A,˜B)=2DE(˜A,C˜A)DE(˜B,C˜B)1
    DM(˜A,˜B) f1(x,y)=x+y2 SM1(˜A,˜B)=DM(˜A,C˜A)+DM(˜B,C˜B)2
    f2(x,y)=sin(x+y)π4 SM2(˜A,˜B)=sin(DM(˜A,C˜A)+DM(˜B,C˜B))π4
    f3(x,y)=kx+y2(k>1) SM3(˜A,˜B)=kDM(˜A,C˜A)+DM(˜B,C˜B)2(k>1)
    f4(x,y)=logx+y+13 SM4(˜A,˜B)=log(DM(˜A,C˜A)+DM(˜B,C˜B))3
    f5(x,y)=2xy1 SM5(˜A,˜B)=2DM(˜A,C˜A)DM(˜B,C˜B)1
    DHH(˜A,˜B) f1(x,y)=x+y2 SHH1(˜A,˜B)=DHH(˜A,C˜A)+DHH(˜B,C˜B)2
    f2(x,y)=sin(x+y)π4 SHH2(˜A,˜B)=sin(DHH(˜A,C˜A)+DHH(˜B,C˜B))π4
    f3(x,y)=kx+y2(k>1) SHH3(˜A,˜B)=kDHH(˜A,C˜A)+DHH(˜B,C˜B)2(k>1)
    f4(x,y)=logx+y+13 SHH4(˜A,˜B)=log(DHH(˜A,C˜A)+DHH(˜B,C˜B))3
    f5(x,y)=2xy1 SHH5(˜A,˜B)=2DHH(˜A,C˜A)DHH(˜B,C˜B)1
    Dh(˜A,˜B) f1(x,y)=x+y2 Sh1(˜A,˜B)=Dh(˜A,C˜A)+Dh(˜B,C˜B)2
    f2(x,y)=sin(x+y)π4 Sh2(˜A,˜B)=sin(Dh(˜A,C˜A)+Dh(˜B,C˜B))π4
    f3(x,y)=kx+y2(k>1) Sh3(˜A,˜B)=kDh(˜A,C˜A)+Dh(˜B,C˜B)2(k>1)
    f4(x,y)=logx+y+13 Sh4(˜A,˜B)=log(Dh(˜A,C˜A)+Dh(˜B,C˜B))3
    f5(x,y)=2xy1 Sh5(˜A,˜B)=2Dh(˜A,C˜A)Dh(˜B,C˜B)1

     | Show Table
    DownLoad: CSV

    The knowledge content of IVIFSs is typically characterized through knowledge measures, with entropy serving as a pivotal quantifying the uncertainty associated with fuzzy variables. In this section, we established an approach of developing similarity measures to devise knowledge measures and entropy for IVIFSs.

    Theorem 3.6. Let f be a function defined in Definition 3.3 and D be an aforementioned distance in Definition 2.3 for IVIFSs. For any ˜AIVIFS(X),

    K(˜A)=1f(D(˜A,C˜A),D(~Ac,C˜Ac))

    is a knowledge measure on IVIFS(X).

    Proof. In the following we will show that the knowledge measure we construct satisfies the four conditions in Definition 2.4.

    (KD1): Let K(˜A)=1, then consider the definition of the function f, that is, D(˜A,C˜A)=D(~Ac,C˜Ac)=0. Therefore, by Definition 2.3 and Corollary 3.2(1), it is easily obtained that ˜AP(X).

    (KD2): It is not difficult to see that if K(˜A)=0, then D(˜A,C˜A)=D(~Ac,C˜Ac)=1, and according to the definition of the distance D,

    μL˜A(x)=μU˜A(x)=νL˜A(x)=νU˜A(x)=0,

    can be obtained. Thus, π˜A(x)=[1,1].

    (KD3): Assume that ˜A˜B for μL˜B(x)νL˜B(x) and μU˜B(x)νU˜B(x). It is certain that

    μL˜A(x)μL˜B(x)νL˜B(x)νL˜A(x),μU˜A(x)μU˜B(x)νU˜B(x)νU˜A(x),

    and from the notion of the closest crisp set, it can be derived that

    μLC˜A(x)=μUC˜A(x)=μLC˜B(x)=μUC˜B(x)=0,
    νLC˜A(x)=νUC˜A(x)=νLC˜B(x)=νUC˜B(x)=1.

    Therefore, let ˜N=<[0,0],[1,1]>, and if ˜N˜A˜B, then

    D(˜A,˜N)=D([μL˜A(x),μU˜A(x)],[νL˜A(x),νU˜A(x)],[0,0],[1,1])D(˜B,˜N)=D([μL˜B(x),μU˜B(x)],[νL˜B(x),νU˜B(x)],[0,0],[1,1]).

    Similarly, it is obvious that D(~Ac,C˜Ac)D(~Bc,C˜Bc). Then, by considering the function f to be component-wise increasing, there is

    f(D(˜A,C˜A),D(~Ac,C˜Ac))f(D(˜B,C˜B),D(~Bc,C˜Bc)).

    Hence, it is easy to know that K(˜A)K(˜B). On the other hand, assume that ˜A˜B for μL˜B(x)νL˜B(x) and μU˜B(x)νU˜B(x). K(˜A)K(˜B) can also be obtained.

    (KD4): In order to verify K(˜A)=K(˜Ac), simply prove that

    f(D(˜A,C˜A),D(~Ac,C˜Ac))=f(D(~Ac,C˜Ac),D(˜A,C˜A)),

    which is obvious from the definition of the function f.

    Next, a series of knowledge measures can be obtained by using different distance measures and different expressions of function f. They are shown in Table 3.

    Table 3.  Different knowledge measures.
    Distance measures Conversion functions Knowledge measures
    DH(˜A,˜B) f1(x,y)=x+y2 KH1(˜A)=1DH(˜A,C˜A)+DH(˜Ac,C˜Ac)2
    f2(x,y)=sin(x+y)π4 KH2(˜A)=1sin(DH(˜A,C˜A)+DH(˜Ac,C˜Ac))π4
    f3(x,y)=kx+y2(k>1) KH3(˜A)=1kDH(˜A,C˜A)+DH(˜Ac,C˜Ac)2(k>1)
    f4(x,y)=logx+y+13 KH4(˜A)=1log(DH(˜A,C˜A)+DH(˜Ac,C˜Ac))3
    f5(x,y)=2xy1 KH5(˜A)=22DH(˜A,C˜A)DH(˜Ac,C˜Ac)
    DE(˜A,˜B) f1(x,y)=x+y2 KE1(˜A)=1DE(˜A,C˜A)+DE(˜Ac,C˜Ac)2
    f2(x,y)=sin(x+y)π4 KE2(˜A)=1sin(DE(˜A,C˜A)+DE(˜Ac,C˜Ac))π4
    f3(x,y)=kx+y2(k>1) KE3(˜A)=1kDE(˜A,C˜A)+DE(˜Ac,C˜Ac)2(k>1)
    f4(x,y)=logx+y+13 KE4(˜A)=1log(DE(˜A,C˜A)+DE(˜Ac,C˜Ac))3
    f5(x,y)=2xy1 KE5(˜A)=22DE(˜A,C˜A)DE(˜Ac,C˜Ac)
    DM(˜A,˜B) f1(x,y)=x+y2 KM1(˜A)=1DM(˜A,C˜A)+DM(˜Ac,C˜Ac)2
    f2(x,y)=sin(x+y)π4 KM2(˜A)=1sin(DM(˜A,C˜A)+DM(˜Ac,C˜Ac))π4
    f3(x,y)=kx+y2(k>1) KM3(˜A)=1kDM(˜A,C˜A)+DM(˜Ac,C˜Ac)2(k>1)
    f4(x,y)=logx+y+13 KM4(˜A)=1log(DM(˜A,C˜A)+DM(˜Ac,C˜Ac))3
    f5(x,y)=2xy1 KM5(˜A)=22DM(˜A,C˜A)DM(˜Ac,C˜Ac)
    DHH(˜A,˜B) f1(x,y)=x+y2 KHH1(˜A)=1DHH(˜A,C˜A)+DHH(˜Ac,C˜Ac)2
    f2(x,y)=sin(x+y)π4 KHH2(˜A,˜B)=sin(DHH(˜A,C˜A)+DHH(˜B,C˜B))π4
    f3(x,y)=kx+y2(k>1) KHH3(˜A,˜B)=kDHH(˜A,C˜A)+DHH(˜B,C˜B)2(k>1)
    f4(x,y)=logx+y+13 KHH4(˜A)=1log(DHH(˜A,C˜A)+DHH(˜Ac,C˜Ac))3
    f5(x,y)=2xy1 KHH5(˜A)=22DHH(˜A,C˜A)DHH(˜Ac,C˜Ac)
    Dh(˜A,˜B) f1(x,y)=x+y2 Kh1(˜A)=1Dh(˜A,C˜A)+Dh(˜Ac,C˜Ac)2
    f2(x,y)=sin(x+y)π4 Kh2(˜A)=1sin(Dh(˜A,C˜A)+Dh(˜Ac,C˜Ac))π4
    f3(x,y)=kx+y2(k>1) Kh3(˜A)=1kDh(˜A,C˜A)+Dh(˜Ac,C˜Ac)2(k>1)
    f4(x,y)=logx+y+13 Kh4(˜A)=1log(Dh(˜A,C˜A)+Dh(˜Ac,C˜Ac))3
    f5(x,y)=2xy1 Kh5(˜A)=22Dh(˜A,C˜A)Dh(˜Ac,C˜Ac)

