An Easy-to-Understand Method to Construct Desired Distance-Like Measures

College of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710062, China Freshman Academy, Xi’an Technological University, Xi’an 710021, China School of Mathematics, %apar Institute of Engineering & Technology, Deemed University, Patiala 147004, Punjab, India School of Science, Guangxi University of Nationalities, Nanning 530006, China Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt School of Psychology, Shaanxi Normal University, Xi’an 710062, China


Introduction and Preliminaries
In many cases, the measure values of true data are not unique (but two or more) for uncertainty or complexity. For example, there are several agents in China who value and order all periodicals published in China. Peking University Library and Nanjing University Library are generally thought to be the best two and incomparable to each other. For a journal J, assume that the orders given by Peking University Library and the Nanjing University Library are m-th and n-th, respectively; then, m and n may be not the same in general.
ere are also many other examples. In 2012, breakthrough selected by the famous journal Science are different from those selected by the famous journal Nature; Gini coefficients in China in 2012 from two different agents are 0.481 and 0.61, respectively; the Chebyshev distance (resp., the Euclidean distance, the Manhattan distance or the city block distance, and the river distance) between two points (0, 1) and (1,2) in the Euclidean plane R 2 is 1 (resp., � 2 √ , 2, 3). Please see Proposition 1 for definitions of these metrics; the effective distances used in cluster analysis are many and varied; a given asymptomatic infected people to Corona Virus Disease (COVID-19 for short) are thought to be highly contagious (which can be represented by a fuzzy number A) by experts in one country but lowly contagious (which can also be represented by a fuzzy number B that is much different from A) by experts in another country.
In practice, most people choose just one of the measure values (or choose the arithmetic mean of these measure values) as the true data; such a kind of dispose can be accepted only in rare cases (e.g., the information loss cannot be avoided or make almost no difference). To make an improvement in the disposal of these uncertain or complex data, at least two better theories (One is theoretically inspirational, another is application-motivated; both are based mainly on the idea of fuzzy set.) have been proposed which are mostly about measuring values of difference between two abstract "points" (precisely, two elements of a set) whose information or data can be provided by at least two different agents (but cannot be provided satisfactorily by one agent, see Example 1).
Example 1 (1) e distance between two fuzzy sets . erefore, all sets involved here can be thought to be finite. (2) e distance between two bodies A � (x, y, z) ∈ R 3 |x 2 + y 2 + z 2 ≤ ϵ 2 } (i.e., the closed ball in the 3dimensional Euclidean space R 3 with center (0, 0, 0) � 0 → and radius ϵ 2 ) and B � (x, y, z) ∈ R 3 |(x − ϵ 2 − ε) 2 + y 2 + z 2 ≤ ε 2 } (i.e., the ball in R 3 with center (ϵ + ε, 0, 0) and radius ϵ) cannot be expressed accurately (or satisfactorily) in a real number but can be expressed satisfactorily in several numbers (where 0 ≤ ϵ ≤ ε). For instance, it can be expressed in a 3-element set a, _ a, € a { } (in this way information loss can be almost avoided) or be expressed by any 2-element subset of a, _ a, € a { } (in this way much more information loss can be avoided), where a � 0 � inf ρ 2 ( x → , y → ) | x → ∈ A, y → ∈ B} (ρ 2 is the ordinary Euclidean metric on R 3 ), _ a � ϵ + ε is the Euclidean distance between the centroid (0, 0, 0) � 0 → of A and the centroid (ϵ + ε, 0, 0) of B, and € a � 2ε is the Hausdorff distance between A and B (notice that a ≤ _ a ≤ € a). (3) During the time of COVID-19 (especially, the first three months), a lot of things (particularly, traveling and meeting) had to stop in China. e expected College back-to-University time can be forecasted rather than accurately based on several (even the first two of them) time series (the time series of suspected cases, the time series of imported cases, the time series of close contact cases, the time series of no symptoms cases, the time series of cluster infection cases, the time series of confirmed cases, etc.) but cannot be forecasted satisfactorily by one of them. e first way to make an improvement is using a fuzzy metric. Several authors have