Certain Types of Covering-Based Multigranulation (<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.081pt" id="M1" height="13.9799pt" version="1.1" viewBox="-0.0657574 -11.8989 38.094 13.9799" width="38.094pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-10" d="M63 95C70 84 85 83 93 83C121 83 136 106 136 131C136 160 113 179 85 179C52 179 26 150 26 107C26 41 92 -15 199 -15C302 -15 399 18 484 106C535 159 576 245 617 333C684 348 739 389 774 457L758 467C722 411 681 381 633 368C679 468 726 563 779 603C807 603 888 604 888 655C888 667 875 675 859 675C848 675 818 671 783 648C758 651 718 656 665 656C605 656 537 649 482 616C434 585 396 536 396 471C396 392 448 345 520 330C511 311 502 292 493 271C400 61 306 16 181 16C110 16 61 68 61 95H63ZM729 601C648 547 595 474 536 362C465 370 437 423 437 462C437 518 477 560 522 580S625 604 676 604C696 604 714 604 729 603V601Z"/></g><g transform="matrix(.017,0,0,-0.017,15.393,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,22.181,0)"><path id="g198-21" d="M896 675C863 652 826 634 783 634C683 634 549 687 439 687C259 687 128 601 128 473C128 383 196 329 279 329C392 329 469 405 496 541H476C440 428 385 359 286 359C224 359 169 400 169 480C169 566 257 641 382 641C475 641 622 584 731 584C796 584 862 620 907 657L896 675ZM658 553C579 484 521 424 454 277C364 81 295 16 184 16C120 16 82 44 78 79H80C87 68 100 67 108 67C136 67 153 89 153 116S131 165 104 165C69 165 43 139 43 91C43 31 110 -15 191 -15C263 -15 321 6 378 45C463 101 526 197 557 304C588 413 618 487 671 539L658 553Z"/></g></svg>)-</span>Fuzzy Rough Sets with Application to Decision-Making

Ma, Jue; Atef, Mohammed; Nada, Shokry; Nawar, Ashraf

doi:https://doi.org/10.1155/2020/6661782

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Dynamic Analysis, Learning, and Robust Control of Complex Systems

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Research Article | Open Access

Volume 2020 | Article ID 6661782 | https://doi.org/10.1155/2020/6661782

Certain Types of Covering-Based Multigranulation ()-Fuzzy Rough Sets with Application to Decision-Making

Jue Ma,¹Mohammed Atef,²Shokry Nada,²and Ashraf Nawar²

Academic Editor: Ahmed Mostafa Khalil

Received18 Oct 2020

Revised31 Oct 2020

Accepted10 Nov 2020

Published29 Nov 2020

Abstract

As a generalization of Zhan’s method (i.e., to increase the lower approximation and decrease the upper approximation), the present paper aims to define the family of complementary fuzzy -neighborhoods and thus three kinds of covering-based multigranulation ()-fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation ()-fuzzy rough sets are defined and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations and illustrates with a numerical example to describe its performance.

1. Introduction

Group decision-making aims at aggregating individual judgments to construct a composite group decision, which must be a true representative of individual preferences. The MAGDM methods choose among a discrete set of alternatives evaluated on multiple attributes and overall utility of the decision makers. MAGDM have some of the popular methods such as the weighted sum and the weighted product method (see, e.g., [1–7]). The theory of rough set was founded by Pawlak [8, 9] for dealing with the vagueness and granularity in information systems and data analysis. This theory has been applied to many different fields (see, e.g., [10–20]). Furthermore, we have noticed a wide range of generalized rough set models (see, e.g., [21–23]). Covering-based rough sets (CRSs) are considered to be one of the most studied generalized models. Pomykala [24, 25] obtained two pairs of dual approximation operators. Yao [26] studied these approximation operators by the concepts of neighborhood and granularity. Couso and Dubois [27] examined the two pairs within the context of incomplete information. Bonikowski et al. [28] established a CRS model based on the notion of minimal description. Zhu and Wang [29–32] presented several CRS models and discussed their relationships. Tsang et al. [33] and Xu and Zhang [34] proposed additional CRS models. Liu and Sai [35] compared Zhus CRS models and Xu and Zhangs CRS models. Ma [36] constructed some types of neighborhood-related covering rough sets by using the definitions of the neighborhood and complementary neighborhood. In 2016, Ma [37] introduced the definition of fuzzy -neighborhood. In 2017, Yang and Hu [38] constructed the definition of the fuzzy -complementary neighborhood to establish some types of fuzzy covering-based rough sets. Zhang et al. [39], in 2019, established the fuzzy covering-based -fuzzy rough set models and applications to multiattribute decision-making. Also, in 2019, Zhan et al. [40] proposed covering-based multigranulation -fuzzy rough set models and applications in multiattribute group decision-making.