     | Show Table
    DownLoad: CSV

    Subsequently, entropy from the perspective of knowledge measures for IVIFSs will be explored.

    Corollary 3.7. Let K be a knowledge measure induced by the aforementioned function f for IVIFSs, then for any ˜AIVIFS(X),

    E(˜A)=1K(˜A)

    is an entropy for IVIFSs.

    Based on Corollary 3.7, it is straightforward to deduce the existence of a distinct numerical correlation between entropy and the newly devised knowledge metric. Put simply, within the context of an IVIFS, a system with a lower entropy value may consistently possess a greater quantity of knowledge, provided that it adheres to the prescribed axioms.

    By carefully examining the intricate relationship between the recently introduced knowledge measure and entropy, we can effortlessly deduce certain entropies.

    Upon scrutinizing the information measures previously discussed for IVIFSs, it becomes evident that they share certain linkages, which we shall subsequently elaborate in the form of corollaries.

    Corollary 3.8. Let S be an aforementioned similarity measure in Theorem 3.5 for IVIFSs, and for any ˜AIVIFS(X),

    K(˜A)=1S(˜A,~Ac)

    is a knowledge measure for IVIFSs.

    Corollary 3.9. Let S be an aforementioned similarity measure in Theorem 3.5 for IVIFSs; assuming that ˜AIVIFS(X),

    E(˜A)=S(˜A,~Ac)

    is an entropy for IVIFSs.

    Regarding the construction method and transformation relationship of information measures, there are some important issues to be further discussed below, and these discussions pave the way for the subsequent research.

    (1) Drawing upon the axiomatic definition of IVIFSs, the intricate relationship among similarity measures, knowledge measures, and the entropy of IVIFSs is comprehensively formulated in general terms, accompanied by explicit expressions. Notably, several key findings are presented in the form of theorems, which underscore the existence of a profound connection between the information measures associated with IVIFSs.

    (2) Among these theorems, through the axiomatic definition of IVIFS, we can easily convert the above information measures, i.e., similarity measure, knowledge measure, and entropy, to each other. Specifically, we consider that the similarity measure and knowledge measure of IVIFS can be converted to each other, and, finally, based on the general relationship between knowledge measure and entropy, the entropy is constructed with the similarity measure, so that the larger the similarity measure, the less the amount of knowledge it contains, and the larger the entropy.

    (3) The result of Corollary 3.7 in this paper shows that there is a dyadic relationship between entropy and knowledge measure, which paves the way for continuing to explore whether there is a dyadic relationship between entropy and knowledge measure for fuzzy sets and their extended forms.

    This section employs the recently proposed knowledge measure to tackle multi-attribute decision-making (MADM) problems under conditions of unknown weights, while utilizing the novel similarity measure to address pattern recognition issues within the framework of IVIFSs. The derived outcomes are subsequently contrasted with several established measures for comprehensive analysis.

    In the complex process of MADM, where the weights of individual attributes remain elusive, this subsection introduces a methodology for determining these weights through the utilization of knowledge measures techniques.

    Construct an MADM process that examines m alternatives ˜A={˜A1,˜A2,,˜Am} using n attributes C={C1,C2,,Cn}. Suppose that w=(w1,w2,,wn)T is the weight vector of attributes, where 0wj1 and nj=1wj=1,j=1,2,,n.

    The MADM problem of IVIFSs can be described in the matrix form below:

    R=˜A1˜A2˜AmC1  C2  Cn(˜a11˜a12˜a1n˜a21˜a22˜a2n˜am1˜am2˜amn), (4.1)

    where ˜aij=(x,[μij˜aL(x),μij˜aU(x)],[νij˜aL(x),νij˜aU(x)])|xX(i=1,2,,m;j=1,2,,n) indicates the assessment of an expert corresponding to attribute Cj(j=1,2,,n) to evaluate the alternative ˜Ai(i=1,2,,m).

    Step 1. By defining the knowledge measure of IVIFSs using KH1, the knowledge measure based decision matrix Q=(˜kij)m×n is generated from the decision matrix R above. The matrix Q=(˜kij)m×n is shown below.

    Q=˜A1˜A2˜AmC1  C2  Cn(˜k11˜k12˜k1n˜k21˜k22˜k2n˜km1˜km2˜kmn), (4.2)

    where ˜kij is the knowledge measure corresponding to each ˜aij.

    Step 2. The following formula can be used to construct the attribute weights based on the knowledge measure when the attribute weights wj are completely unknown:

    wj=mi=1˜kijnj=1mi=1˜kij, (4.3)

    where 0wj1 and nj=1wj=1.

    Step 3. In an interval-valued intuitionistic fuzzy environment, the interval-valued intuitionistic fuzzy arithmetic mean operator [3] is used to calculate the weighted aggregation value of each alternative ˜Ai after the weights of each attribute have been determined and the operator is expressed as follows:

    Zi=[μLi,μUi],[νLi,νUi]=[1nj=1(1μij˜aL(x))wj,1nj=1(1μij˜aU(x))wj],[nj=1(νij˜aL(x))wj,nj=1(νij˜aU(x))wj], (4.4)

    where i=1,2,,m;j=1,2,,n. The calculated Zi is an IVIFS.

    Step 4. In order to differentiate between the alternatives, the alternatives will be ranked below by calculating a score function for each alternative, with the following equation:

    ϕ(Zi)=λ(μLi+μUi)(1λ)(νLi+νUi),i=1,2,...,m,λ(0,1). (4.5)

    Subsequently, arrange the alternatives in a nonincreasing sequence of score function, the larger the score function of the alternative, the better the alternative is. In order to realize the whole decision-making process, Figure 5 describes a framework.

    Figure 5.  A methodological framework for solving MADM problems with unknown weights based on knowledge measures.