introduced the concepts of fuzzy metric and fuzzy metric space from different points of view [1][2][3][4]. Erceg [1] defined a fuzzy pseudo-metric on a set consisting of fuzzy sets (e.g., fuzzifications of data or information to be disposed) by generalizing the Hausdorff distance between usual sets. e motivation behind the fuzzy metric of Kramosil and Michalek [3] was a statistical metric; recently, these kinds of fuzzy metric spaces have stimulated a lot of interest (see [5] and references herein for details). Kaleva and Seikkala's metric [2] defined the distance between two points as a fuzzy number for the reason that sometimes uncertainty is due to fuzziness rather than randomness. As is known, Kaleva and Seikkala's fuzzy metric spaces possess rich structures with suitable choices of binary operations. Much theoretical work related to Kaleva and Seikkala's fuzzy metric spaces has been done in recent years (see [2,[5][6][7][8][9] and references herein). By presenting intuitive level forms for the triangle inequalities in the definition of Kaleva and Seikkala's fuzzy metric, Huang and Wu [6] offered a new and convenient tool for the description and analysis of fuzzy metric spaces (cf. their subsequent work [7] on the existence and uniqueness of completion of special fuzzy metric spaces). Under some intuitive and mild assumptions, Fang [5] (as an improvement on Huang and Wu [7]) also proved the existence and the uniqueness of completion of fuzzy metric spaces. Xiao et al. [10] took a closer step toward possible applications of fuzzy metrics (in integral equations, differential equations, qualitative behavior, dynamic systems, and other nonlinear problems), in which the authors studied the existence and uniqueness of fixed points (under a weaker assumption) for nonlinear contractions in fuzzy metrics spaces in the sense of Kaleva and Seikkala [2]. By virtue of a level-cut method, they established relationships between a fuzzy metric and a family of quasi-metrics (so that the utilizing of results and skills in metric spaces becomes more and more possible). e second way is using a dissimilarity measure or a distance measure, which is defined on a set consisting of special fuzzy sets (e.g., fuzzifications of data or information to be disposed); it takes many steps toward practical applications of metrics. Balopoulos et al. [11] defined a new family of distance measures (based on matrix norms) for binary operators on [0, 1], which can also be used to measure the difference between two fuzzy sets on two finite sets. Bustince et al. [12] constructed distance measures, proximity measures, and fuzzy entropies by aggregating restricted dissimilarity functions in a special way. Liu [13] gave systematically an axiom definition of entropy, distance measure, and similarity measure of fuzzy sets; he also discussed basic relations between these measures. Fan et al. [14] defined a new divergence measure, study the relations between fuzzy entropy, fuzzy Hamming distance, and divergence measures defined. ey obtain quite a general conclusion and solve a problem proposed by Liu [13]. Dissimilarity measure and distance measure have already found wide applications for their intuitiveness (see [15][16][17][18][19][20][21]).
As far as data are concerned, fuzzy metrics mentioned above have some shortcomings. It seems not very convenient 2 Complexity to apply these fuzzy metrics in practical problems because axioms satisfied by these metrics are complex; implementation of computing related to these fuzzy metric spaces is obviously difficult because operations (even if addition or subtraction) or sorting of fuzzy numbers cannot be realized in computers unless the fuzzy numbers involved are extremely simple (for example, they are triangle fuzzy numbers or interval numbers). Dissimilarity measures and distance measures mentioned above also have some faults. For example, they have few nice topological properties (see [8,[11][12][13][16][17][18][19][20][21]) because they do not satisfy the triangle inequality in general; axioms they satisfy are also complex. ese naturally urge people to find a kind of metrics (or their weaker forms) which are down to earth, i.e., they are theoretically good, intuitive (because intuition and examples always provide inspirations for both theoretical study itself and its applications in the real world), and easyto-use. For example, try to find a metric (or its weaker form) such that the complexity of the distance value of two points is between that of a real number and that of a fuzzy number.