The concept of a family fuzzy -neighborhoods was defined and their properties were studied by Zhan et al. [40]. Hence, to increase the lower approximation and decrease the upper approximation of Zhan’s model, this article’s contribution is to introduce three kinds of covering-based multigranulation -fuzzy rough sets models and explore the properties of these models with their relationships. Also, six kinds of covering-based variable precision multigranulation -fuzzy rough sets are demonstrated. An application to a practical problem illustrates their ability to help practitioners to make decisions. The outline of this paper is as follows. Section 2 gives technical preliminaries. Section 3 describes our three new types of covering-based multigranulation -fuzzy rough sets and also we introduce variable precision in order to produce the six types of covering-based variable precision multigranulation -fuzzy rough sets. Section 4 establishes relationships among our models. Section 5 puts forward a decision-making procedure that takes advantage of our theoretical framework. The conclusion is written in Section 6.

2. Preliminaries

In this section, we provide a brief survey of some notions used throughout the paper.

Definition 1 (see [1]). Suppose that the mapping is commutative, associative, and satisfies the increasing laws plus the boundary condition for each . We say that such a is a t-norm (for triangular norm) of .
Among the most important continuous t-norms, we can cite(1)The minimum operator (2)The algebraic product (3)The Lukasiewicz t-norm

Definition 2 (see [1]). Suppose that the mapping is commutative, associative, and satisfies the increasing laws plus the boundary condition for each . We say that such an is an s-norm or a t-conorm of .
Continuous s-norms are(1)The maximum operator (2)The algebraic summation (3)The bounded summation (4)The probabilistic summation

Definition 3 (see [1]). A negator operator is , an order-reversing mapping with the properties and .
We say that is involutive when for every .
The standard negator operator is defined as , for any .
Involutive negators are continuous.
Negators produce fuzzy complements. Involutive negators assure that when and , we always obtain .
We say that , a t-norm, and , a t-conorm, are dual with respect to negator , when for each , it must be the case that and .

Definition 4 (see [1]). A fuzzy implicator operator is , and a mapping with the properties and .
If, in addition, is such that , respectively, , for every , then is left monotonic decreasing, respectively, increasing.
We say that is hybrid monotonic when it is both left and right monotonic.
An implicator is a border implicator when for each .
Next, we recall three relevant classes of implicator operators [1].

Definition 5. (1)The -implicator defined by and is given: for each , (2)The -implicator defined by a continuous t-norm is given: for each , (3)If and are dual with respect to , the -implicator defined from , , and is given: for all , Well-known -implicators are(1), according to and (2), according to and (3), according to and

Definition 6 (see [42, 43]). Let be an arbitrary universal set, and be the fuzzy power set of . We call , with , a fuzzy covering of , if for each .
As a generalization of fuzzy covering, Ma [37] defined a fuzzy -covering by replacing 1 with a parameter , that is, we call , with , a fuzzy -covering of , if for each . Moreover, is called a fuzzy -covering approximation space (briefly, FCAS).