    In this section, a practical case study, adapted from [5], will be presented to demonstrate the application of the newly introduced knowledge measure in addressing MADM problems involving unknown weights.

    An investment company is about to make an investment, and the four possible choices for the investment capital are (1) ˜A1, a car company; (2) ˜A2, a food company; (3) ˜A3, a computer company; and (4) ˜A4, a branch office. The investment firm is trying to decide which of these is the best option. The following four attributes must also be taken into consideration by the investment company when making its choice: C1: Risk analysis; C2: Growth analysis; C3: Environmental impact analysis; and C4: Social and political impact analysis.

    Let w=(w1,w2,w3,w4)T be the weight vector of the attributes, which is completely unknown. Additionally, the ratings of the alternatives over the attributes are listed in the following Table 4.

    Table 4.  Rating of alternatives in terms of attributes.
    C1 C2 C3 C4
    ˜A1 [0.1,0.2],[0.1,0.2] [0.25,0.5],[0.25,0.5] [0.4,0.5],[0.3,0.5] [0.5,0.5],[0.5,0.5]
    ˜A2 [0.5,0.6],[0.2,0.3] [0.2,0.5],[0.2,0.5] [0.0,0.0],[0.25,0.75] [0.3,0.4],[0.4,0.6]
    ˜A3 [0.25,0.5],[0.25,0.5] [0.2,0.4],[0.2,0.4] [0.2,0.3],[0.4,0.7] [0.2,0.3],[0.5,0.6]
    ˜A4 [0.2,0.3],[0.6,0.7] [0.4,0.7],[0.2,0.3] [0.2,0.5],[0.2,0.5] [0.5,0.7],[0.1,0.3]

     | Show Table
    DownLoad: CSV

    Step 1. Based on KH1, the knowledge measure matrix is calculated as follows:

    Q=˜A1˜A2˜A3˜A4C1   C2   C3   C4(0.50.50.4750.50.650.50.750.5750.50.50.650.650.70.650.50.7). (4.6)

    Step 2. Next, the weight of attributes are calculated by applying Eq (4.3):

    w1=0.2527,w2=0.2312,w3=0.2554,w4=0.2608.

    Step 3. The interval-valued intuitionistic fuzzy weighted average operator can be computed by utilizing the formula presented in Eq (4.4). As a result, the combined results of the schemes are obtained as follows:

    Z1=[0.3326,0.4370],[0.2486,0.3966];Z2=[0.2737,0.4085],[0.2536,0.5111];
    Z3=[0.2129,0.3796],[0.3207,0.5426];Z4=[0.3379,0.5766],[0.2203,0.4234].

    Step 4. The scoring values of the alternatives are calculated with Eq (4.5) when λ=12 as follows:

    ϕ(Z1)=1.1922,ϕ(Z2)=0.8921,ϕ(Z3)=0.6863,ϕ(Z4)=1.4207.

    Therefore, the sorting order is ˜A4˜A1˜A2˜A3, and the optimal alternative is ˜A4. Furthermore, to underscore the practicality of the newly introduced knowledge measure in addressing the MADM problem, we contrast the derived outcomes with diverse entropy and knowledge measures. Evidently, the majority of the ranking outcomes closely align with our findings, as evident from the subsequent Table 5 and Figure 6. Notably, ˜A4 emerges as the optimal solution.

    Table 5.  Comparison results of methods.
    Methods Weighting vector Alternatives ranking
    EJPCZ(E1) [29] w=(0.2274,0.2860,0.2370,0.2496)T ˜A4˜A1˜A2˜A3
    ELZX(E2) [30] w=(0.2274,0.2860,0.2370,0.2496)T ˜A4˜A1˜A2˜A3
    EWWZ(E3) [31] w=(0.2274,0.2860,0.2370,0.2496)T \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    E_{ZJJL}(E4) [32] w={(0.2366, 0.3159, 0.1859, 0.2616)^{T}} \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    Das et al. [27] w={(0.2506, 0.2211, 0.2472, 0.2811)^{T}} \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    Proposed method w={(0.2527, 0.2312, 0.2554, 0.2608)^{T}} \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3

     | Show Table
    DownLoad: CSV
    Figure 6.  Comparison results of methods.

    The aforementioned examples demonstrate the remarkable effectiveness of applying knowledge measures to tackle MADM problems that involve unknown weights. Additionally, they underscore the usefulness and practical applicability of the newly introduced knowledge measures in our daily lives.

    It is evident that the parameter \lambda in Eq (4.5) can be interpreted as the weight of membership degree, and its magnitude significantly impacts the ultimate ranking of alternatives. Sensitivity analysis is performed by selecting various \lambda values. Table 6 presents the score function values \phi(Z_{i}) and the rankings of each alternative, which are calculated based on different \lambda values.

    Table 6.  The score function value \phi(Z_{i}) and alternatives order.
    \phi(Z_{1}) \phi(Z_{2}) \phi(Z_{3}) \phi(Z_{4}) Alternatives order
    \lambda=0.1 0.1325 0.0991 0.0763 0.1579 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.2 0.2980 0.2230 0.1716 0.3552 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.3 0.5109 0.3823 0.2941 0.6089 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.4 0.7948 0.5947 0.4576 0.9472 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.5 1.1922 0.8920 0.6863 1.4207 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.6 1.7882 1.3380 1.0295 2.1311 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.7 2.7817 2.0813 1.6015 3.3150 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.8 4.7686 3.5680 2.7454 5.6829 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3
    \lambda=0.9 10.7295 8.0280 6.1771 12.7865 \tilde{A}_4\succ \tilde{A}_1\succ \tilde{A}_2\succ \tilde{A}_3

     | Show Table
    DownLoad: CSV

    We randomly select 9 values for \lambda within the interval (0, 1) , and it is evident that regardless of how \lambda varies, the ranking of alternatives remains consistent throughout Table 6. Additionally, we will visualize the data in Figure 7, which illustrates that the ultimate ranking remains unaltered despite changes in \lambda . As evident from Table 6, when \lambda > 0.5 , the score function values of the alternative solutions exhibit a substantial increase, primarily due to a greater preference for membership degree under such conditions.

    Figure 7.  The change of the score function of the alternatives under different \lambda values.

    To clearly demonstrate the disparities among alternative solutions as \lambda escalates, consider the following definitions: T_{1} = \phi(Z_{4})-\phi(Z_{1}), T_{2} = \phi(Z_{1})-\phi(Z_{2}), T_{3} = \phi(Z_{2})-\phi(Z_{3}). It can be found that with the increase of \lambda , the discrimination between the alternatives increases in Table 7 and Figure 8. That is to say, with the increase of \lambda , the score function \phi(Z_{i}) can better distinguish the alternatives. Therefore, the ranking results should be minimally influenced by subjective factors, which further demonstrates the stability of the method proposed in this paper.

    Table 7.  Discrimination between alternatives.
    T_{1} T_{2} T_{3}
    \lambda=0.1 0.0254 0.0334 0.0229
    \lambda=0.2 0.0571 0.0750 0.0514
    \lambda=0.3 0.0980 0.1286 0.0881
    \lambda=0.4 0.1524 0.2001 0.1371
    \lambda=0.5 0.2286 0.3002 0.2057
    \lambda=0.6 0.3428 0.4502 0.3085
    \lambda=0.7 0.5333 0.7004 0.4799
    \lambda=0.8 0.9143 1.2007 0.8226
    \lambda=0.9 2.0571 2.7015 1.8509

     | Show Table
    DownLoad: CSV
    Figure 8.  Discrimination between alternatives under different \lambda values.