An immediately thought way to construct such a metric (or its weaker form) ρ is to take it as a weighted mean of some well-known or usually used metrics ρ 1 , . . . , ρ n , i.e., let ρ � k 1 ρ 1 + · · · + k n ρ n , where k 1 , . . . , k n ∈ (0, 1) satisfies k 1 + · · · + k n � 1. However, information lost in this way may be too much compared to a way based on interval numbers (unless the set k 1 , . . . , k n } is chosen as the optimal one such that ρ(x, y) is the fittest value for each (x, y) ∈ X 2 ).
Interval numbers or interval data appear in many cases (such as numerical analysis-handling rounding errors, computer-assisted proofs, global optimization, particularly, modeling uncertainty) because the data involved there cannot be accurately expressed in real numbers but can be expressed in interval numbers. e measure values of true data can be looked as an interval data, i.e., one can use an interval number to denote the measure values of true data. For example, the distance between A and B in Example 1 (2) can be expressed as one of the following six interval numbers: [a, a] (or 〈a, a〉), Motivated by Polya's plausible reasoning [22], the present paper will consider a fusion of easy-to-obtain measure data (e.g., values of various easy-to-obtain pseudosemi-metrics, pseudo-metrics, or metrics) by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators coined by practitioners; of course, interval numbers and triangular fuzzy numbers can also be used. In Section 2, a distance-like notion, called weak interval-valued pseudo-metrics (briefly, WIVP-metrics), is defined by using known notions of pseudo-semi-metrics, pseudo-metrics, and metrics (this notion and its special cases are also characterized by using axioms and connections between these notions and other known and related notions); propositions and detailed examples are given to illustrate how to fabricate (including using what "material") an expected or demanded WIVP-metric (even intervalvalued metric) in practical problems. Sections 3 and 4 are on theoretical applications of WIVP-metrics, where we demonstrate how to construct, by using some logic implication operators, some WIVP-metrics which may be useful in quantitative logic (cf. [23]) and quantitative reasoning (cf. [24]), and how to define well matched interval-valued metrics on interval-valued fuzzy graphs. Fixed point theorems and common fixed point theorems in WIVP-metrics are presented in Section 5. We end the paper with concluding remarks in Section 6. Now, we present some preliminary notions needed in this paper.
Let [0, 1] R be the set of all fuzzy sets on R (i.e., the set of all mappings from R to [0, 1]), where R is the set of all real numbers (also called the real line) and [0, 1] is the closed unit interval of R. For each subset Y ⊆R and each b ∈ [0, 1], we use b Y to denote the fuzzy set on R taking value b on Y and 0 elsewhere; we will make no distinction between 1 Y and Y. If a fuzzy set μ ∈ [0, 1] R satisfies μ(r) � 1 for some r ∈ R and [μ a − , μ a + ] (i.e., a closed interval of R ) for any a ∈ (0, 1], then we call μ a fuzzy number (we will regard μ − and μ + as functions on (0, 1], cf. [14,24,25]). e set of all fuzzy numbers is denoted by F. A fuzzy number μ is called e set of all nonnegative fuzzy numbers is denoted by F +. Fuzzy numbers are the most commonly used fuzzy sets on the real line (see [25][26][27][28][29][30] and references herein); one of their applications is that they can be used to define fuzzy metric space (an important notion in fuzzy analysis). A binary operation * on [0, 1], which is commutative (or symmetric), associative, and nondecreasing coordinately, is said to be a t-norm on [0, 1] if it satisfies a * 1 � a( ∀a ∈ [0, 1]); * is said to be a t-conorm on [0, 1] if it satisfies a * 0 � a( ∀a ∈ [0, 1]).
Definition 1 (see [2]). A fuzzy metric space is a quadruple (X, d, L, R) (in this case, d : X ×X ⟶ F + is called a fuzzy metric) which satisfies the following two conditions ( ∀(x, y, z) ∈ X 3 ): Here, d : X ×X ⟶ F + is a mapping, L and R are operations on [0, 1] which are symmetric, nondecreasing coordinately and satisfy L(0, 0) � 0 and R(1, Definition 2 (see [31]). An interval number (i.e., a special kind of fuzzy number) is a point a � 〈a − , a +〉 in the 2dimensional Euclidean space R 2 which satisfies a − ≤ a +.