Definition 7 (see [37]). Let be a FCAS with for some . Then, for each , define the fuzzy -neighborhood of as follows:

Definition 8 (see [38]). Let be a FCAS for some . Then, for each , define the fuzzy complementary -neighborhood of as follows:

Definition 9. (see [39]). Let be a FCAS for some . For each and , define the fuzzy set (resp., , , , , , , and ) which is called the first type fuzzy lower approximation (resp., the first type of the fuzzy upper approximation, the second type of the fuzzy lower approximation, the second type of the fuzzy upper approximation, the third type of the fuzzy lower approximation, the third type of the fuzzy upper approximation, the fourth type of the fuzzy lower approximation, and the fourth type of the fuzzy upper approximation), briefly, 1-FCITFLA (resp., 1-FCITFUA, 2-FCITFLA, 2-FCITFUA, 3-FCITFLA, 3-FCITFUA, 4-FCITFLA, and 4-FCITFUA), whereIf (resp., , , ) (resp., , , ), then is called the first type of a fuzzy -covering-based -fuzzy rough set (resp., the second type of a fuzzy -covering-based -fuzzy rough set, the third type of a fuzzy -covering-based -fuzzy rough set, and the fourth type of a fuzzy -covering-based -fuzzy rough set), briefly, 1-FCITFRS (resp., 2-FCITFRS, 3-FCITFRS, and 4-FCITFRS).
Zhan et al. [40] defined the covering-based multigranulation -fuzzy rough set models (briefly, CMGITFRSs). So, in the following, some basic notions related to CMGITFRSs are given.
Suppose that be fuzzy -coverings of for some , where , for all . Then, for each , define the family of fuzzy -neighborhoods as follows:

Definition 10 (see [40]). Let be an FCAS and be fuzzy -coverings of for some , where , for all . For each , the set (resp., , , and ) is called the optimistic multigranulation -fuzzy rough lower approximation (resp., the optimistic multigranulation -fuzzy rough upper approximation, the pessimistic multigranulation -fuzzy rough lower approximation, and the pessimistic multigranulation -fuzzy rough upper approximation), briefly, 0-OMGITFRLA (resp., 0-OMGITFRUA, 0-PMGITFRLA, and 0-PMGITFRUA), whereIf (resp., ) (resp., ), then is called a covering-based optimistic multigranulation -fuzzy rough set (resp., a covering-based pessimistic multigranulation -fuzzy rough set), briefly, 0-COMGITFRS (resp., 0-CPMGITFRS).

Definition 11 (see [40]). Let be an FCAS and be fuzzy -coverings of for some , where , for all . For any and a variable precision parameter . Then, the set (resp., , , and ) is called the first type of the variable precision multigranulation -fuzzy lower approximation (resp., the first type of the variable precision multigranulation -fuzzy upper approximation, the second type of the variable precision multigranulation -fuzzy lower approximation, and the second type of the variable precision multigranulation -fuzzy upper approximation), briefly, I-VPMGITFLA (resp., I-VPMGITFUA, II-VPMGITFLA, and II-VPMGITFUA), whereIf (resp., ) (resp., ), then is called a covering-based I-variable precision multigranulation -fuzzy rough set (resp., a covering-based II-variable precision multigranulation -fuzzy rough set), briefly, I-VPMGITFRS (resp., II-VPMGITFRS); otherwise, it is I-variable precision multigranulation fuzzy definable (resp., II-variable precision multigranulation fuzzy definable).

3. Three Types of Covering-Based Multigranulation -Fuzzy Rough Sets

Here, three new kinds of covering-based multigranulation -fuzzy rough sets models (briefly, CMGITFRS) are introduced. Also, some of their properties are investigated.

Assume that be fuzzy -coverings of for some , where , for all . Then, for each , define the family of fuzzy -neighborhoods as follows:

Example 1. Let be a FCAS and be 2 fuzzy -coverings of , where , , and and as in Tables 1 and 2.
Then, we introduce the family of a fuzzy -neighborhood and fuzzy complementary -neighborhood for as follows in Tables 3–6.

3.1. Three Types of the Optimistic Multigranulation -Fuzzy Rough Sets

In the following, three kinds of COMGITFRS models are given and some of their properties are presented.