    Novel coronavirus pneumonia (COVID-19) is an acute respiratory infectious disease caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Since its outbreak in Pakistan, the COVID-19 situation there has garnered widespread attention. The epidemic's spread in Pakistan has exhibited certain fluctuations, with a gradual rise in confirmed cases, exerting a profound impact on public health, socioeconomic conditions, and the daily lives of citizens.

    This paper refers to the example of healthcare facilities in Khyber Pakhtunkhwa, Pakistan in [33]. There is currently a review panel comprising three experts, e_{1}, e_{2} , and e_{3} , tasked with evaluating four hospitals A_{l} (l = 1, 2, 3, 4) in Khyber Pakhtunkhwa based on four attributes: c_{1} : Reverse Transcription-Polymerase Chain Reaction Facilities (RT-PCR), c_{2} : Personal protective equipment (PPE), c_{3} : Shortage of Isolation Ward (SIW), c_{4} : Hight Qualified Staff (HQS).

    Next, the specific steps of the decision-making method are as follows.

    Step 1. We construct interval valued intuitionistic fuzzy decision matrices, as shown in Tables 810.

    Table 8.  Interval-valued intuitionistic fuzzy decision matrix of e_{1} .
    c_1 c_2 c_3 c_4
    A_1 \langle[0.3, 0.4], [0.1, 0.3]\rangle \langle[0.2, 0.5], [0.2, 0.3]\rangle \langle[0.4, 0.5], [0.3, 0.5]\rangle \langle[0.1, 0.3], [0.3, 0.5]\rangle
    A_2 \langle[0.1, 0.3], [0.3, 0.4]\rangle \langle[0.3, 0.4], [0.1, 0.4]\rangle \langle[0.2, 0.4], [0.1, 0.2]\rangle \langle[0.3, 0.4], [0.1, 0.2]\rangle
    A_3 \langle[0.3, 0.4], [0.1, 0.2]\rangle \langle[0.2, 0.4], [0.1, 0.2]\rangle \langle[0.3, 0.4], [0.1, 0.4]\rangle \langle[0.1, 0.3], [0.3, 0.4]\rangle
    A_4 \langle[0.1, 0.3], [0.3, 0.5]\rangle \langle[0.4, 0.5], [0.3, 0.5]\rangle \langle[0.2, 0.5], [0.2, 0.3]\rangle \langle[0.3, 0.4], [0.1, 0.3]\rangle

     | Show Table
    DownLoad: CSV
    Table 9.  Interval-valued intuitionistic fuzzy decision matrix of e_{2} .
    c_1 c_2 c_3 c_4
    A_1 \langle[0.1, 0.3], [0.3, 0.5]\rangle \langle[0.4, 0.5], [0.3, 0.5]\rangle \langle[0.2, 0.5], [0.2, 0.3]\rangle \langle[0.2, 0.5], [0.2, 0.3]\rangle
    A_2 \langle[0.3, 0.4], [0.1, 0.2]\rangle \langle[0.2, 0.4], [0.1, 0.2]\rangle \langle[0.3, 0.4], [0.1, 0.4]\rangle \langle[0.3, 0.4], [0.1, 0.4]\rangle
    A_3 \langle[0.1, 0.3], [0.3, 0.4]\rangle \langle[0.3, 0.4], [0.1, 0.4]\rangle \langle[0.2, 0.4], [0.1, 0.2]\rangle \langle[0.2, 0.4], [0.1, 0.2]\rangle
    A_4 \langle[0.3, 0.4], [0.1, 0.3]\rangle \langle[0.2, 0.5], [0.2, 0.3]\rangle \langle[0.4, 0.5], [0.3, 0.5]\rangle \langle[0.4, 0.5], [0.3, 0.5]\rangle

     | Show Table
    DownLoad: CSV
    Table 10.  Interval-valued intuitionistic fuzzy decision matrix of e_{3} .
    c_1 c_2 c_3 c_4
    A_1 \langle[0.2, 0.5], [0.2, 0.3]\rangle \langle[0.3, 0.5], [0.2, 0.3]\rangle \langle[0.1, 0.3], [0.3, 0.5]\rangle \langle[0.4, 0.5], [0.3, 0.5]\rangle
    A_2 \langle[0.3, 0.4], [0.1, 0.4]\rangle \langle[0.1, 0.4], [0.2, 0.4]\rangle \langle[0.3, 0.4], [0.1, 0.2]\rangle \langle[0.2, 0.4], [0.1, 0.2]\rangle
    A_3 \langle[0.2, 0.4], [0.1, 0.2]\rangle \langle[0.3, 0.4], [0.1, 0.2]\rangle \langle[0.1, 0.3], [0.3, 0.4]\rangle \langle[0.3, 0.4], [0.1, 0.4]\rangle
    A_4 \langle[0.4, 0.5], [0.3, 0.5]\rangle \langle[0.1, 0.5], [0.2, 0.5]\rangle \langle[0.3, 0.4], [0.1, 0.3]\rangle \langle[0.2, 0.5], [0.2, 0.3]\rangle

     | Show Table
    DownLoad: CSV

    Step 2. By utilizing K_{h1} to define the knowledge measure of IVIFSs, a decision matrix Q^{k} (k = 1, 2, 3) , grounded in this knowledge measure, is derived from the decision matrix. The matrix Q^{k}(k = 1, 2, 3) is presented as follows.

    Q^{1} = \begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {}\\ {A_1}\\ {A_2}\\ {A_3}\\ {A_4} \end{array}}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} c_1 & \ \ c_2& \ \ c_3& \ \ c_4 \end{array}}\\ {\left( {\begin{array}{*{20}{c}} 0.6&0.8&0.6&0.7\\ 0.7&0.7&0.8&0.7\\ 0.7&0.8&0.7&0.7\\ 0.7&0.6&0.8&0.7 \end{array}} \right).} \end{array}} \end{array}
    Q^{2} = \begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {}\\ {A_1}\\ {A_2}\\ {A_3}\\ {A_4} \end{array}}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} c_1 & \ \ c_2& \ \ c_3&\ \ c_4 \end{array}}\\ {\left( {\begin{array}{*{20}{c}} 0.7&0.6&0.8&0.8\\ 0.7&0.8&0.7&0.7\\ 0.7&0.7&0.8&0.8\\ 0.7&0.8&0.6&0.6 \end{array}} \right).} \end{array}} \end{array}
    Q^{3} = \begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {}\\ {A_1}\\ {A_2}\\ {A_3}\\ {A_4} \end{array}}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} c_1 & \ \ c_2& \ \ c_3& \ \ c_4 \end{array}}\\ {\left( {\begin{array}{*{20}{c}} 0.8&0.7&0.7&0.6\\ 0.7&0.8&0.7&0.8\\ 0.8&0.7&0.7&0.7\\ 0.6&0.9&0.7&0.8 \end{array}} \right).} \end{array}} \end{array}

    Step 3. We assign equal weights to the three experts and aggregate their knowledge measure decision matrices into one using the formula below:

    \begin{equation} \bar{Q} = \frac{Q^{1}+Q^{2}+Q^{3}}{3}. \end{equation} (4.7)

    Obtain the following knowledge measure aggregation matrix \bar{Q} :

    \bar{Q} = \begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {}\\ {A_1}\\ {A_2}\\ {A_3}\\ {A_4} \end{array}}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} c_1 &\ \ \ c_2& \ \ \ c_3& \ \ \ c_4 \end{array}}\\ {\left( {\begin{array}{*{20}{c}} 0.7&0.7&0.7&0.7\\ 0.7&0.77&0.73&0.73\\ 0.73&0.73&0.73&0.73\\ 0.67&0.77&0.7&0.7 \end{array}} \right).} \end{array}} \end{array}

    Step 4. Determine attribute weights by applying Eq (4.3).

    w_1 = 0.2437, w_2 = 0.2585, w_3 = 0.2489, w_4 = 0.2489.