e set of all interval numbers (with the point-wise order ≤ ) is denoted by I(R) (Notice that a total order [32,33] ≤ was also defined on Moreover, we will identify a closed interval [a, b] of R, a point 〈a, b〉 in R 2 , and a fuzzy number 1 [a, b] since there exists a natural one-to-one correspondence between the set of all closed intervals of R and I(R).). For any 〈a − , a +〉, 〈b − , b +〉 ∈ I(R) and each nonnegative real number r,

Definition and Examples of WIVP-Metric
In this section, we will define the notion of weak intervalvalued pseudo-metrics (shortly, WIVP-metrics), exemplify in detail how to construct distance-like measures (including WIVP-metrics) desired in practice by fusing easy-to-obtain or easy-to-coin pseudo-semi-metrics, pseudo-metrics, or metrics based on operators ∧, ∨, and simple aggregation operators. We also characterize WIVP-metric and its special forms intuitively so that practitioners can understand them easily.
x)|x ∈ X} is the diagonal of X 2 .) (resp., a pseudo-metric, a semi-metric, a metric) on X and ρ + � p 2 ∘ ρ : X ×X ⟶ [0, +∞) is a pseudo-metric (resp., a pseudo-metric, a metric, a metric) on X, then ρ is called a WIVP-metric on X (resp., an intervalvalued pseudo-metric on X, a weak interval-valued metric on X, an interval-valued metric on X) and (X, ρ) is called a WIVP-metric space (resp., an interval-valued pseudo-metric space, a weak interval-valued metric space, an intervalvalued metric space), where p 1 : R 2 ⟶ R and p 2 : R 2 ⟶ R are the first coordinate projection and the second coordinate projection, respectively (For a WIVP-metric on for some a ≫ 0 whenever x ∈ W} ⊆T 1, ρ ⊆T 2, ρ, T ρ is a topology on X, and both T 1, ρ and T 2, ρ are pre-topologies on X. It can be proved that T ρ , T 1, ρ, and T 2, ρ have good topological properties.).

Proposition 1.
Let R 3 be the 3-dimension Euclidean space and ρ 1 , ρ 2 , ρ 3 , and ρ 4 be the Chebyshev metric, the Euclidean metric, the Manhattan metric or the city block metric, and the river metric on R 3 , respectively (We will consider them to be metrics on R 2 by identifying a point (a 1 , Again, let ρ 0 � (1 /3)ρ 3 . en, the following premises hold: is an interval-valued metric (actually, a metric) (Each of these interval-valued metrics can be used to measure the difference between any two bodies A and B with the centroid set a . e last two are WIVP-metrics on Y 1 (but the first is not in general); all of them can be used to measure the difference between any two fuzzy sets Φ, Ψ ∈ Y 1 .
, is a pseudometric; and d 2 , defined by is a metric (called the Hausdorff metric induced by ρ 2 and written also as ρ H 2 ). erefore, we obtain three WIVP-metrics d i :

Example 4
(1) Consider the two bodies A and B in Example 1 (2). en, the differences between A and B can be taken as (3)Analogously, let A be the set of all nonempty compact sets in the ordinary Euclidean space (R 3 , ϱ), ρ � ϱ H be the Hausdorff metric on A induced by ϱ, ∈ A( ∀d ∈ (0, 1])}, (a, c) � (0.4, 0.6), and (Φ, Ψ) ∈ A be two clouds (in which Φ(x, y, z) stands for the degree of concentration of cloud at the point (x, y, z)), defined by Φ(x, y, z) � φ(x) ∧φ(y) ∧φ(z) and Ψ(x, y, z) � ψ(x) ∧ψ(y) ∧ψ(z) ((x, y, z) Example 5. Now, we exemplify an application of the idea of WIVP-metrics in pattern recognition (i.e., COVID-19 diagnosis). Consider the following symptom data (see Table 1), involving 5 symptoms (asthma, sore throat, cough, fever, age), and diagnosis results, involving yes (briefly, Y) and no (briefly, N), of 7 patients given by expert physicians in a Chinese hospital in Nanjing: From this, we can get the following 7 mappings (patients): Take other 2 patients in the same hospital for test: Notice that both P 8 and P 9 have two asthma values (we will take their center in the following computation for simplicity) because he/she ate a kind of TCL-like food before taking the two values. As the group of experts give two different weight vectors e → � (e 1 , e 2 , e 3 , e 4 , e 5 ) � (2, 1, 5, 30, 1) and v → � (v 1 , v 2 , v 3 , v 4 , v 5 ) � (4, 3, 10, 60, 1) for importance of vector of symptom s → �(asthma, sore throat, cough, fever, age), we finish this decision-making problem in the following nine steps (see Figure 1).