Definition 12. Let be a FCAS and be fuzzy -coverings of for some , for each . Then, the first type of the optimistic multigranulation -fuzzy lower approximation (briefly, 1-OMGITFLA) and the first type of the optimistic multigranulation -fuzzy upper approximation (briefly, 1-OMGITFUA) are, respectively, defined as follows:If , then is called a covering-based optimistic multigranulation ()-fuzzy rough set (briefly, 1-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.

Example 2. Let us consider Example 1. If , then we have the following.Case 1Let us fix based on and , and :Case 2Let us fix based on and , and . Then, we have the following results:

Theorem 1. Let be a FCAS and be fuzzy -coverings of for some , for each . Then, we have the following properties:(1)If is an -implicator based on , a continuous -conorm, and , an involutive negator,(i)(ii)(2)If either is an -implicator based on , a continuous -norm, and , a negator induced by , or is a -implicator based on , a continuous -norm, and , an involutive negator,(i)(ii)(3)If satisfies left monotonicity, then(i)(ii)(4)If satisfies right monotonicity and , then(i)(ii)(5)If satisfies right monotonicity, then . Furthermore, if and satisfy , for all , then . Also, if satisfies for all , then .(6)If and satisfy for every , then . Also, if and satisfy the weakened distributivity laws, then . Specially, .

Proof. We only need to prove (1)–(4) (i), (5), and (6), since the other proofs are similar:(1).(2)(3)Since is left monotonic, we have . Also, we have .(4)Since is right monotonic, we have . Thus, holds.(5)Since is right monotonic, we have . Since and satisfy , for all . Then, we have . Thus, . Also, since and , from (3) above, we have and . Therefore, .(6)Since and satisfy , for all . We conclude that . Hence, . Also, , we have . Since the weakened distributivity laws are satisfied, In particular, if , we have

Definition 13. Let be a FCAS and be fuzzy -coverings of for some . Then, for each , the second type of the optimistic multigranulation -fuzzy lower approximation (briefly, 2-OMGITFLA) and the second type of the optimistic multigranulation -fuzzy upper approximation (briefly, 2-OMGITFUA) are, respectively, defined as follows:If , then is called a covering-based optimistic multigranulation -fuzzy rough set (briefly, 2-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.

Example 3. (continued from Example 2). We compute for all , where for some , as shown in Tables 7 and 8 as follows.
Now, we calculate the 2-OMGITFLA and 2-OMGITFUA as explored in the following two cases.Case 1Let us fix based on and and . So,Case 2Let us fix based on and and . Thus,

Remark 1. Definition 13 satisfies Theorem 1.

Definition 14. Let be a FCAS and be fuzzy -coverings of for some . For each , the third type of the optimistic multigranulation -fuzzy lower approximation (briefly, 3-OMGITFLA) and the third type of the optimistic multigranulation -fuzzy upper approximation (briefly, 3-OMGITFUA) are, respectively, defined as follows:If , then is called a covering-based optimistic multigranulation ()-fuzzy rough set (briefly, 3-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.

Example 4. (continued from Example 2). We compute for all , where for some , as shown in Tables 9 and 10.
The following two cases calculate the 3-OMGITFLA and 3-OMGITFUA, respectively.Case 1Let us fix based on and and . So,Case 2Let us fix based on and and . Thus,

Remark 2. Definition 14 satisfies Theorem 1.

3.2. Three Types of the Pessimistic Multigranulation -Fuzzy Rough Sets

In the following, we introduce three kinds of CPMGITFRS models and study some of their properties.

Let be a FCAS and be fuzzy -coverings of for some . For each . We have three models of the pessimistic multigranulation -fuzzy lower approximation (briefly, 1-PMGITFLA, 2-PMGITFLA, and 3-PMGITFLA) and three model of the pessimistic multigranulation -fuzzy upper approximation (briefly, 1-PMGITFUA, 2-PMGITFUA, and 3-PMGITFUA) are, respectively, defined as follows. Model 1: Model 2: Model 3:

If (resp., , ) (resp., , ), then is called a covering-based pessimistic multigranulation ()-fuzzy rough set (briefly, 1-CPMGITFRS, 2-CPMGITFRS, and 3-CPMGITFRS); otherwise, it is pessimistic multigranulation fuzzy definable.