    Step 5. Employ Eq (4.4) to apply weights and aggregate the attributes of the alternative (Table 11).

    Table 11.  Weighted aggregation of alternative attributes.
    e_1 e_2 e_3
    z_1 \langle[0.2577, 0.4316], [0.2067, 0.3869]\rangle \langle[0.2357, 0.4573], [0.2352, 0.3877]\rangle \langle[0.2591, 0.4563], [0.2447, 0.3869]\rangle
    z_2 \langle[0.2306, 0.3770], [0.1307, 0.2833]\rangle \langle[0.2754, 0.4000], [0.1000, 0.2824]\rangle \langle[0.2278, 0.4000], [0.1196, 0.2833]\rangle
    z_3 \langle[0.2286, 0.3765], [0.1314, 0.2824]\rangle \langle[0.2046, 0.3770], [0.1307, 0.2833]\rangle \langle[0.2302, 0.3765], [0.1314, 0.2824]\rangle
    z_4 \langle[0.2607, 0.4321], [0.2063, 0.3877]\rangle \langle[0.3289, 0.4773], [0.2067, 0.3869]\rangle \langle[0.2562, 0.4768], [0.1858, 0.3877]\rangle

     | Show Table
    DownLoad: CSV

    Step 6. When \lambda = \frac{1}{2} , the score values by each expert to the alternative are calculated using Eq (4.5) as shown in Table 12.

    Table 12.  The score values by each expert.
    \varphi_1 \varphi_2 \varphi_3 \varphi_4
    e_1 1.1613 1.4679 1.4623 1.1662
    e_2 1.0949 1.7662 1.4051 1.3583
    e_3 1.1328 1.5581 1.4660 1.2781

     | Show Table
    DownLoad: CSV

    Step 7. Aggregate the score values of three experts to obtain the final ranking of alternative solutions.

    \bar{\varphi}_{1} = 1.1297, \bar{\varphi}_{2} = 1.5974, \bar{\varphi}_{3} = 1.4445, \bar{\varphi}_{4} = 1.2675.

    It is not hard to observe that \bar{\varphi}_{2} > \bar{\varphi}_{3} > \bar{\varphi}_{4} > \bar{\varphi}_{1} . Thus, yielding the alternative ranking result: A_2\succ A_3\succ A_4\succ A_1 . This is consistent with the result in [33], which also demonstrates the effectiveness and practicality of the method proposed in this paper.

    Suppose X is a finite universe of discourse. There are m patterns which are denoted by IVIFSs

    \begin{equation*} \begin{split} \tilde{M}_{j} = &\{\langle x_{1}, [\mu^{L}_{\tilde{M}_{j}}(x_{1}), \mu^{U}_{\tilde{M}_{j}}(x_{1})], [\nu^{L}_{\tilde{M}_{j}}(x_{1}), \nu^{U}_{\tilde{M}_{j}}(x_{1})]\rangle, ..., \\ &\langle x_{n}, [\mu^{L}_{\tilde{M}_{j}}(x_{n}), \mu^{U}_{\tilde{M}_{j}}(x_{n})], [\nu^{L}_{\tilde{M}_{j}}(x_{n}), \nu^{U}_{\tilde{M}_{j}}(x_{n})]\rangle\; |\; x\in X\}\; (j = 1, 2, ..., m) \end{split} \end{equation*}

    and a test sample that needs to be categorized exists, which is denoted by IVIFS

    \begin{equation*} \begin{split} \tilde{B} = &\{\langle x_{1}, [\mu^{L}_{\tilde{B}}(x_{1}), \mu^{U}_{\tilde{B}}(x_{1})], [\nu^{L}_{\tilde{B}}(x_{1}), \nu^{U}_{\tilde{B}}(x_{1})]\rangle, ..., \\ &\langle x_{n}, [\mu^{L}_{\tilde{B}}(x_{n}), \mu^{U}_{\tilde{B}}(x_{n})], [\nu^{L}_{\tilde{B}}(x_{n}), \nu^{U}_{\tilde{B}}(x_{n})]\rangle\; |\; x\in X\}. \end{split} \end{equation*}

    The following is the recognition process:

    (a) Compute the similarity measure S(\tilde{M}_{j}, \tilde{B}) between \tilde{M}_{j}(j = 1, ..., m) and \tilde{B} with S_{H1} and S_{h1} .

    (b) Sort S(\tilde{M}_{j}, \tilde{B}) in ascending order, where a larger value of S(\tilde{M}_{j}, \tilde{B}) indicates a closer proximity of \tilde{B} to category \tilde{M}_{j} .

    A diverse array of scholars have proposed numerous approaches to tackle medical diagnosis from varying perspectives. In this section, we employ pattern recognition techniques to address the medical diagnosis challenge: the collection of symptoms serves as the discourse universe, the patient functions as an unidentified test specimen, and the various diseases correspond to multiple modalities. The objective is to categorize patients into distinct groups based on their respective diseases.

    Suppose X = \{x_{1}\text{(Temperature)}, \; x_{2}\text{(Cough)}, \; x_{3} \text{(Headache)}, \; x_{4}\text{(Stomach }\; \text{pain)}\} is a group of symptoms and \tilde{M} = \{\tilde{M}_{1}\text{(Viral fever)}, \; \tilde{M}_{2}\text{(Typhoid)}, \; \tilde{M}_{3}{(Pneumonia)}, \; \tilde{M}_{4}\text{(Stomach problem)}\} is a group of the diseases. Then, the disease-symptom matrix expressed by IVIFSs is shown in Table 13 and data is derived from [16].

    Table 13.  The disease-symptom matrix.
    x_{1} (Temperature) x_{2} (Cough) x_{3} (Headache) x_{4} (Stomach pain)
    \tilde{M}_{1} \langle[0.8, 0.9], [0.0, 0.1]\rangle \langle[0.7, 0.8], [0.1, 0.2]\rangle \langle[0.5, 0.6], [0.2, 0.3]\rangle \langle[0.6, 0.8], [0.1, 0.2]\rangle
    \tilde{M}_{2} \langle[0.5, 0.6], [0.1, 0.3]\rangle \langle[0.8, 0.9], [0.0, 0.1]\rangle \langle[0.6, 0.8], [0.1, 0.2]\rangle \langle[0.4, 0.6], [0.1, 0.2]\rangle
    \tilde{M}_{3} \langle[0.7, 0.8], [0.1, 0.2]\rangle \langle[0.7, 0.9], [0.0, 0.1]\rangle \langle[0.4, 0.6], [0.2, 0.4]\rangle \langle[0.3, 0.5], [0.2, 0.4]\rangle
    \tilde{M}_{4} \langle[0.8, 0.9], [0.0, 0.1]\rangle \langle[0.7, 0.8], [0.1, 0.2]\rangle \langle[0.7, 0.9], [0.0, 0.1]\rangle \langle[0.8, 0.9], [0.0, 0.1]\rangle

     | Show Table
    DownLoad: CSV

    Let the patient B be expressed as:

    \begin{equation} \begin{split} \notag B = &\{ \langle x_{1}, [0.4, 0.5], [0.1, 0.2]\rangle, \langle x_{2}, [0.7, 0.8], [0.1, 0.2]\rangle, \\ &\langle x_{3}, [0.9, 0.9], [0.0, 0.1]\rangle, \langle x_{4}, [0.3, 0.5], [0.2, 0.4]\rangle\}. \end{split} \end{equation}

    The aim is to categorize patient B into one of diseases \tilde{M}_{1}, \tilde{M}_{2}, \tilde{M}_{3}, \tilde{M}_{4} . We can then obtain the following results on IVIFSs, as shown in Table 14.