Step 1. Compute the difference between P 8 and those with diagnosis results Y using a metric rely on e → (there are many cases on choices of ρ e ). ρ e (P 2 , � 77. us, we obtain the average d 8, e, Y � 132.6.
Step 2. Compute the difference between P 8 and those with diagnosis results N using a metric rely on e → .
Step 3. Compute the difference between P 9 and those with diagnosis results Y using a metric rely on e → .
Step 4. Compute the difference between P 9 and those with diagnosis results N using a metric rely on e → .
Step 5. Compute the difference between P 8 and those with diagnosis results Y using a metric rely on v → .
Step 6. Compute the difference between P 8 and those with diagnosis results N using a metric rely on v → .
Step 7. Compute the difference between P 9 and those with diagnosis results Y using a metric rely on v → .
Step 8. Compute the difference between P 9 and those with diagnosis results N using a metric rely on v → .
Step 9. As 〈d 8, , N, d 9, v, N〉, P 9 is Y (which is also the same as the diagnosis result given by expert physicians in that hospital).
, is a pseudo-metric on F(S).
(2) A WIVP-metric space (X, ρ) is said to be complete if every Cauchy sequence in (X, ρ) is convergent.
It is not difficult to verify the following.

Concluding Remarks
Since data from many real-world problems are not only from multi agents but also becoming more and more big and complex for vagueness and uncertainty, measurement by a single metric does not meet the needs of some practical problems. Motivated by Polya's plausible reasoning and artificial neural networks, this paper consider a distance-like Complexity 11 notion, called weak interval-valued pseudo-metric (WIVPmetric for short), which, as a generalization of the notion of metric, is still topologically good. To benefit practitioners, easy-to-understand propositions and much detailed examples are given (in the first half of the paper) to illustrate how to fabricate (including using what "material") an expected or demanded WIVP-metric (even interval-valued metric) in practical problems. To show theoretical applications of WIVP-metrics, we exemplify how to construct (by using some logic implication operators, some WIVP-metrics which may be useful in quantitative logic [23] and quantitative reasoning [24]) and how to define well-matched interval-valued metrics on interval-valued fuzzy graphs. As these WIVP-metrics are relatively precise, flexible, and compatible than single pseudo-semi-metric, pseudo-metric, and metric, more applications should be investigated (even put forward) based on plausible reasoning. Practitioners are also suggested to explore (in the plausible reasoning manner) other complex and more fitted methods to fabricate more desired distance-like measures, for example, to fuse easy-to-obtain pseudo-semi-metrics, pseudo-metrics, or metrics by making full use of well-known or frequently used t-norms, t-conorms, aggregation operators, and similar operators coined by practitioners or others; strategies also contain making full use of interval numbers and very special triangular fuzzy numbers. Of course, these (including also determination of weighted vector) are not always easy to practitioners. How to overcome this limitation will be one of our future works. Our future work also includes completion of WIVPmetric spaces, interval-valued truth degrees of formulas based on deferent logic implication operators, intervalvalued similarity degrees of formulas based on deferent logic implication operators, related approximate reasoning, dynamic systems on interval-valued metric spaces (even on interval-valued pseudo-metric spaces), and applications of weak interval-valued pseudo-metrics in medical diagnosis and decision-making problems (see related works [45][46][47][48][49][50][51] for details); for decision-making problems with data given by a fuzzy set (resp., interval-valued fuzzy sets, picture fuzzy sets, generalized pythagorean fuzzy sets), one can use the WIVP-metrics used in Example 2 to replace distance measures used in published papers and use the linear orders in [32,33,52] to replace the ordinary order in the real line.