It is obvious that the properties of these mentioned models satisfy Theorem 1.

Example 5. (continued from Examples 2 and 3 and Remark 2). We have the following results.Case 1Let us fix based on and and . So,(i) and (ii) and (iii) and Case 2Let us fix based on and and . Thus,(i) and (ii) and (iii) and

3.3. Some Types of Covering-Based Variable Precision Multigranulation -Fuzzy Rough Sets

In the following, six new kinds of CVPMITFRS are defined and their properties are investigated. It is clear that the properties of these mentioned models satisfy Theorem 1. So, we only present the concepts and omit the properties.

Definition 15. Let be a FCAS and be fuzzy -coverings of for some . For each and a variable precision parameter . Define the -variable precision multigranulation -fuzzy lower approximation (briefly, i-VPMGITFLA) and the -variable precision multigranulation -fuzzy upper approximation (briefly, i-VPMGITFUA), are, respectively, defined as follows:(1) and (2) and (3) and (4) and (5) and (6) and If , then is called a covering-based i-variable precision multigranulation -fuzzy rough set (briefly, i-VPMGITFRS); otherwise, it is i-variable precision multigranulation fuzzy definable.
The best way to explain the above definition is to give the following example.

Example 6. (continued from Example 2). Assume that . Then, we have the following results:(1)(2)(3)(4)(5)(6)

4. The Relationships between COMGITFRS Models and CPMGITFRS Models

In this section, we explain relationships among our models. Through the proposed study, we have the following results.

From Definitions 10 and 12, we conclude the following results.

Proposition 1. Let be a FCAS and be fuzzy -coverings of for some . Then, for each , we have the following properties: (i) and . (ii) and .
By Definitions 11 and 13, we have the following results.

Proposition 2. Let be a FCAS and be fuzzy -coverings of for some . Then, for each , we have the following properties:(i) and (ii) and From Definitions 11 and 14, we have the following results.

Proposition 3. Let be a FCAS and be fuzzy -coverings of for some . Then, for each , we have the following properties.(i) and (ii) and

Proposition 4. Let be a FCAS and be fuzzy -coverings of for some . Then, for each , we have the following properties:(i)(ii)(iii)(iv)

Proposition 5. Let be a FCAS and be fuzzy -coverings of for some . Then, for each , we have the following properties:(i)(ii)(iii)(iv)

Remark 3. Let be a FCAS and be fuzzy -coverings of for some . Then, for each , we have the following properties:(1) and (2) and (3) and (4) and (5) and (6) and

5. An Application to Decision-Making

In this section, we apply the proposed method to make a decision on a real-life problem.

5.1. Description and Process

Let be alternatives and be decision makers. Suppose for each is a weighted vector correspondingly to , where for and . Hence, , for all is a set of attributes. A family of mappings , where . So, we construct the MAGDM with fuzzy information system . Based on the proposed covering methods, we present a decision-making algorithm to find the best alternative through the following steps:Step 1:Construct the decision-making object with fuzzy information of the universe of discourse. Through the rule of fuzzy TOPSIS method, we have where and denote “max” and “min,” respectively.Step 2:Compute the respective distances and as follows: where and is the cardinality of .Step 3:Calculate the lower and upper approximations of the best and worst decision-making objects with fuzzy information by Definition 13 (2-OMGITFLA and 2-OMGITFUA).Step 4:Calculate the closeness coefficient degree by , where be the worst and the best decision-making objects for individual ranking function of expert for the candidates , and .Step 5:Calculate the group ranking function by the following equation , and hence rank the alternatives.

According to these steps, we give an algorithm to solve the decision-making problems based on the 2-COMGITFRS model. The steps corresponding to it are summarized in Algorithm 1.