    Table 14.  Comparison of results with different similarity measures.
    S(\tilde{M}_{1}, B) S(\tilde{M}_{2}, B) S(\tilde{M}_{3}, B) S(\tilde{M}_{4}, B) Recognition Result
    S_{1} [11] 0.73 0.80 0.78 0.73 \tilde{M}_{2}
    S_{D} [6] 0.82 0.91 0.86 0.84 \tilde{M}_{2}
    S_{H1} 0.32 0.36 0.33 0.28 \tilde{M}_{2}
    S_{h1} 0.38 0.45 0.43 0.34 \tilde{M}_{2}

     | Show Table
    DownLoad: CSV

    We may determine that there is the most similarity between \tilde{M}_{2} and B by taking into account the pattern recognition principle of IVIFSs, but the similarity measure S_{1} is unable to determine which of \tilde{M}_{1} and \tilde{M}_{4} is greater. Furthermore, the newly suggested similarity measures are more practical and capable of resolving this point. In addition, by using the similarity measures S_{H1} and S_{h1} proposed in this paper, the same recognition results can be obtained. Hence, based on the recognition principle, we can classify patient B as suffering from disease \tilde{M}_{2} , thereby conclusively diagnosing them with typhoid illness.

    This paper introduces the axiomatic definition of the closest crisp set for IVIFSs, along with its pertinent properties. Subsequently, leveraging the distance measure to the closest crisp set, a novel knowledge measure and entropy for IVIFSs are presented. Furthermore, several theorems are proven, and the interplay between knowledge measures and entropy is examined. Ultimately, the efficacy of the proposed method is validated through two concrete examples: The application of knowledge measures in MADM and similarity measures in pattern recognition.

    In the future, we will persist in exploring the application of IVIFS information measures in other domains and strive to develop superior and more rational models for the mutual conversion of these information measures.

    Chunfeng Suo: Writing-review & editing, supervision, project administration, funding acquisition; Le Fu: Conceptualization, Methodology, software, validation, writing-original draft, writing-review & editing; Jingxuan Chen: Software, validation, supervision, conceptualization; Xuanchen Li: Validation, writing-review. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This paper was supported by National Science Foundation of China (Grant Nos: 11671244, 12071271), Natural Science Foundation of Jilin Province (Grant Nos: YDZJ202201ZYTS320) and the Sixth Batch of Jilin Province Youth Science and Technology Talent Lifting Project.

    The authors declare that there are no conflicts of interest in the publication of this paper.



    [1] Sima V, Gheorghe IG, Subić J, et al. (2020) Influences of the Industry 4.0 Revolution on the human capital development and consumer behavior: A systematic review. Sustainability 12: 4035. https://doi.org/10.3390/su12104035 doi: 10.3390/su12104035
    [2] Bringezu S, Schütz H, Steger S, et al. (2004) International comparison of resource use and its relation to economic growth. Ecol Econ 51: 97–124. https://doi.org/10.1016/j.ecolecon.2004.04.010 doi: 10.1016/j.ecolecon.2004.04.010
    [3] Venables AJ (2016) Using natural resources for development: Why has it proven so difficult? J Econ Perspect 30: 161–184. https://doi.org/10.1257/jep.30.1.161 doi: 10.1257/jep.30.1.161
    [4] Almeida TAdN, Cruz L, Barata E, et al. (2017) Economic growth and environmental impacts: An analysis based on a composite index of environmental damage. Ecol Indic 76: 119–130. https://doi.org/10.1016/j.ecolind.2016.12.028 doi: 10.1016/j.ecolind.2016.12.028
    [5] Schwarze P, Stilianakis N, Momas I, et al. (2005) Health effects of transport-related air pollution. https://doi.org/10.1038/s12276-020-0403-3
    [6] Schraufnagel DE (2020) The health effects of ultrafine particles. Exp Mol Med 52: 311–317. https://doi.org/10.1038/s12276-020-0403-3 doi: 10.1038/s12276-020-0403-3
    [7] Kwon HS, Ryu MH, Carlsten C (2020) Ultrafine particles: Unique physicochemical properties relevant to health and disease. Exp Mol Med 52: 318–328. https://doi.org/10.1038/s12276-020-0405-1 doi: 10.1038/s12276-020-0405-1
    [8] Moreno-Ríos AL, Tejeda-Benítez LP, Bustillo-Lecompte CF (2022) Sources, characteristics, toxicity, and control of ultrafine particles: An overview. Geosci Front 13: 101147. https://doi.org/10.1016/j.gsf.2021.101147 doi: 10.1016/j.gsf.2021.101147
    [9] Li HL, Zheng Y, Yu Q, et al. (2022) Optimization for enhanced electrokinetic treatment of air pollution control residues using response surface methodology focusing on heavy metals leaching risk and extractability. Process Saf Environ 159: 534–546. https://doi.org/10.1016/j.psep.2022.01.033 doi: 10.1016/j.psep.2022.01.033
    [10] Sawasdee V, Pisutpaisal N (2016) Simultaneous pollution treatment and electricity generation of tannery wastewater in air-cathode single chamber MFC. Int J Hydrogen Energ 41: 15632–15637. https://doi.org/10.1016/j.ijhydene.2016.04.179 doi: 10.1016/j.ijhydene.2016.04.179
    [11] Panahi Y, Mellatyar H, Farshbaf M, et al. (2018) Biotechnological applications of nanomaterials for air pollution and water/wastewater treatment. Mater Today: Proc 5: 15550–15558. https://doi.org/10.1016/j.matpr.2018.04.162 doi: 10.1016/j.matpr.2018.04.162
    [12] Bhatt P, Pandey SC, Joshi S, et al. (2022) Nanobioremediation: A sustainable approach for the removal of toxic pollutants from the environment. J Hazard Mater 427: 128033. https://doi.org/10.1016/j.jhazmat.2021.128033 doi: 10.1016/j.jhazmat.2021.128033
    [13] Mohamed EF (2017) Nanotechnology: Future of environmental air pollution control. Environ Manag Sust Dev 6: 429–454. https://doi.org/10.5296/emsd.v6i2.12047 doi: 10.5296/emsd.v6i2.12047
    [14] García-Mayagoitia S, Pérez-Hernández H, Medina-Pérez G, et al. (2020) Bio-nanomaterials in the air pollution treatment, In: Nanomaterials for air remediation, Amsterdam: Elsevier, 227–248.
    [15] Yadav KK, Singh JK, Gupta N, et al. (2017) A review of nanobioremediation technologies for environmental cleanup: A novel biological approach. J Mater Environ Sci 8: 740–757.
    [16] Singh BSM, Singh D, Dhal NK (2022) Enhanced phytoremediation strategy for sustainable management of heavy metals and radionuclides. Case Stud Chem Environ Eng 5: 100176. https://doi.org/10.1016/j.cscee.2021.100176 doi: 10.1016/j.cscee.2021.100176
    [17] World Health Organization, WHO Collaborating Centre on Air Pollution Control, WHO Collaborating Centre on Clinical and Epidemiological Aspects of Air Pollution (1976) Selected methods of measuring air pollutants/prepared in cooperation with the WHO Collaborating Centre on Air Pollution Control, Office of Research and Development, U.S. Environmental Protection Agency, Washington, DC, USA and WHO Collaborating Centre on Clinical and Epidemiological Aspects of Air Pollution, Medical Research Council Air Pollution Unit, St. Bartholomew's Hospital Medical College, London, England. Available from: https://apps.who.int/iris/handle/10665/37047.
    [18] Language B, Piketh SJ, Burger RP (2016) Correcting respirable photometric particulate measurements using a gravimetric sampling method. Clean Air J 26: 10–14. https://doi.org/10.17159/2410-972X/2016/v26n1a7 doi: 10.17159/2410-972X/2016/v26n1a7
    [19] Amaral S, de Carvalho J, Costa M, et al. (2015) An overview of particulate matter measurement instruments. Atmosphere 6: 1327–1345. https://doi.org/10.3390/atmos6091327 doi: 10.3390/atmos6091327
    [20] Smodiš B, Parr RM (1999) Biomonitoring of air pollution as exemplified by recent IAEA programs. Biol Trace Elem Res 71: 257–266. https://doi.org/10.1007/BF02784211 doi: 10.1007/BF02784211
    [21] Costa C, Teixeira JP (2014) Biomonitoring, In: Encyclopedia of toxicology, 3 Eds., Oxford: Academic Press, 483–484. https://doi.org/10.1016/B978-0-12-386454-3.01000-9
    [22] Kousehlar M, Widom E (2019) Sources of metals in atmospheric particulate matter in Tehran, Iran: Tree bark biomonitoring. Appl Geochem 104: 71–82. https://doi.org/10.1016/j.apgeochem.2019.03.018 doi: 10.1016/j.apgeochem.2019.03.018
    [23] Gueguen F, Stille P, Lahd Geagea M, et al. (2012) Atmospheric pollution in an urban environment by tree bark biomonitoring-Part Ⅰ: Trace element analysis. Chemosphere 86: 1013–1019. https://doi.org/10.1016/j.chemosphere.2011.11.040 doi: 10.1016/j.chemosphere.2011.11.040
    [24] Berlizov AN, Blum OB, Filby RH, et al. (2007) Testing applicability of black poplar (Populus nigra L.) bark to heavy metal air pollution monitoring in urban and industrial regions. Sci Total Environ 372: 693–706. https://doi.org/10.1016/j.scitotenv.2006.10.029 doi: 10.1016/j.scitotenv.2006.10.029
    [25] Zampieri MCT, Sarkis JES, Pestana RCB, et al. (2013) Characterization of Tibouchina granulosa (Desr.) Cong. (Melastomataceae) as a biomonitor of air pollution and quantification of particulate matter adsorbed by leaves. Ecol Eng 61: 316–327. https://doi.org/10.1016/j.ecoleng.2013.09.050 doi: 10.1016/j.ecoleng.2013.09.050
    [26] González CM, Casanovas SS, Pignata ML (1996) Biomonitoring of air pollutants from traffic and industries employing Ramalina ecklonii (Spreng.) Mey. and Flot. in Córdoba, Argentina. Environl Pollut 91: 269–277. https://doi.org/10.1016/0269-7491(95)00076-3 doi: 10.1016/0269-7491(95)00076-3
    [27] Garty J, Garty-Spitz RL (2015) Lichens and particulate matter: Inter-relations and biomonitoring with lichens, In: Recent advances in lichenology, New Delhi: Springer, 47–85. https://doi.org/10.1007/978-81-322-2181-4_3
    [28] Massimi L, Castellani F, Protano C, et al. (2021) Lichen transplants for high spatial resolution biomonitoring of Persistent Organic Pollutants (POPs) in a multi-source polluted area of Central Italy. Ecol Indic 120: 106921. https://doi.org/10.1016/j.ecolind.2020.106921 doi: 10.1016/j.ecolind.2020.106921
    [29] Kousehlar M, Widom E (2020) Identifying the sources of air pollution in an urban-industrial setting by lichen biomonitoring-A multi-tracer approach. Appl Geochem 121: 104695. https://doi.org/10.1016/j.apgeochem.2020.104695 doi: 10.1016/j.apgeochem.2020.104695
    [30] Thomas W (1986) Representativity of mosses as biomonitor organisms for the accumulation of environmental chemicals in plants and soils. Ecotoxicol Environ Saf 11: 339–346. https://doi.org/10.1016/0147-6513(86)90106-5 doi: 10.1016/0147-6513(86)90106-5
    [31] Tak AA, Kakde UB (2017) Assessment of air pollution tolerance index of plants: A comparative study. Int J Pharm Pharm Sci 9: 83–89. https://doi.org/10.22159/ijpps.2017v9i7.18447 doi: 10.22159/ijpps.2017v9i7.18447
    [32] Isaac-Olive K, Solis C, Martinez-Carrillo MA, et al. (2012) Tillandsia usneoides L, a biomonitor in the determination of Ce, La and Sm by neutron activation analysis in an industrial corridor in Central Mexico. Appl Radiat Isotopes 70: 589–594. https://doi.org/10.1016/j.apradiso.2012.01.007 doi: 10.1016/j.apradiso.2012.01.007
    [33] Techato K, Salaeh A, van Beem NC (2014) Use of Atmospheric Epiphyte Tillandsia usneoides (Bromeliaceae) as Biomonitor. APCBEE Proc 10: 49–53. https://doi.org/10.1016/j.apcbee.2014.10.014 doi: 10.1016/j.apcbee.2014.10.014