	Input: Fuzzy information systems .
	Output: Decision-Making.
(1)	Enter , , and .
(2)	From Definition 7, calculate fuzzy -neighborhood.
(3)	From Step 2 and by Definition 8, calculate complementary fuzzy -neighborhood
(4)	From Steps 2 and 3, calculate .
(5)	Enter and .
(6)	Calculate the distances and .
(7)	From Definition 13, calculate the lower approximation and the upper approximation .
(8)	Calculate the worst and the best decision-making objects and for each individual decision-maker.
(9)	Calculate the individual ranking function .
(10)	Calculate the group ranking function .
(11)	Obtain the decision.

5.2. Applied Example

The abovementioned steps have been illustrated with a numerical example as shown next.

Example 7 (see [40]). Let be six system analysis engineers and emotional steadiness , oral communication skill , personality , past experience , self-confidence be the attribute set of the basic description of the candidates. Suppose that three experts , and are invited to evaluate the system analysis engineers according to their specialized knowledge. The weights of every expert are , and . The following steps of the stated algorithm are implemented here.Step 1:Experts evaluate each candidate under the set of the attribute and present their judgments with the real values. These values are summarized in Tables 11–13.Step 2:According to the importance of these five attributes, we give the following results for each expert:Step 3:If the threshold , it produces and as displayed in Tables 14–22.Step 4:Calculate the distances and as follows:Step 5:Calculate the lower and upper approximations of the best and worst decision-making objects as follows. Take , and we have Take , and we have Take , and we haveStep 6Based on the importance of these five attributes, we give the worst and the best decision-making objects as follows: Thus, we evaluate a closeness coefficient as follows:Step 7Based on these results, we calculate the group optimal index as follows. , and hence get the ranking order as . From the calculations, we conclude that the 4th system analysis engineer is the best alternative among the others. Furthermore, we get the solution for Case 2 by the same analysis in Case 1. Therefore, we have the group optimal index as follows: and hence get the ranking order as . Through the previous computation, we obtain the 4th system analysis engineer is the best alternative among the others.

5.3. Comparative Analysis

The main aim of the current work is to present a method that increases the lower approximation and decreases the upper approximation of Zhan’s methods in [40]. This can be seen easily from Examples 2– 4. Moreover and by looking at Tables 23 and 24, we can see that the ranking results of the two decision-making models. It is obvious that the optimal selected alternative is the same, although there exist some differences in the ranking results because we choose different decision-making methods.

An easy way to see the effectiveness of our method and the differences between the four models (i.e., our three proposed models and Zhan’s model) are shown in Figures 1 and 2.

(a)

(b)

(a)

(b)

Figure 1 explained the comparisons between the lower approximations for the four models (i.e., 0-OMGITFLA, 1-OMGITFLA, 2-OMGITFLA, and 3-OMGITFLA) for the two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). This figure justifies that the 2-OMGITFLA is better than the others.

Figure 2 clarified the differences between the upper approximations for the four models (i.e., 0-OMGITFUA, 1-OMGITFUA, 2-OMGITFUA, and 3-OMGITFUA) for the two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). This figure illustrates that the 2-OMGITFUA is lower than the others.

6. Conclusion and Future Work

The main aim of the present work is to increase the effectiveness of Zhan’s method by increasing the lower approximation and decreasing the upper approximation. So, based on the concepts of a family of fuzzy -neighborhood (and a family of fuzzy complementary -neighborhood), we introduced new three types of covering-based multigranulation -fuzzy rough sets models and their properties. Furthermore, we give six kinds of covering-based variable precision multigranulation ([[1008]])-fuzzy rough sets. The relationships among these models are investigated. Also, an illustrative example with algorithm is given. Therefore, it is clear to see that 2-COMGITFRS is better than the other models (i.e., 0-COMGITFRS, 1-COMGITFRS, and 3-COMGITFRS).

In future research, we plan to further investigate along with the following: (1) topological properties of the presented methods [44, 45], (2) combination with the soft set and the proposed methods [46, 47], and (3) combination with the neutrosophic set and the current methods [48].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Yulin University-Industry Collaboration Project (2019-75-2).

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Copyright © 2020 Jue Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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