    [34] Gallego-Cartagena E, Morillas H, Carrero JA, et al. (2021) Naturally growing grimmiaceae family mosses as passive biomonitors of heavy metals pollution in urban-industrial atmospheres from the Bilbao Metropolitan area. Chemosphere 263: 128190. https://doi.org/10.1016/j.chemosphere.2020.128190 doi: 10.1016/j.chemosphere.2020.128190
    [35] Nikam J, Archer D, Nopsert C (2021) Air quality in Thailand: Understanding the regulatory context, Stockholm: Academic Press.
    [36] Boonpeng C, Sangiamdee D, Noikrad S, et al. (2020) Metal accumulation in lichens as a tool for assessing atmospheric contamination in a natural park. Environ Nat Resour J 18: 166–176. https://doi.org/10.32526/ennrj.18.2.2020.16 doi: 10.32526/ennrj.18.2.2020.16
    [37] Kayee P, Songphim W, Parkpein A (2015) Using Thai native moss as bio-adsorbent for contaminated heavy metal in air. Proc-Soc Behav Sci 197: 1037–1042. https://doi.org/10.1016/j.sbspro.2015.07.312 doi: 10.1016/j.sbspro.2015.07.312
    [38] Meesang W, Baothong E, Poojeera S, et al. (2022) Model feasibility of air pollution treatment using plants as filter by computational fluid dynamic (CFD) analysis: A case study in laboratory. Environ Asia 15: 142–153. http://doi.org/10.14456/ea.2022.13 doi: 10.14456/ea.2022.13
    [39] Cargill DC, Beckmann K, Seppelt R (2021) Taxonomic revision of Riccia (Ricciaceae, Marchantiophyta) in the monsoon tropics of the Northern Territory, Australia. Aust Syst Bot 34: 336–430. https://doi.org/10.1071/sb20030 doi: 10.1071/sb20030
    [40] Lee GE, Gradstein S (2021) Guide to the liverworts and hornworts of Malaysia, Tokyo: Hattori Botanical Laboratory.
    [41] Świsłowski P, Vergel K, Zinicovscaia I, et al. (2022) Mosses as a biomonitor to identify elements released into the air as a result of car workshop activities. Ecol Indic 138: 108849. https://doi.org/10.1016/j.ecolind.2022.108849 doi: 10.1016/j.ecolind.2022.108849
    [42] Kosior G, Samecka-Cymerman A, Brudzinska-Kosior A (2018) Transplanted Moss Hylocomium splendens as a Bbioaccumulator of trace elements from different categories of sampling sites in the Upper Silesia area (SW Poland): Bulk and dry deposition impact. Bull Environ Contam Toxicol 101: 479–485. https://doi.org/10.1007/s00128-018-2429-y doi: 10.1007/s00128-018-2429-y
    [43] Capozzi F, Adamo P, Spagnuolo V, et al. (2021) Field comparison between moss and lichen PAHs uptake abilities based on deposition fluxes and diagnostic ratios. Ecol Indic 120: 106954. https://doi.org/10.1016/j.ecolind.2020.106954 doi: 10.1016/j.ecolind.2020.106954
    [44] Read DJ, Edwards D, Read DJ, et al. (2000) Symbiotic fungal associations in 'lower' land plants. Phil Trans R Soc Lond B 355: 815–831. https://doi.org/10.1098/rstb.2000.0617 doi: 10.1098/rstb.2000.0617
    [45] Ghafari S, Kaviani B, Sedaghathoor S, et al. (2020) Assessment of air pollution tolerance index (APTI) for some ornamental woody species in green space of humid temperate region (Rasht, Iran). Environ Dev Sust 23: 1579–1600. https://doi.org/10.1007/s10668-020-00640-1 doi: 10.1007/s10668-020-00640-1
    [46] Ferella F, Zueva S, Innocenzi V, et al. (2018) New scrubber for air purification: Abatement of particulate matter and treatment of the resulting wastewater. Int J Environ Sci Technol 16: 1677–1690. https://doi.org/10.1007/s13762-018-1826-4 doi: 10.1007/s13762-018-1826-4
    [47] Vianna NA, DG, Brandao F, de Barros RP, et al. (2011) Assessment of heavy metals in the particulate matter of two Brazilian metropolitan areas by using Tillandsia usneoides as atmospheric biomonitor. Environ Sci Pollut Res Int 18: 416–427. https://doi.org/10.1007/s11356-010-0387-y doi: 10.1007/s11356-010-0387-y
    [48] Che YH, Yao TT, Wang HR, et al. (2022) Potassium ion regulates hormone, Ca2+ and H2O2 signal transduction and antioxidant activities to improve salt stress resistance in tobacco. Plant Physiol Biochem 186: 40–51. https://doi.org/DOI:10.1016/j.plaphy.2022.06.027
    [49] Nwajinka CO, Okonjo EO, Amaefule DO, et al. (2020) Effects of microwave power and slice thickness on fiber and ash contents of dried sweet potato (Ipomoea batata). NIJOTECH 39: 932–941. https://doi.org/10.4314/njt.v39i3.36 doi: 10.4314/njt.v39i3.36
    [50] Wherry ET, Buchanan R (1926) Composition of the ash of Spanish-moss. Ecology 7: 303–306. https://doi.org/10.2307/1929312 doi: 10.2307/1929312
    [51] Gortner RA, Hoffman WF (1992) Determination of moisture content of expressed plant tissue fluids. Bot Gaz 74: 308–313.
    [52] Ahmad S, Amin MN, Mosaddik MA (2002) Studies on moisture content, biomass yield (crude plant extract) and alkaloid estimation of in vitro and field grown plants of Rauvolfia serpentina. Pak J Biol Sci 5: 416–418.
    [53] Yoshikawa K, Overduin PP, Harden JW (2004) Moisture content measurements of moss (Sphagnum spp.) using commercial sensors. Permafrost Periglacial Processes 15: 309–318. https://doi.org/10.1002/ppp.505 doi: 10.1002/ppp.505
    [54] Kim DM, Zhang HR, Zhou HY, et al. (2015) Highly sensitive image-derived indices of water-stressed plants using hyperspectral imaging in SWIR and histogram analysis. Sci Rep 5: 15919. https://doi.org/10.1038/srep15919 doi: 10.1038/srep15919
    [55] Krishnaveni M, Chandrasekar R, Amsavalli L, et al. (2013) Air pollution tolerance index of plants at Perumalmalai Hills, Salem, Tamil Nadu, India. Int J Pharm Sci Rev Res 20: 234–239.
    [56] Wang HX, Shi H, Li YY (2011) Leaf dust capturing capacity of urban greening plant species in relation to leaf micromorphology. 2011 International Symposium on Water Resource and Environmental Protection, 2198–2201.
    [57] Hamontree C, Ibrahim IZ, Chong W-T, et al. (2018) The design of the botanical indoor air biofilter system for the atmospheric particle removal. MATEC Web Conf 192: 02035. https://doi.org/10.1051/matecconf/201819202035 doi: 10.1051/matecconf/201819202035
    [58] Kim JJ, Park J, Jung SY, et al. (2020) Effect of trichome structure of Tillandsia usneoides on deposition of particulate matter under flow conditions. J Hazard Mater 393: 122401. https://doi.org/10.1016/j.jhazmat.2020.122401 doi: 10.1016/j.jhazmat.2020.122401
    [59] Shen L, Zhang X, Yue DKP, et al. (2003) Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J Fluid Mech 484: 197–221. https://doi.org/10.1017/s0022112003004294 doi: 10.1017/s0022112003004294
    [60] Gul A, Tezcan Un U (2022) Effect of temperature and gas flow rate on CO2 capture. Eur J Sust Dev Res 6: em0181. https://doi.org/10.21601/ejosdr/11727 doi: 10.21601/ejosdr/11727
    [61] Saysroy A, Eiamsa-ard S (2017) Enhancing convective heat transfer in laminar and turbulent flow regions using multi-channel twisted tape inserts. Int J Therm Sci 121: 55–74. https://doi.org/10.1016/j.ijthermalsci.2017.07.002 doi: 10.1016/j.ijthermalsci.2017.07.002
    [62] Choi YK, Song HJ, Jo JW, et al. (2021) Morphological and chemical evaluations of leaf surface on particulate matter2.5 (PM2.5) removal in a botanical plant-based biofilter system. Plants 10: 2761. https://doi.org/10.3390/plants10122761 doi: 10.3390/plants10122761
    [63] Wang YC, Chen B (2021) Dust capturing capacity of woody plants in clean air zones throughout Taiwan. Atmosphere 12: 696. https://doi.org/10.3390/atmos12060696 doi: 10.3390/atmos12060696
    [64] Fernandez V, Gil-Pelegrin E, Eichert T (2021) Foliar water and solute absorption: An update. Plant J 105: 870–883. https://doi.org/10.1111/tpj.15090 doi: 10.1111/tpj.15090
    [65] Palma AD (2016) Mosses for monitoring air pollution: Towards the standardization of moss-bag technique and the set-up of a new biomaterial, PhD thesis, University of Naples Federico Ⅱ.
    [66] Usman F, Bakar ARA, Mohamed Nazri F (2020) Thermal comfort study using CFD analysis in residential house with mechanical ventilation system, In: Proceedings of AICCE'19, Cham: Springer.
    [67] Liu J, Heidarinejad M, Pitchurov G, et al. (2018) An extensive comparison of modified zero-equation, standard k-ε, and LES models in predicting urban airflow. Sust Cities Soc 40: 28–43. https://doi.org/10.1016/j.scs.2018.03.010 doi: 10.1016/j.scs.2018.03.010
    [68] Muhsin F, Yusoff WFM, Mohamed MF et al. (2017) CFD modeling of natural ventilation in a void connected to the living units of multi-storey housing for thermal comfort. Energ Build 144: 1–16. https://doi.org/10.1016/j.enbuild.2017.03.035 doi: 10.1016/j.enbuild.2017.03.035
    [69] Zhang Y, Yu WX, Li YL, et al. (2021) Comparative research on the air pollutant prevention and thermal comfort for different types of ventilation. Indoor Built Environ 30: 1092–1105. https://doi.org/10.1177/1420326x20925521 doi: 10.1177/1420326x20925521
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3127) PDF downloads(147) Cited by(0)

Figures and Tables

Figures(3)  /  Tables(6)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog