Multiple Criteria Group Decision-Making Method with Dempster–Shafer Theory and Probabilistic Linguistic Term Sets

+e motivation of this study is to propose a novel multiple criteria group decision-making (MCDGM) method based on Dempster–Shafer theory (DST) and probabilistic linguistic term sets (PLTSs) to handle the distinctions between compensatory information at the criterion level and noncompensatory information at the individual level in the process of information fusion. Initially, the information at the individual level is extracted by BPA functions. +en, they are fused with DST considering ignorance and DMs’ reliabilities. Next, the obtained BPA functions are transformed into interval-valued PLTSs with the assistance of intermediate belief and plausibility. Subsequently, the interval-valued PLTSs are converted into standard PLTSs. After normalization, the holistic PLTS is obtained with weighted addition operation and the round function is applied to determine the ultimate evaluation result. Finally, a case simulation study of evaluating the marine ranching ecological security is presented to verify and improve the validity and feasibility of the proposed method and algorithm in practical application. +e proposed method and its relevant algorithm are both innovative combination of DSTand PLTSs from the perspective of compensatory and noncompensatory features of information, which provides a new angle of view for the development of probabilistic preference theory and is beneficial to apply probabilistic preference theory in practice.

Introduction
Multiple criteria decision-making (MCDM) theory is composed of a series of methods that deal with alternative ranking or selection problems that are characterized by multiple attributes and conflicting noncommensurable objectives. Introduced in the 1950s [1,2], MCDM has since received extensive attention from two kinds of methods: the multiple attribute decision-making (MADM) method and the multiple objective decision-making (MODM) method. MCGDM is an indispensable branch of modern decisionmaking theory. e representative methods of MCDM, such as multiobjective programming, analytic hierarchy process (AHP), ELimination Et Choice Translating REality (ELECTRE), and technique for order preference by similarity to an ideal solution (TOPSIS), have been applied in solving diversified practical MCDM problems [3][4][5][6][7][8]. As decision-making problems are getting more complicated, researchers have studied MCDM from the perspectives of utility achievement, weight determination, consistency measurement, fuzziness information, and so on [9][10][11][12][13].
MCDM methods usually comprehensively analyze MCDM problems from all angles. eir applications are achieved by balancing multiple attributes or objectives to find the best one among several given alternatives. In the implementation process of classical MCDM, an individual decision maker (DM) is the single source providing the decision information that involves all the factors and forms the final decision results. However, limited by the incomplete knowledge reserve, restricted ability of judgment, inadequate experience, and other such issues, an individual DM cannot provide sufficient valid information for today's complex and synthetic MCDM. Consequently, multiple criteria group decision-making (MCGDM) methods [14][15][16] have been proposed to solve MCDM problems that arise under the group decision-making setting. e major MCGDM methods provide a basic framework for implementation that comprises extracting individual information first and then fusing it into group information. Here, two kinds of problems are of vital importance. One is about information expression and extraction, and the other is about information disposal and fusion. With respect to the former, existing literature has a lot of focus on enhancing the accuracy of the information given by each individual DM. Linguistic term, interval value, fuzzy number, membership, and other forms of information [17][18][19][20] are incorporated to facilitate DMs determining or expressing their preference. e information characteristics of uncertainty, fuzziness, and incompleteness [21][22][23][24] are also key concerns of MCGDM. With respect to the problem of information disposal and fusion, researchers have proposed several fusion methods to fuse individual information, such as the consensus model, fuzzy operators, Dempster-Shafer theory (DST), evidence reasoning (ER) approach, and probabilistic linguistic term set (PLTS) operations [25][26][27]. Different disposal and fusion processes usually cause different degrees of information loss, which affects the feasibility and validity of the decision results. To address this issue, several solutions, such as information conflict, reliability and weight of single information, consistency measure, and correlation measure [28][29][30][31] have been proposed in the published literature.
e motivation of this study is to propose a novel MCGDM method based on DST and PLTSs to overcome the problem of distinctions between information at different dimensions potentially disturbing the process of information fusion and thus influencing the final result of the MCGDM. ese distinctions are performed since the information at the individual level is noncompensatory and that at the criterion level is compensatory in the process of information fusion. Note that MCGDM is the combination of MCDM and group decision making (GDM). In MCDM, the information at the criterion level is fused into a holistic score or value pertaining to the whole alternative, and according to this, the alternatives are ranked or selected [32]. In GDM, the final group decisions are derived through fusing information at the individual level [33]. e information involved in MCGDM is related to both the criterion and individual level. erefore, the above distinctions should be considered carefully in MCGDM and distinct fusion rules corresponding to different information dimensions should be utilized.
In the process of implementing MCGDM, a set of criteria are selected to characterize the alternative in different aspects that are essential for the MCGDM problem [34]. For a specific MCGDM problem, the alternative always has good or poor performance (information at the criterion level) on different criteria with different weights, and these performances can compensate for each other [34][35][36][37][38] in additive fusion. Additive fusion is a process of trade-off where poor (good) performance on one criterion can be offset by good (poor) performance on other criteria [34] and performance on a high (low) weighted criterion can be offset by performance on a low (high) weighted criterion [36]. Consequently, the advantages or weaknesses of the alternative may not be revealed in the final comprehensive results [35]. Based on the above features, we believe that the fusion of information at the criterion level should follow compensatory strategies, which means that the holistic performance of the alternative is derived by adding performances on the criteria with equal or unequal weights [38]. In the published literature, common compensatory models are weighted additive, including but not limited to simple additive weighting (SAW), TOPSIS, AHP, and fuzzy AHP [39]. Among the existing compensatory methods, PLTSs [27] are an effective technique owing to their advantages of linguistic preference expression, probability measure, and multiple operations [40,41]. PLTSs and its variants have solved various practical problems. For example, the reliable participant selection problem in mobile crowdsensing has been solved by a probabilistic linguistic VIKOR method based on TODIM [42], the edge node selection problem in edge computing has been solved using the probabilistic linguistic ELECTRE II method [43], and the problem of evaluating the Internet of things platforms has been solved using an integrated probabilistic linguistic MCDM method [44].
In addition to the criteria, individual DMs constitute another dimension in the process of MCGDM. DMs in a group are invited to give judgment information based on their preference about the performance of the alternative on a single criterion, and information at the individual level is then fused into group judgment information on a single criterion or an alternative (after information fusion at the criterion level). However, limited by different degrees of bounded rationality [38,45], the information given by different DMs is usually uncertain and has different degrees of reliability, which is a key factor in the fusion of information at the individual level. When a DM is definitely reliable, their judgment information is assigned total belief in the process of fusion. If a DM is not entirely reliable, their judgment information is assigned a certain degree of belief instead of total belief in the process of information fusion. In other words, the fusion result is relative to the most reliable judgment and cannot be reserved by others [34,37,38]. Influenced by lower reliable information, information at the individual level is incomplete, and therefore, the compensatory strategies cannot be applied in the process of fusion [38]. Hence, noncompensatory strategies are employed. Compared with the analytic cognition in compensatory strategies, noncompensatory strategies are akin to intuitive cognition [46]. Noncompensatory strategies are more suitable for explaining DMs' decision-making behavior because of their lower computational demand and dependence on the most reliable information [36,38]. In the published literature, classical noncompensatory models include conjunctive, disjunctive, elimination by aspects, lexicographic, and "take the best" [37]. DST [26,47] is a superior method for noncompensatory strategies [48] because of its ability of expressing and fusing uncertain information. DST and its subsequent versions have been employed to solve multiple practical problems, such as mapping flood susceptibility [49], predicting rolling bearing faults [50], and determining artificial recharge location [51], among others.
To sum up, compensatory and noncompensatory strategies are two indispensable information fusion strategies that are suitable when information belongs to different dimensions. However, the published MCGDM methods have not considered the distinctions between compensatory information at the criterion level in MCDM and noncompensatory information at the individual level in GDM, which could result in compromised decision validity and quality. As MCGDM problems are getting more and more complex, it is advisable and effective to choose both type of strategies in the process of information fusion. We propose a new MCGDM method that considers both compensatory and noncompensatory strategies. e rest of this paper is organized as follows. In Section 2, we introduce the background knowledge about DST and PLTSs. In Section 3, a novel MCGDM method is proposed considering both compensatory and noncompensatory strategies based on DST and PLTSs, and the corresponding algorithm is constructed. A case simulation study of evaluating marine ranching ecological security (MRES) is presented to test the scientific validity and practical feasibility of the proposed method and algorithm in Section 4. Section 5 concludes the paper.

Dempster-Shafer eory.
DST is an uncertainty reasoning technique proposed by Dempster [47] and Shafer [26]. It can handle uncertain information at the individual level on the basis of a frame of discernment, which is composed of a set of mutually exclusive and collectively exhaustive propositions. e basic probability assignment (BPA) function is applied to extract the uncertain information.
Definition 1 (see [47]). Suppose a possible proposition is θ n (n � 1, 2, . . . , N), and each of the proposition is exclusive. en, a finite nonempty exhaustive set of all propositions Θ � θ 1 , . . . , θ N is called a frame of discernment, and its power set that consists of 2 N subsets of Θ is usually expressed as Definition 2 (see [47]). Suppose θ is a nonempty subset of Θ, and its belief is p(θ). If the mapping function p: where ∅ represents the empty set, then p(θ) is called the BPA function. If p(θ) > 0, θ is named a focal element. p(θ) is a basic probability that is assigned exactly to θ and not to any smaller subset. Considering the situation that counterintuitive problems [52,53] may impede the combination of evidences with the orthogonal sum operator in Dempster's rule, Shafer proposed a discounting method, named Shafer discounting, to solve these kinds of problems.
Definition 3 (see [26]). Suppose the belief distribution given by a piece of evidence e i that points to a proposition θ is p θ,i , and w i is the weight of evidence e i that is used to discount p θ,i with 0 ≤ w i ≤ 1, ∀i and I i�1 w i � 1. en, the Shafer discounting can be defined to modify the BPA function for the evidence as follows.
Definition 4 (see [26]). Suppose the discounted BPA functions of two pieces of evidence e 1 and e 2 are, respectively, m 1 and m 2 on Θ, and ⊗ is the orthogonal sum operator. For any θ ⊆ Θ, Dempster's rule is described as follows.
A belief measure associated with the BPA function and composed of the belief function and the plausibility function is necessary. e belief function measures the total mass that must be allocated to the elements of a certain subset of Θ, and the plausibility function measures the maximal amount of mass that can be allocated to the elements of a certain subset of Θ.
Definition 5 (see [47]). Suppose m(θ) is a piece of BPA function on the frame of discernment Θ. en, the belief Mathematical Problems in Engineering 3 function and the plausibility function of a subset θ of Θ can be denoted by where Pl(θ) ≥ Bel(θ) and Pl(θ) � 1 − Bel(θ). θ is the classical complement of θ. In equation (5), Bel(θ) is known as the belief function that represents the lower limit of the belief level of θ, and Pl(θ) is known as the plausibility function that represents the upper limit of the belief level of θ. erefore, the confidence interval [Bel(θ), Pl(θ)] can describe the uncertainty about θ, where Bel(θ) and Pl(θ), respectively, describe the minimal and maximal uncertainty about θ.

Probabilistic Linguistic Term Sets.
PLTSs [27] are an extension of hesitant fuzzy linguistic term sets (HFLTSs) [54] with the intention of providing a more convenient way for DMs to express their preference with probability. A series of operational laws and aggregation operations are also introduced to facilitate the fusion of information. We firstly introduce the definition of linguistic term sets (LTSs).
Actually, HFLTSs mainly consider the situation wherein DMs hesitate among several optional linguistic terms in the moment of making their decision. HFLTSs ignore the probabilistic messages associated with linguistic terms [27]. erefore, PLTSs are proposed by introducing probabilistic messages in LTSs as follows.
Definition 7 (see [27]). Suppose S � s α |α � 0, 1, . . . , τ is an LTS. en, its relative PLTS is denoted by where L (k) (p (k) ) means that the associated probability of the kth linguistic term L (k) in L(p) is p (k) and #L(p) is the total number of the linguistic terms in PLTS L(p). It is worth noting that #L(p) k�1 p (k) represents the degree of information completeness with respect to the probabilistic distribution on all possible linguistic terms. We have #L(p) k�1 p (k) � 1 under complete information, #L(p) k�1 p (k) � 0 under complete ignorance, and 0 < #L(p) k�1 p (k) < 1 under partial ignorance, where the ignorance represented by 1 − #L(p) k�1 p (k) should be averagely assigned to all of the linguistic terms in L(p).
e PLTSs in equation (6) are called standard PLTSs [56] with a certain point value of probability. e interval value of probability is more suitable for solving practical decisionmaking problems because of its consideration of DMs' vagueness. However, it is also less computable. erefore, Gu et al. [56] provided a method for converting intervalvalued PLTSs into standard PLTSs as follows.
Definition 8 (see [56]). Suppose an interval-valued PLTS is given as en, the method for converting interval-valued PLTSs into standard PLTSs is denoted by where l (k) and u (k) are, respectively, the lower limit and upper limit of the given interval value of probability. Obviously, standard PLTSs are special interval-valued PLTSs when l (k) � u (k) . Certain problems are caused by the different numbers of linguistic terms in PLTSs. erefore, an extension rule for PLTSs is introduced. e normalization of PLTSs to avoid information distortion [57] follows the rules of association and extension, as below.

Definition
11 (see [27]). Given a PLTS L(p) � L (k) (p (k) )| k � 1, 2, . . . , #L(p) , let r (k) be the subscript of the linguistic term L (k) . en, the ordered PLTS is obtained by arranging the elements in the normalized PLTS in accordance to the values of r (k) p (k) in descending order.

Remark 1.
Considering the situation wherein two or more elements in a PLTS have equal values of r (k) p (k) , a complement for the ordering rule as in Definition 11 is introduced. Generally, when the values of r (k) in r (k) p (k) are unequal, the chosen elements in L(p) are arranged according to the values of r (k) in descending order. Otherwise, the chosen elements in L(p) are arranged according to the values of p (k) in descending order [57].
Based on the above definitions, some operations are defined as follows.
Definition 12 (see [27]). Given two ordered PLTSs L 1 (p) and , then the addition operation is defined as where L (k) 1 and L (k) 2 are the kth linguistic terms in L 1 (p) and L 2 (p), respectively, and the condition λ 1 S α ⊕ λ 2 S β � S λ 1 α+λ 2 β is satisfied [58]. en, the uniqueness of PLTSs compared with the ordinary LTSs, that is, probability, is not revealed. Zhang et al. [59] defined a new operation for normalized and ordered PLTSs as where L are the k 1 th and k 2 th linguistic terms in L is a normalized PLTS. Virtual linguistic terms may exist when PLTSs satisfy λ 1 S α ⊕ λ 2 S β � S λ 1 α+λ 2 β . en, the round function is as follows.

Definition
13. (see [57]). Given a PLTS L(p) � L (k) (p (k) )| k � 1, 2, . . . , #L(p) , the integer score (linguistic term) of L(p) is denoted by is the subscript of the kth linguistic term L (k) , andround(s α ) is the classical round function that is used to determine the linguistic term that nearest to s α with the rounding-off method.

Extraction of Information.
MCGDM is usually conducted by a group of DMs who evaluate a set of given alternatives according to their own preference, with the aim to seek a satisfactory alternative or alternative rank based on a common criteria system. e only information sources in MCGDM are the DMs in the group, who may come from various fields with different degrees of knowledge reserves, experience, and cognitive styles, leading to diverse judgment information at the beginning of the MCGDM. e judgment information is subjectively given by DMs with different degrees of epistemic uncertainty, which could generate inaccurate decisions. erefore, considering the uncertainty along with the preference expression of DMs is of vital importance in the extraction of information. In this regard, the BPA function in DST is an appropriate tool for extracting information, as it provides a unified way to represent preference and model uncertainty [60,61].
For a specific MCGDM problem of evaluating alternatives, we give the following description. In order to solve the MCGDM problem, a group of DMs are invited to make judgment about the alternative on each given criterion. e DM set and criterion set are denoted by E � e i | i � 1, 2, . . . , I and C � c n | n � 1, 2, . . . , N , where I and N represent the number of DMs and criteria, respectively. A set of linguistic terms denoted by Θ � θ j | j � 1, 2, . . . , J is given for facilitating DMs to make their judgment based on their own preference with uncertainty. θ j refers to the jth linguistic term in the given J ones. Linguistic terms in Θ are actually a series of variable values expressed in a way that conforms to human language, and each of them is mutually exclusive and exhaustive. erefore, Θ is a finite nonempty set, called the frame of discernment, according to Definition 1. en, the preference information given by DM e i on criterion c n can be extracted by a piece of the individual BPA function satisfying Definition 2 as Mathematical Problems in Engineering where θ is an element of the power set of the frame of discernment P(Θ) as in equation (1) and p n i (θ) is the probability related to θ. In terms of extracting and expressing information at the individual level, the BPA function is more suitable than PLTSs to reflect the actual cognitive competence of the DMs. Since PLTSs can just reflect judgment using single linguistic term with associated probability, the BPA function additionally has the ability of presenting judgment information with local ignorance. As shown in equation (12), the value of θ describes the completeness of the DMs' recognition. When θ � θ j and P n i � (θ j , p n i (θ j ))| p n i (θ j ) � 1}, we believe that DM e i has complete recognition about the alternative's performance on criterion c n , and he/she can give a certain judgment with no ignorance [63]. When θ � Θ and P n i � (Θ, p n i (Θ))| p n i (Θ) � 1 , it is believed that DM e i can recognize nothing about the alternative's performance on criterion c n , and he/she makes judgment with global ignorance [62]. P n i � (θ, { p n i (θ))| p n i (θ) � 1} means that DM e i has incomplete recognition about the alternative's performance on criterion c n , and he/she can make judgment with local ignorance [62]; that is, DM e i believes that the alternative's performance on criterion c n can be depicted by linguistic terms in θ, but he/ she cannot decide which one in θ is the best. Equation (12) represents a more general case wherein DM e i believes that the alternative's performance on criterion c n can be depicted by θ with several different values, and he/she can give relative probabilities to each of them.

Fusion of Information at the Individual Level.
With the application of the BPA function, the DMs' judgment information about the alternative's performance on each criterion with corresponding probability is obtained. en, the information at the individual level needs to be fused to get the group judgment about the alternative's performance on each criterion. It is worth noting that the BPA functions from different DMs are always associated with different degrees of uncertainty caused by objective environmental factors, subjective cognitive factors, and other factors. Even though the BPA function has abilities to express the judgments of "uncertain" or "unaware," the uncertainty caused by differences among DMs still needs to be attended. Generally, the differences among DMs are reflected in various aspects of the DMs, such as knowledge, experience, background, and authority, resulting in different degrees of reliabilities that play an important role in the process of information fusion for a particular MCGDM problem. Accordingly, the reliability of a DM is a key factor that needs to be considered in the fusion of information at the individual level.
From a group point of view, the reliability of DM e i actually measures the degree of influence brought by his/ her individual BPA function on the group result relative to the BPA functions given by other DMs. However, as mentioned in Section 1, this kind of influence is dominant in the process of information fusion, and influence with a low degree of reliability cannot offset that with a high degree of reliability. In other words, the information at the individual level, which is formalized as the individual BPA function, is noncompensatory, and the group result depends more on the individual BPA function given by a DM with greater reliability. For example, in the problem of evaluating the school performance of a student by three teachers, the final evaluation result is usually influenced more by (or even determined by) the teacher who knows the evaluated student more because their judgment is considered the most reliable. Suppose that the three invited teachers are e 1 , e 2 , and e 3 . Teachers e 2 and e 3 just have information about the evaluated student in terms of learning and activities, and their judgments are given with local ignorance as (poor, average) and average, respectively. Teacher e 1 , who has more information about the evaluated student in terms of various aspects, gives judgment as poor because he/she knows about a punishment the evaluated student has received. As a result, the final evaluation result should be poor because the judgment given by the most reliable teacher e 1 has vote power. In this example, the information given by teacher e 1 has dominant influence on the evaluation result and cannot be compensated by information given by teachers e 2 and e 3 . Similarly, the judgments given by each DM in an MCGDM problem are also noncompensatory. In this scenario, DST is an effective tool for handling noncompensatory information in a more rational way [63]. In DST, an evidence with 100% reliability is totally supported; thus, it can have dominant influence on the fusion result. Otherwise, residual supports are assigned on the frame of discernment [63], which makes it possible that the residual supports are assigned on any element in the frame of discernment rather than to compensate other evidences.
Consequently, we applied DST to fuse information at the individual level. Suppose that the reliability set of DMs is R � r i | i � 1, 2, . . . , I . Subsequently, the extracted individual BPA functions as in equation (12) are first discounted with Shafer's discounting by taking P n i and R into equation (3). e discounted individual BPA functions can be represented as where m n i (θ) is the discounted probability associated with the linguistic terms in θ, and we have m n i (θ) � r i p n i (θ) + (1 − r i ) when θ � Θ; otherwise, m n i (θ) � r i p n i (θ). Next, the discounted individual BPA functions are fused with Dempster's rule by taking M n i into equation (4). For any two given individual BPA functions from DM e i and e i′ , the fusion formula is described as 6 Mathematical Problems in Engineering Accordingly, I pieces of individual BPA functions as demonstrated in equation (12) given by DM e 1 , e 2 , . . . , e I can be recursively fused into a group BPA function relative to criterion c n . In the above process of applying DST, the reliabilities of DMs are used to discount the extracted information, where the dominant influences of DMs with higher reliabilities on the group BPA function are enhanced. Furthermore, the group judgment depends more on BPA functions with higher degrees of reliabilities under the application of the orthogonal sum operator in Dempster' rule. As a result, the influence of each individual BPA function on the group BPA function cannot compensate one another in the process of fusion, and the influences of the total reliable individual BPA functions could be entirely brought to the group BPA function with no discount. Eventually, the group BPA function relative to criterion c n is derived based on noncompensatory strategies as follows.

Transformation from BPA Functions into PLTSs.
Based on the demonstration of DST, we have obtained N pieces of group BPA functions that describe the group judgments about the alternative's performance on each criterion. In fact, the expression of information by the BPA functions facilitates handling uncertainty in DMs' judgments through assigning probabilities on the power set of the frame of discernment. When probabilities are assigned to single elements of the frame of discernment like θ j , we believe that the judgment is precise with no ignorance. When probabilities are assigned to a subset including more than one element like θ, we believe that the judgment is "uncertain" with local ignorance. When probabilities are assigned to the frame of discernment itself like Θ, we believe that the judgment is "unaware" with global ignorance. However, these kinds of data make it difficult and complicated to fuse the group judgments on each criterion. As mentioned above, the information at the criterion level, that is, the group BPA functions demonstrated by equation (15), is actually compensatory in the process of fusion. us, the influence given by the group BPA function on some criterion can offset that given by others in the process of fusion. erefore, the fusion result of the information at the criterion level can reflect the comprehensive performance on all criteria and needs to be obtained with the weighted additive fusion rule. e information expression form of the BPA function is not suitable for the weighted additive fusion rule because of its assignment of probabilities. In order to establish links between information at the criterion level (group BPA functions) and the application of the weighted additive operations, the transformation method is proposed to convert the group BPA functions as in equation (15) into a simple expression appropriate for compensatory model, such as PLTSs.
In general, the probability in the BPA function usually describes the support degree of a proposition obtained from DMs. e higher the degree of support, the more certain is the proposition. In other words, the probabilities in the BPA functions are actually a kind of representation of uncertainty arising from DMs' judgments. In DST, this kind of uncertainty is measured by two functions with respect to each single linguistic term, namely, the belief function and plausibility function, which separately measure the lowest and highest probability of the proposition being true [64]. By taking the group BPA function as in equation (15) into equation (5), we can derive the belief and plausibility of the group BPA function M n as Bel n (θ j ) and Pl n (θ j ), which, respectively, depict the lower and upper limits of probabilities relative to single linguistic term θ j on criterion c n . From the perspective of PLTSs, Bel n (θ j ) and Pl n (θ j ) actually constitute the probability associated with linguistic term θ j in the form of an interval value. For convenience, illustration, and understanding, we redefine the frame of discernment Θ � θ j | j � 1, 2, . . . , J as an LTS S � s α | α � 0, 1, . . . , τ that satisfies Definition 6, where s 0 � θ 1 , s 1 � θ 2 , . . ., s τ � θ J . Bel n (θ j ) and Pl n (θ j ) are redefined as l (k) n and u (k) n , which separately represent the lower and upper limits of probability relative to the kth linguistic term in LTS S on criterion c n . Accordingly, the group BPA function M n as in equation (15) can be transformed into a PLTS L n (p) as follows according to Definition 8.
Mathematical Problems in Engineering 7 where L (k) n represents the kth linguistic term relative to criterion c n and #L n (p) represents the number of linguistic terms in L n (p). Note that standard PLTSs are a special case of interval-valued PLTSs as in equation (16) when l (k) n � u (k) n .

Fusion of Information at the Criterion Level.
e obtained group judgments, which are demonstrated in the form of interval-valued PLTSs on each criterion, need to be further fused to derive the holistic judgment about the whole alternative. As mentioned in Section 1, the group judgments on each criterion are actually information at the criterion level and compensatory to each other. For the whole alternative or the MCGDM problem, each criterion is selected according to a set of principles, such as systematic, typical, scientific, and comprehensive, so that the constructed criterion system can consider aspects that are integrated and valid enough for describing the MCGDM problem. In other words, each criterion in an MCGDM problem describes an essential aspect of the alternative, and the final decision is made by comprehensively combining the alternative's performance on all aspects, which is a process of trade-off. For example, in the problem of evaluating a student from the aspects of academic record (c 1 ), student leader work (c 2 ), and activity achievements (c 3 ), bad performance on one aspect can usually be compensated by good work on other aspects because the school always train students following the principle of integrated development and any single aspect cannot decide the evaluation results. After a trade-off, a student with good performance on criterion c 1 , excellent performance on criterion c 2 , and poor performance on criterion c 3 may be evaluated as a good student because of the mutual compensation of performance on each aspect. Similarly, the information at the criterion level in an MCGDM problem is also compensatory. erefore, the obtained PLTSs on each criterion as in equation (16) need to be fused into a holistic result in accordance with compensatory strategies.
As demonstrated in equation (16), the obtained PLTSs contain the information of criterion c n , associated linguistic term L (k) n , and its corresponding interval value of probability [l (k) n , u (k) n ]. As mentioned above, the interval value of probability is more suitable to demonstrate the vagueness of the linguistic term. However, it is not applicable for fusion operations on PLTSs. In order to solve this problem, several researchers have made strong attempts [65][66][67] by introducing new aggregation operators and comparison laws for interval values, which provide new calculation methods to accommodate the vagueness of linguistic terms and complicated calculations. We believe that the method as in equation (7) proposed by Gu et al. [56] is simple and applicable to deal with interval-valued probability. Additionally, it should be noticed that the situation of Bel n (θ j ) � Pl n (θ j ) actually exists in practice when DMs only definitely assign the probabilities with no ignorance on single elements of the frame of discernment rather than any subset including more than one element. us, it is possible to have l (k) n � u (k) n when the interval value of probability in the PLTSs can be regarded as a certain point value, that is, (7) does not work under this circumstance. Accordingly, the obtained interval values of probabilities in the PLTSs are first converted into equivalent certain point values of probabilities as follows.
Hence, the interval-valued PLTSs as in equation (16) can be converted into standard PLTSs as L n (p) � L (k) Afterwards, the procedure of fusing information at the criterion level can be implemented through normalization of the standard PLTSs L n (p). In conformity to the procedure of normalization of PLTSs, association and extension need to be carried out. However, the situation explained in Definition 9 is out of consideration here because #L n (p) � #L n (p) and #L n (p) k�1 p (k) n � 1, which can be easily proven. erefore, there is no need for association. en, the PLTSs L n (p) need to be extended according to Definition 10. e probabilities assigned on the added linguistic terms are zero. en, the extended PLTSs are arranged according to the value of r (k) n p (k) n in descending order, where r (k) n is the subscript of linguistic term L (k) n (see Definition 11). As a result, the normalized and ordered PLTSs are obtained as n ≥ 0, k � 1, 2, . . . , #L n (p),

Mathematical Problems in Engineering
Subsequently, we need to choose an operation to derive the fusion result of information at the criterion level, where the weight of the criterion is also a necessary factor that has substantial effect on the fusion result. erefore, we apply the redefined addition operation [19] to fuse the normalized and ordered PLTSs. Let W � w n |0t ≤ nw n q ≤ h1 , x N n�1 w n � C1; , n � 1, 2, . . . , N} be the weight set corresponding to criterion set C � c n | n � 1, 2, . . . , N . en, any two given PLTSs L n (p) and L n′ (p) satisfy w n S α ⊕ w n′ S β � S w n α+w n′ β , α, β � 0, 1, . . . , τ. Based on Definition 12, the weighted addition operation of any two given PLTSs L n (p) and L n′ (p) is as follows.
where (20) in which L (k δ ) δ , k δ � 1, 2, . . . , #L δ (p) represents the k δ th linguistic term in the fusion result of PLTSs L 1 (p) and L 2 (p). e value of L is the probability relative to the linguistic term L (k δ ) δ in the fusion result, which is determined based on the redefined addition operation as in equation (10). For convenience of understanding, here we introduce a simple example to illustrate.  (19) and (20) is as follows.
Subsequently, the information at the criterion level is fused into a piece of holistic PLTS relative to the whole alternative by recursively repeating the fusion as in equations (19) and (20) en, for n � 2, we conduct the first fusion and have w Δ � w 1 , and 2 . For n � 3, we conduct the second fusion and have w Δ � w Δ + w 2 , and 3 . In general, for ∀n � 2, . . . , N, Finally, the obtained holistic PLTS that describes the group judgment on all criteria is denoted by Obviously, virtual linguistic terms may exist in L(p) (see Example 1). e ultimate result of MCGDM is calculated by taking L(p) into equation (11), and we have

MCGDM Process and Algorithm.
For convenience of illustration, the operational process of the proposed method is summarized in Figure 1. In summary, we present a novel MCGDM method using DST and PLTSs for distinguishing the noncompensatory information at the individual level and compensatory information at the criterion level in the process of information fusion. Firstly, the extraction of information based on the BPA function in DST is introduced considering ignorance that exists in DMs' judgments. en, DST is applied to discount and obtain the group judgment on each criterion, where the reliabilities of DMs are considered. e information at the individual level is fused based on the noncompensatory strategy. In order to establish links between the group judgment formed in the group BPA function and the compensatory fusion strategy for information at the criterion level, we propose a transformation from BPA functions into PLTSs based on the belief and plausibility function in DST. Consequently, we obtain the information at the criterion level, which is in the form of interval-valued PLTSs. us, a method for converting interval-valued PLTSs into standard PLTSs is constructed. Finally, the standard PLTSs on each criterion are fused into a holistic PLTS with weighted addition operations, and the final MCGDM result is determined based on the round function. To facilitate practical demonstration, the above process is simplified as in the following Algorithm 1.

Case Simulation Study
In order to verify and improve the scientific validity and practical feasibility of the proposed method and algorithm, we give a case simulation study of evaluating marine ranching ecological security in this section to offer a simulative demonstration of the proposed MCGDM method. According to the definition given by the Chinese Academy of Aquatic Sciences, marine ranching is a fishery mode based on the principle of marine organisms to breed, grow bait, or avoid enemies, through releasing artificial reefs and stock enhancement. It is conductive to increase and protect fishery resources, improve the marine ecological environment, and realize sustainable utilization of fishery resources [57] (Aquatic Industry SC/T9111-2017). As a fishery mode that safeguards economic, social, and environmental revenue, marine ranching has received substantial attention in terms of various aspects, especially MRES. e objective of the case simulation study is to evaluate the MRES of a marine ranch by applying the proposed method and algorithm.
MRES is a kind of integral balance of the marine resources, marine environment, fishery activities, and other human activities. In order to keep this kind of balance, humans perform actions of releasing artificial reef and cultivating seaweed and other measures to develop a habitat for the target species. Other environmental actions are also taken to protect and improve the marine ranching ecosystem. Enhancement of fishery resources and sustainable increase of the fishery output are two main components of MRES. Consequently, we believe that the problem of evaluating MRES of a specific marine ranch includes five indispensable criteria: special funds acquisition and management (c 1 ), cost and profit of the marine ranch (c 2 ), monitoring system and regular inspections (c 3 ), main biomass index (c 4 ), and seawater and seaweed bed quality (c 5 ). Suppose five DMs, namely, marine environmental monitoring technician (e 1 ), aquaculture expert (e 2 ), manager of the evaluated marine ranching (e 3 ), marine environment and resource specialist (e 4 ), and local governmental personnel (e 5 ), are invited to participate in the evaluation of MRES. We denote the criterion set and DM set by C � c n | n � 1, 2, . . . , 5 and E � e i | i � 1, 2, . . . , 5}. Correspondingly, the weight set of the criteria and reliability set of the DMs are obtained as W � (0.150, 0.050, 0.100, 0.400, 0.300) and R �(0.680, 0.920, 0.600, 1.000, 0.800), respectively.
e DMs need to make judgment about the performance of the specific marine ranch on each criterion based on the given frame of discernment that includes three linguistic terms: Θ � (θ 1 � bad, θ 2 � average, θ 3 � good).

Extracting Information from the Five DMs.
Based on the above description, the individual information on each criterion given by each DM is extracted with the BPA function as in equation (12), as shown in Table 1.
It is clearly shown in Table 1 that DMs e 1 , e 2 , . . ., e 5 give their judgments about the MRES performance of the evaluated marine ranch on five criteria relative to the power set of frame of discernment with probabilities. For example, the extracted information given by DM e 1 on criterion c 1 can be expressed as P 1 1 � (θ 2 , 0.300), (θ 3 , 0.600), (Θ, 0.100) , which means that DM e 1 believes that the MRES performance of the evaluated marine ranch on criterion c 1 has 30% probability to be average (θ 2 ), 60% probability to be good (θ 3 ), and 10% probability that the DM cannot make judgment with global ignorance. As another example, DM e 5 believes that the MRES performance of the evaluated marine ranch on criterion c 2 can be evaluated as average (θ 2 ) or good (θ 3 ); he/she only has 10% probability to hold average (θ 2 ). For the remaining 90% probability, he/she cannot Information at criterion level Noncompensatory strategy: Dempster's rule R Compensatory strategy: weighted addition operation Mathematical Problems in Engineering decide which one of the two linguistic terms ( θ 2 , θ 3 ) is the best. Accordingly, the extracted information is P 2 5 � (θ 2 , 0.100), ( θ 2 , θ 3 , 0.900 ) , which includes local ignorance. DM e 3 thinks that the MRES performance of the evaluated marine ranching on criterion c 1 can definitely be evaluated as average (θ 2 ); thus, his/her individual BPA function on criterion c 1 is P 1 3 � (θ 2 , 1.000) with no ignorance.

Fusing Information at the Individual Level.
As mentioned in Section 3.2, the information given by different DMs has different degrees of reliabilities in the problem of evaluating MRES. To be specific, MRES is a complicated MCGDM problem with various aspects that are represented as the five criteria in this case. However, not every DM has enough ability, knowledge, and experience to give reliable judgment, as MRES involves many professional fields, such as government policy standards, marine sediment quality, marine environment monitoring technology, and so on. For the problem of evaluating MRES in this case, the invited marine environment and resource specialist (e 4 ) is authoritative in the field of marine environmental protection and marine resource management, and his/her judgment is the most reliable in the judgments given by the five DMs. is DM's reliability is 1.000. e marine environmental monitoring technician (e 1 ), aquaculture expert (e 2 ), and local governmental personnel (e 5 ), respectively, have a certain degree of abilities to make judgment with reliabilities of 0.680, 0.920, and 0.800. Even though the manager of the evaluated marine ranch (e 3 ) knows a lot about the operation and management of the evaluated marine ranch, such as the cost and profit of the marine ranch (c 2 ), he/she has the lowest reliability of 0.600 in the synthetic evaluation of the MRES. By taking the DM reliability set R and the information at the individual level in Table 1 into equation (13), we obtain the discounted individual BPA functions as follows.
In the process of fusion, the information from the DM with lower reliability (e.g., DM e 3 ) cannot compensate for the information from a DM with a higher reliability (e.g., DM e 4 ). So, the fusion result is effective and reliable as much as possible.
erefore, the fusion rule based on noncompensatory strategies, such as Dempster's rule, is applied, and the group BPA functions representing the group judgment of the five DMs on each criterion are obtained by taking the discounted BPA functions in Table 2 into equation (14). e obtained group BPA functions are shown in Table 3.

Transforming Group BPA Functions into PLTSs.
As mentioned in Section 3.3, the BPA function provides a unified way to model and dispose the uncertainty existing in EndFor % Fuse information at the criterion level based on compensatory strategies % Convert interval-valued PLTSs into standard PLTSs For n � 1 to N For k � 1 to #L n (p) EndFor % Extend and arrange the linguistic terms in standard PLTSs and form the normalized and ordered PLTSs #L n (p) k�1 p (k) n � 1 EndFor % Fuse the normalized and ordered PLTSs with the weighted addition operation EndFor End ALGORITHM 1: MCGDM with DST and PLTSs. Table 3, the group BPA function on criterion c 1 is M 1 � (θ 1 , 0.011), (θ 2 , 0.924), (θ 3 , 0.031), ( θ 1 , θ 2 , 0.022 ), ( θ 2 , θ 3 , 0.008), (Θ, 0.003) }, which contains both local ignorance and global ignorance of the DM group. e group BPA function on criterion c 5 is M 5 � (θ 1 , 0.097), (θ 2 , 0.903) , which assigns the probabilities to single elements with no ignorance. Irrespective of the kind of ignorance contained, all information in the group BPA functions is associated with a degree of uncertainty, which is measured by the belief and plausibility function in DST. erefore, we calculate the belief and plausibility of the group BPA functions by taking the group BPA functions in Table 3 into equation (5). e calculation result is provided in Table 4.

DMs' judgments. As is shown in
As a result, we obtain the lower and upper limits of the probability assigned to each linguistic term. For convenience of illustration and understandability, we redefine the LTS and its associated value limit of probability. According to the explanation in Section 3.3, the frame of discernment Θ � (θ 1 � bad, θ 2 � average, θ 3 � good) is redefined as an LTS S � (s 0 � bad, s 1 � average, s 2 � good), and the associated value limit of the probability is redefined each to each. For example, the belief (Bel 2 (θ 1 )) and plausibility (Pl 2 (θ 1 )) of the group BPA function M 2 relative to linguistic term θ 1 are redefined as l (1) 2 and u (1) 2 , respectively. Accordingly, the group BPA functions in Table 3 are transformed into interval-valued PLTSs as shown in Table 5.  Table 5 into equation (17), we derive the standard PLTSs. After normalization and arrangement, the normalized and ordered PLTSs are obtained (Table 6), where r (k) n is the subscript of the kth linguistic term in the normalized and ordered PLTSs and r (k) n p (k) n is the value that is used to arrange the linguistic terms in descending order.
Generally, each PLTS in Table 6 describes the information at the criterion level. As illustrated in Section 3.4, each criterion of MRES is an independent and indispensable aspect, and all criteria compose the MCGDM problem jointly. In this case, the evaluation of MRES includes five criteria, and the information at the criterion level is Table 4: e belief and plausibility of the group BPA functions.
L (50) 0.3136 0.0000054 L (51) 0.3147 0.0001764 L (52) 0.3157 0.0001194 L (53) 0.3168 0.0004043 L (54) 0.3178 0.0002736 L (55) 0.3210 0.0000713 L (56) 0.3220 0.0000001 L (57) 0.3231 0.0015821 L (58) 0.3241 0.0000028 L (59) 0.3252 0.0036268 L (60) 0.3262 0.0000065 L (61) 0.3273 0.0028885 L (62) 0.3283 0.0000052 L (63) 0.3294 0.0641101 L (64) 0.3304 0.0001153 L (65) 0.3315 0.1469629 L (66) 0.3325 0.0002644 L (67)  Mathematical Problems in Engineering synthetically fused to derive the final evaluation result, which cannot be decided by a single criterion. In this process, any criterion cannot decide the evaluation result of MRES. Even though the information values on monitoring system and regular inspections (c 3 ) and main biomass index (c 4 ) are both definitely good, the final evaluation result is still influenced by three other criteria. us, the fusion of information at the criterion level is actually a process of tradeoff, which should follow compensatory strategies. Accordingly, we derive the holistic PLTSs (Table 7) by taking the criterion weight set W � (0.150, 0.050, 0.100, 0.400, 0.300) and the normalized and ordered PLTSs as shown in Table 6 into equation (22). As demonstrated in Table 7, values in the column of r (k) represent the subscripts of 162 virtual linguistic terms in the holistic PLTS, and values in the column of p (k) are their associated probabilities. en, we apply the round function as in equation (11) to compute the integer score of the holistic PLTS and obtain α � 0.670. en, we have s * � round(s α ) � s 1 . us, the ultimate evaluation result is that the MRES is evaluated as average.

Conclusion
MCGDM is a decision-making theory that has gained wide attention and applications. As a combination of MCDM and GDM, MCGDM contains two dimensions of information: individual and criterion. Distinction between these two kinds of information is important. Specifically, information at the individual level is noncompensatory in the process of fusion and influence brought by information with a lower degree of reliability cannot offset that brought by information with a higher degree of reliability. Information at the criterion level is compensatory in the process of fusion; the information offsets one another and jointly forms the fusion result. Unfortunately, existing MCGDM methods do not pay attention to this kind of distinction. erefore, this paper proposes a novel MCGDM method with DST and PLTSs under the consideration of both compensatory and noncompensatory strategies. To conclude, this paper provides contributions in terms of the following aspects.
Firstly, this paper proposes that the information in the two dimensions in MCGDM is distinct, that is, information at the individual level is noncompensatory in the process of fusion, and information at the criterion level is compensatory. Accordingly, the proposed MCGDM method provides a comprehensive information fusion procedure that can fuse information at different levels.
Secondly, the proposed method provides a new orientation for the development of probabilistic preference theory in MCGDM by distinguishing noncompensatory information at the individual level and compensatory information at the criterion level. With the assistance of DST, PLTSs were developed to support information fusion and improve the scientific validity of the MCGDM.
irdly, this paper provides a method of building links between group BPA functions and PLTSs. Consequently, a framework based on belief and plausibility for transforming the certain point-valued probabilities assigned on power set  Mathematical Problems in Engineering of the frame of discernment into interval-valued probabilities associated with single linguistic terms is proposed, which also provides way of correlating BPA functions with PLTSs. Finally, this paper contributes to improving the application of probabilistic preference theory for evaluating MRES.
is paper provides a relative algorithm for the proposed method, which can be easily converted into a core algorithm of a decision system to promote its application. A case simulation study is also presented by applying the proposed method to evaluate the MRES.
It is worth noting that this paper just uses DST and PLTSs for reference to support noncompensatory and compensatory fusion, respectively. Other effective noncompensatory and compensatory modes could also be applied following the framework in this paper according to different problem characteristics. In the process of fusing compensatory information at the criterion level, the weights of the criteria are assumed to be known in advance. is paper does not address the way of determining the criteria weights. However, as the weights of the criteria influence the value of the virtual linguistic terms in the holistic PLTS a lot during the application of the weighted addition operation, they may lead to distortion of the fusion result. erefore, the criteria weights should be determined carefully and an appropriate determination principle of criteria weights that suits the weighted addition operation of PLTSs in MCGDM is a worthy future research direction.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.


extfieldsofstudy:
Computer Science
Medicine

text:
COVID-19 image classification using deep features and fractional-order marine predators algorithm

Currently, we witness the severe spread of the pandemic of the new Corona virus, COVID-19, which causes dangerous symptoms to humans and animals, its complications may lead to death. Although convolutional neural networks (CNNs) is considered the current state-of-the-art image classification technique, it needs massive computational cost for deployment and training. In this paper, we propose an improved hybrid classification approach for COVID-19 images by combining the strengths of CNNs (using a powerful architecture called Inception) to extract features and a swarm-based feature selection algorithm (Marine Predators Algorithm) to select the most relevant features. A combination of fractional-order and marine predators algorithm (FO-MPA) is considered an integration among a robust tool in mathematics named fractional-order calculus (FO). The proposed approach was evaluated on two public COVID-19 X-ray datasets which achieves both high performance and reduction of computational complexity. The two datasets consist of X-ray COVID-19 images by international Cardiothoracic radiologist, researchers and others published on Kaggle. The proposed approach selected successfully 130 and 86 out of 51 K features extracted by inception from dataset 1 and dataset 2, while improving classification accuracy at the same time. The results are the best achieved on these datasets when compared to a set of recent feature selection algorithms. By achieving 98.7%, 98.2% and 99.6%, 99% of classification accuracy and F-Score for dataset 1 and dataset 2, respectively, the proposed approach outperforms several CNNs and all recent works on COVID-19 images.

Medical imaging techniques are very important for diagnosing diseases. Image segmentation is a necessary image processing task that applied to discriminate region of interests (ROIs) from the area of outsides. Also, image segmentation can extract critical features, including the shape of tissues, and texture 5,6 .
In general, feature selection (FS) methods are widely employed in various applications of medical imaging applications. For example, Lambin et al. 7 proposed an efficient approach called Radiomics to extract medical image features. They showed that analyzing image features resulted in more information that improved medical imaging. Chong et al. 8 proposed an FS model, called Robustness-Driven FS (RDFS) to select futures from lung CT images to classify the patterns of fibrotic interstitial lung diseases. They applied the SVM classifier with and without RDFS. The evaluation showed that the RDFS improved SVM robustness against reconstruction kernel and slice thickness. In 9 , to classify ultrasound medical images, the authors used distance-based FS methods and a Fuzzy Support Vector Machine (FSVM). Moreover, a multi-objective genetic algorithm was applied to search for the optimal features subset.
More so, a combination of partial differential equations and deep learning was applied for medical image classification by 10 . They employed partial differential equations for extracting texture features of medical images. Acharya et al. 11 applied different FS methods to classify Alzheimer's disease using MRI images. The Shearlet transform FS method showed better performances compared to several FS methods. Also, in 12 , an Fs method based on SVM was proposed to detect Alzheimer's disease from SPECT images. Duan et al. 13 applied the Gaussian mixture model (GMM) to extract features from pulmonary nodules from CT images. The optimum path forest (OPF) classifier was applied to classify pulmonary nodules based on CT images. In 14 , the authors proposed an FS method based on a convolutional neural network (CNN) to detect pneumonia from lung X-ray images.
Afzali et al. 15 proposed an FS method based on principal component analysis and contour-based shape descriptors to detect Tuberculosis from lung X-Ray Images. They used K-Nearest Neighbor (kNN) to classify x-ray images collected from Montgomery dataset, and it showed good performances. Zhang et al. 16 proposed a kernel feature selection method to segment brain tumors from MRI images. They applied the SVM classifier for new MRI images to segment brain tumors, automatically. To segment brain tissues from MRI images, Kong et al. 17 proposed an FS method using two methods, called a discriminative clustering method and the information theoretic discriminative segmentation. Harikumar et al. 18 proposed an FS method based on wavelets to classify normality or abnormality of different types of medical images, such as CT, MRI, ultrasound, and mammographic images. It can be concluded that FS methods have proven their advantages in different medical imaging applications 19 .
Furthermore, deep learning using CNN is considered one of the best choices in medical imaging applications 20 , especially classification. CNNs are more appropriate for large datasets. Also, they require a lot of computational resources (memory & storage) for building & training. In some cases (as exists in this work), the dataset is limited, so it is not sufficient for building & training a CNN. In such a case, in order to get the advantage of the power of CNN and also, transfer learning can be applied to minimize the computational costs 21,22 . In transfer learning, a CNN which was previously trained on a large & diverse image dataset can be applied to perform a specific classification task by 23 . Therefore, several pre-trained models have won many international image classification competitions such as VGGNet 24 , Resnet 25 , Nasnet 26 , Mobilenet 27 , Inception 28 and Xception 29 .
However, some of the extracted features by CNN might not be sufficient, which may affect negatively the quality of the classification images. Therefore, a feature selection technique can be applied to perform this task by removing those irrelevant features. Among the FS methods, the metaheuristic techniques have been established their performance overall other FS methods when applied to classify medical images. For example, Da Silva et al. 30 used the genetic algorithm (GA) to develop feature selection methods for ranking the quality of medical images. They used different images of lung nodules and breast to evaluate their FS methods. Evaluation outcomes showed that GA based FS methods outperformed traditional approaches, such as filter based FS and traditional wrapper methods. Johnson et al. 31 applied the flower pollination algorithm (FPA) to select features from CT images of the lung, to detect lung cancers. They also used the SVM to classify lung CT images. The evaluation confirmed that FPA based FS enhanced classification accuracy. kharrat and Mahmoud 32 proposed an FS method based on a hybrid of Simulated Annealing (SA) and GA to classify brain tumors using MRI. The combination of SA and GA showed better performances than the original SA and GA. Narayanan et al. 33 proposed a fuzzy particle swarm optimization (PSO) as an FS method to enhance the classification of CT images of emphysema. They applied a fuzzy decision tree classifier, and they found that fuzzy PSO improved the classification accuracy. Li et al. 34 proposed a self-adaptive bat algorithm (BA) to address two problems in lung X-ray images, rebalancing, and feature selection. They compared the BA to PSO, and the comparison outcomes showed that BA had better performance. Dhanachandra and Chanu 35 proposed a hybrid method of dynamic PSO and fuzzy c-means to segment two types of medical images, MRI and synthetic images. They concluded that the hybrid method outperformed original fuzzy c-means, and it had less sensitive to noises. Li et al. 36 proposed an FS method using a discrete artificial bee colony (ABC) to improve the classification of Parkinson's disease. The evaluation outcomes demonstrate that ABC enhanced precision, and also it reduced the size of the features.
In this paper, we proposed a novel COVID-19 X-ray classification approach, which combines a CNN as a sufficient tool to extract features from COVID-19 X-ray images. Then, using an enhanced version of Marine Predators Algorithm to select only relevant features. In general, MPA is a meta-heuristic technique that simulates the behavior of the prey and predator in nature 37 . This algorithm is tested over a global optimization problem. However, it has some limitations that affect its quality. In addition, up to our knowledge, MPA has not applied to any real applications yet. So, based on this motivation, we apply MPA as a feature selector from deep features that produced from CNN (largely redundant), which, accordingly minimize capacity and resources consumption and can improve the classification of COVID-19 X-ray images. The proposed COVID-19 X-ray classification approach starts by applying a CNN (especially, a powerful architecture called Inception which pre-trained on Imagnet dataset) to extract the discriminant features from raw images (with no pre-processing or segmentation) from the dataset that contains positive and negative COVID-19 images. Then, applying the FO-MPA to select the relevant features from the images. This task is achieved by FO-MPA which randomly generates a set of solutions, each of them represents a subset of potential features. The next process is to compute the performance of each solution using fitness value and determine which one is the best solution. Thereafter, the FO-MPA parameters are applied to update the solutions of the current population. The updating operation repeated until reaching the stop condition. Then the best solutions are reached which determine the optimal/relevant features that should be used to address the desired output via several performance measures. Inspired by our recent work 38 , where VGG-19 besides statistically enhanced Salp Swarm Algorithm was applied to select the best features for White Blood Cell Leukaemia classification. Also, other recent published works 39 , who combined a CNN architecture with Weighted Symmetric Uncertainty (WSU) to select optimal features for traffic classification. It is obvious that such a combination between deep features and a feature selection algorithm can be efficient in several image classification tasks.
The main contributions of this study are elaborated as follows: 1. Propose an efficient hybrid classification approach for COVID-19 using a combination of CNN and an improved swarm-based feature selection algorithm. This combination should achieve two main targets; high performance and resource consumption, storage capacity which consequently minimize processing time. 2. Propose a novel robust optimizer called Fractional-order Marine Predators Algorithm (FO-MPA) to select efficiently the huge feature vector produced from the CNN. 3. Test the proposed Inception Fractional-order Marine Predators Algorithm (IFM) approach on two publicity available datasets contain a number of positive negative chest X-ray scan images of COVID-19. 4. Evaluate the proposed approach by performing extensive comparisons to several state-of-art feature selection algorithms, most recent CNN architectures and most recent relevant works and existing classification methods of COVID-19 images.
We do not present a usable clinical tool for COVID-19 diagnosis, but offer a new, efficient approach to optimize deep learning-based architectures for medical image classification purposes. Such methods might play a significant role as a computer-aided tool for image-based clinical diagnosis soon. Remainder sections are organized as follows: "Material and methods" section presents the methodology and the techniques used in this work including model structure and description. The experimental results and comparisons with other works are presented in "Results and discussion" section, while they are discussed in "Discussion" section Finally, the conclusion is described in "Conclusion" section.

Material and methods
Features extraction using convolutional neural networks. In this paper, we apply a convolutional neural network (CNN) to extract features from COVID-19 X-Ray images. We adopt a special type of CNN called a pre-trained model where the network is previously trained on the ImageNet dataset, which contains millions of variety of images (animal, plants, transports, objects,..) on 1000 classe categories. So, transfer learning is applied by transferring weights that were already learned and reserved into the structure of the pre-trained model, such as Inception, in this paper. In Inception, there are different sizes scales convolutions (conv.), such as 5 × 5 , 3 × 3 , 1 × 1 . For instance,1 × 1 conv. is applied before larger sized kernels are applied to reduce the dimension of the channels, which accordingly, reduces the computation cost. Pool layers are used mainly to reduce the input's size, which accelerates the computation as well. So, for a 4 × 4 matrix, will result in 2 × 2 matrix after applying max pooling. There are three www.nature.com/scientificreports/ main parameters for pooling, Filter size, Stride, and Max pool. In this paper, filters of size 2, besides a stride of 2 and 2 × 2 as Max pool, were adopted. Inception architecture is described in Fig. 1.
The main purpose of Conv. layers is to extract features from input images. In this paper, different Conv. Layers are applied to extract different types of features such as edges, texture, colors, and high-lighted patterns from the images. The combination of Conv. and pool layers, three fully connected layers, the last one performs classification. The Softmax activation function is used for this purpose because the output should be binary (positive COVID-19 negative COVID-19). Inception's layer details and layer parameters of are given in Table 1.
As seen in Table 1, we keep the last concatenation layer which contains the extracted features, so we removed the top layers such as the Flatten, Drop out and the Dense layers which the later performs classification (named as FC layer). We have used RMSprop optimizer for weight updates, cross entropy loss function and selected learning rate as 0.0001.
In this paper, Inception is applied as a feature extractor, where the input image shape is (229, 229, 3). Since its structure consists of some parallel paths, all the paths use padding of 1 pixel to preserve the same height & width for the inputs and the outputs.
One of the drawbacks of pre-trained models, such as Inception, is that its architecture required large memory requirements as well as storage capacity (92 M.B), which makes deployment exhausting and a tiresome task. The shape of the output from the Inception is (5, 5, 2048), which represents a feature vector of size 51200. So some statistical operations have been added to exclude irrelevant and noisy features, and by making it more computationally efficient and stable, they are summarized as follows: • Chi-square is applied to remove the features which have a high correlation values by computing the dependence between them. It is calculated between each feature for all classes, as in Eq. (1): where O k and E k refer to the actual and the expected feature value, respectively. In this paper, after applying Chi-square, the feature vector is minimized for both datasets from 51200 to 2000. • Tree based classifier are the most popular method to calculate feature importance to improve the classification since they have high accuracy, robustness, and simple 38 . For each decision tree, node importance is calculated using Gini importance, Eq. (2) calculated two child nodes.
where ni j is the importance of node j, while w j refers to the weighted number of samples reaches the node j, also C j determines the impurity value of node j. left(j) and right(j) are the child nodes from the left split and the right split on node j, respectively. In Eq. (3), the importance of each feature is then calculated.
(1) www.nature.com/scientificreports/ where fi i represents the importance of feature I, while ni j refers to the importance of node j. In order to normalize the values between 0 and 1 by dividing by the sum of all feature importance values, as in Eq. (4).
Finally, the sum of the feature's importance value on each tree is calculated then divided by the total number of trees as in Eq. (5).
where REfi i represents the importance of feature i that were calculated from all trees, where normfi ij is the normalized feature importance for feature i in tree j, also T is the total number of trees. After applying this technique, the feature vector is minimized from 2000 to 459 and from 2000 to 462 for Dataset1 and Dataset 2, respectively.

Feature selection by fractional-order calculus with Marine Predators Algorithm (FO-MPA). Fractional calculus (FC).
Fractional-order calculus (FC) gains the interest of many researchers in different fields not only in the modeling sectors but also in developing the optimization algorithms. The memory properties of Fc calculus makes it applicable to the fields that required non-locality and memory effect. FC provides a clear interpretation of the memory and hereditary features of the process. Accordingly, the FC is an efficient tool for enhancing the performance of the meta-heuristic algorithms by considering the memory perspective during updating the solutions. One from the well-know definitions of FC is the Grunwald-Letnikov (GL), which can be mathematically formulated as below 40 : where where D δ (U(t)) refers to the GL fractional derivative of order δ . Ŵ(t) indicates gamma function.
The GL in the discrete-time form can be modeled as below: where T is the sampling period, and m is the length of the memory terms (memory window). The δ symbol refers to the derivative order coefficient. For the special case of δ = 1 , the definition of Eq. (8) can be remodeled as below: where D 1 [x(t)] represents the difference between the two followed events.
Marine Predators Algorithm. The Marine Predators Algorithm (MPA)is a recently developed meta-heuristic algorithm that emulates the relation among the prey and predator in nature 37 . MPA simulates the main aim for most creatures that is searching for their foods, where a predator contiguously searches for food as well as the prey. Inspired by this concept, Faramarzi et al. 37 developed the MPA algorithm by considering both of a predator a prey as solutions. The MPA starts with the initialization phase and then passing by other three phases with respect to the rational velocity among the prey and the predator.
• Initialization phase: this phase devotes for providing a random set of solutions for both the prey and predator via the following formulas: where the Lower and Upper are the lower and upper boundaries in the search space, rand 1 is a random vector ∈ the interval of (0,1). According to the formula 10, the initial locations of the prey and predator can be defined as below: (3) fi i = j:node j splits on feature i ni j k∈all nodes ni k www.nature.com/scientificreports/ where the Elite matrix refers to the fittest predators. • Stage 1: After the initialization, the exploration phase is implemented to discover the search space. Therefore in MPA, for the first third of the total iterations, i.e., 1 3 t max ). Accordingly, the prey position is upgraded based the following equations.
where R ∈ [0, 1] is a random vector drawn from a uniform distribution and P = 0.5 is a constant number. The symbol R B refers to Brownian motion. indicates the process of element-wise multiplications.
• Stage 2: The prey/predator in this stage begin exploiting the best location that detects for their foods. Stage 2 has been executed in the second third of the total number of iterations when 1 3 t max < t < 2 3 t max . Faramarzi et al. 37 divided the agents for two halves and formulated Eqs. (14)- (15) to emulate the motion of the first half of the population (prey) and Eqs. (18)- (19) for the second half (predator) as represented below.
where R L has random numbers that follow Lévy distribution. Eq. (14)- (15) are implemented in the first half of the agents that represent the exploitation. While the second half of the agents perform the following equations.
where CF is the parameter that controls the step size of movement for the predator.
• Stage 3: This stage executed on the last third of the iteration numbers ( t > 2 3 t max ) where based on the following formula: • Eddy formation and Fish Aggregating Devices' effect: Faramarzi et al. 37 considered the external impacts from the environment, such as the eddy formation or Fish Aggregating Devices (FADs) effects to avoid the local optimum solutions. This stage can be mathematically implemented as below: In Eq. (20), FAD = 0.2 , and W is a binary solution (0 or 1) that corresponded to random solutions. If the random solution is less than 0.2, it converted to 0 while the random solution becomes 1 when the solutions are greater than 0.2. The symbol r ∈ [0, 1] represents a random number. r 1 and r 2 are the random index of the prey. • Marine memory: This is the main feature of the marine predators and it helps in catching the optimal solution very fast and avoid local solutions. Faramarzi et al. 37 implement this feature via saving the previous best solutions of a prior iteration, and compared with the current ones; the solutions are modified based on the best one during the comparison stage.

Fractional-order Marine Predators Algorithm (FO-MPA).
Recently, a combination between the fractional calculus tool and the meta-heuristics opens new doors in providing robust and reliable variants 41 . For this motivation, we utilize the FC concept with the MPA algorithm to boost the second step of the standard version of the algorithm. Hence, the FC memory is applied during updating the prey locating in the second step of the algorithm to enhance the exploitation stage. Moreover, the R B parameter has been changed to depend on weibull distribution as described below.  By taking into account the early mentioned relation in Eq. (23), the general formulation for the solutions of FO-MPA based on FC memory perspective can be written as follows: After checking the previous formula, it can be detected that the motion of the prey becomes based on some terms from the previous solutions with a length of (m), as depicted in Fig. 2 (left). With accounting the first four previous events ( m = 4 ) from the memory data with derivative order δ , the position of prey can be modified as follow; • Second: Adjusting R B random parameter based on weibull distribution. For the exploration stage, the weibull distribution has been applied rather than Brownian to bost the performance of the predator in stage 2 and the prey velocity in stage 1 based on the following formula: Where k, and ζ are the scale and shape parameters. The Weibull Distribution is a heavy-tied distribution which presented as in Fig. 2 (right). In the current work, the values of k, and ζ are set to 2, and 2, respectively.
Our proposed approach is called Inception Fractional-order Marine Predators Algorithm (IFM), where we combine Inception (I) with Fractional-order Marine Predators Algorithm (FO-MPA). The proposed IFM approach is summarized as follows: 1. Extracting deep features from Inception, where about 51 K features were extracted. 2. Initialize solutions for the prey and predator. The prey follows Weibull distribution during discovering the search space to detect potential locations of its food. 3. The predator tries to catch the prey while the prey exploits the locations of its food. The predator uses the Weibull distribution to improve the exploration capability. Meanwhile, the prey moves effectively based on its memory for the previous events to catch its food, as presented in Eq. (24). 4. Finally, the predator follows the levy flight distribution to exploit its prey location. all above stages are repeated until the termination criteria is satisfied. These datasets contain hundreds of frontal view X-rays and considered the largest public resource for COVID-19 image data. They were manually aggregated from various web based repositories into a machine learning (ML) friendly format with accompanying data loader code. They were also collected frontal and lateral view imagery and metadata such as the time since first symptoms, intensive care unit (ICU) status, survival status, intubation status, or hospital location.
Both datasets shared some characteristics regarding the collecting sources. For both datasets, the Covid19 images were collected from patients with ages ranging from 40-84 from both genders. It is also noted that both datasets contain a small number of positive COVID-19 images, and up to our knowledge, there is no other sufficient available published dataset for COVID-19. Table 2 shows some samples from two datasets. Table 2 depicts the variation in morphology of the image, lighting, structure, black spaces, shape, and zoom level among the same dataset, as well as with the other dataset.  • Best accuracy: • Best fitness value: • Worst fitness value: • Average of fitness value: • Standard deviation of fitness value where r is the run numbers. Fit i denotes a fitness function value.

Implementation environment. Convolutional neural networks were implemented in Python 3 under
Google Colaboratory 46 , commonly referred to as "Google Colab, " which is a research project for prototyping machine learning models on powerful hardware options such as GPUs and TPUs. In this paper, we used TPUs for powerful computation, which is more appropriate for CNN. The model was developed using Keras library 47 with Tensorflow backend 48 .

Results and discussion
Performance of the proposed approach. As Inception examines all X-ray images over and over again in each epoch during the training, these rapid ups and downs are slowly minimized in the later part of the training. After feature extraction, we applied FO-MPA to select the most significant features. In this subsection, the results of FO-MPA are compared against most popular and recent feature selection algorithms, such as Whale Optimization Algorithm (WOA) 49 , Henry Gas Solubility optimization (HGSO) 50 , Sine cosine Algorithm (SCA), Slime Mould Algorithm (SMA) 51 , Particle Swarm Optimization (PSO), Grey Wolf Optimization (GWO) 52 , Harris Hawks Optimization (HHO) 53 , Genetic Algorithm (GA), and basic MPA. In this paper, each feature selection algorithm were exposed to select the produced feature vector from Inception aiming (30) F Score = 2 × Specificity × Sensitivity Specificity + Sensitivity   Tables 3 and 4. Table 3 shows the numerical results of the feature selection phase for both datasets. Four measures for the proposed method and the compared algorithms are listed. As seen in Table 3, on Dataset 1, the FO-MPA outperformed the other algorithms in the mean of fitness value as it achieved the smallest average fitness function value followed by SMA, HHO, HGSO, SCA, BGWO, MPA, and BPSO, respectively whereas, the SGA and WOA showed the worst results. The results of max measure (as Table 3. Results of the feature selection phase based on fitness function. Highest results are in bold.  www.nature.com/scientificreports/ in Eq. (33)), showed that FO-MPA also achieved the best value of the fitness function compared to others. SMA is on the second place, While HGSO, SCA, and HHO came in the third to fifth place, respectively. According to the best measure, the FO-MPA performed similarly to the HHO algorithm, followed by SMA, HGSO, and SCA, respectively. Although the performance of the MPA and bGWO was slightly similar, the performance of SGA and WOA were the worst in both max and min measures. Generally, the most stable algorithms On dataset 1 are WOA, SCA, HGSO, FO-MPA, and SGA, respectively. However, WOA showed the worst performances in these measures; which means that if it is run in the same conditions several times, the same results will be obtained. For Dataset 2, FO-MPA showed acceptable (not the best) performance, as it achieved slightly similar results to the first and second ranked algorithm (i.e., MPA and SMA) on mean, best, max, and STD measures. Also, WOA algorithm showed good results in all measures, unlike dataset 1, which can conclude that no algorithm can solve all kinds of problems. Whereas, the worst algorithm was BPSO. For more analysis of feature selection algorithms based on the number of selected features (S.F) and consuming time, Fig. 4 and Table 4 list these results for all algorithms. Regarding the consuming time as in Fig. 4a, the SMA was considered as the fastest algorithm among all algorithms followed by BPSO, FO-MPA, and HHO, respectively, while MPA was the slowest algorithm. Also, As seen in Fig. 4b, FO-MPA algorithm selected successfully fewer features than other algorithms, as it selected 130 and 86 features from Dataset 1 and Dataset 2, respectively. HGSO was ranked second with 146 and 87 selected features from Dataset 1 and Dataset 2, respectively. The largest features were selected by SMA and SGA, respectively.
The convergence behaviour of FO-MPA was evaluated over 25 independent runs and compared to other algorithms, where the x-axis and the y-axis represent the iterations and the fitness value, respectively. Figure 5 illustrates the convergence curves for FO-MPA and other algorithms in both datasets. Figure 5, shows that FO-MPA shows an efficient and faster convergence than the other optimization algorithms on both datasets. Whereas, the slowest and the insufficient convergences were reported by both SGA and WOA in Dataset 1 and by SGA in Dataset 2.
To further analyze the proposed algorithm, we evaluate the selected features by FO-MPA by performing classification. In this experiment, the selected features by FO-MPA were classified using KNN. Table 4 show classification accuracy of FO-MPA compared to other feature selection algorithms, where the best, mean, and STD for classification accuracy were calculated for each one, besides time consumption and the number of selected features (SF). In Table 4, for Dataset 1, the proposed FO-MPA approach achieved the highest accuracy in the best and mean measures, as it reached 98.7%, and 97.2% of correctly classified samples, respectively. While, MPA, BPSO, SCA, and SGA obtained almost the same accuracy, followed by both bGWO, WOA, and SMA. The lowest accuracy was obtained by HGSO in both measures. Based on Standard Deviation measure (STD), the most stable algorithms were SCA, SGA, BPSO, and bGWO, respectively. Whereas, FO-MPA, MPA, HGSO, and WOA showed similar STD results. The HGSO also was ranked last. In Dataset 2, FO-MPA also is reported as the highest classification accuracy with the best and mean measures followed by the BPSO. The classification accuracy of MPA, WOA, SCA, and SGA are almost the same. Whereas the worst one was SMA algorithm. Besides, all algorithms showed the same statistical stability in STD measure, except for HHO and HGSO. Generally, the proposed FO-MPA approach showed satisfying performance in both the feature selection ratio and the classification rate. www.nature.com/scientificreports/ Moreover, from Table 4, it can be seen that the proposed FO-MPA provides better results in terms of F-Score, as it has the highest value in datatset1 and datatset2 which are 0.9821 and 0.99079, respectively.
Comparison with other CNN architectures. In this subsection, the performance of the proposed COVID-19 classification approach is compared to other CNN architectures. It noted that all produced feature vectors by CNNs used in this paper are at least bigger by more than 300 times compared to that produced by FO-MPA in terms of the size of the featureset. For example, as our input image has the shape 224 × 224 × 3 , Nasnet 26 produces 487 K features, Resnet 25 and Xception 29 produce about 100 K features and Mobilenet 27 produces 50 K features, while FO-MPA produces 130 and 86 features for both dataset1 and dataset 2, respectively. Figure 6 shows a comparison between our FO-MPA approach and other CNN architectures. From Fig. 6 (left), for dataset 1, it can be seen that our proposed FO-MPA approach outperforms other CNN models like VGGNet, Xception, Inception, Mobilenet, Nasnet, and Resnet. It also shows that FO-MPA can select the smallest subset of features, which reflects positively on performance. Accordingly, that reflects on efficient usage of memory, and less resource consumption. On the second dataset, dataset 2 (Fig. 6, right), our approach still provides an overall accuracy of 99.68%, putting it first with a slight advantage over MobileNet (99.67 %).
Comparison with related works. In this subsection, a comparison with relevant works is discussed. Figure 7 shows the most recent published works as in 54-57 and 44 on both dataset 1 and dataset 2. In 54 , AlexNet pre-trained network was used to extract deep features then applied PCA to select the best features by eliminating highly correlated features. Based on 54 , the later step reduces the memory requirements, and improve the efficiency of the framework. While 55 used different CNN structures. However, it was clear that VGG19 and MobileNet achieved the best performance over other CNNs. Also, in 58 a new CNN architecture called EfficientNet was proposed, where more blocks were added on top of the model after applying normalization of images pixels intensity to the range (0 to 1). Also, some image transformations were applied, such as rotation, horizontal flip, and scaling. In 57 , ResNet-50 CNN has been applied after applying horizontal flipping, random rotation, random zooming, random lighting, and random wrapping on raw images. As seen in Fig. 7, most works are pre-prints for two main reasons; COVID-19 is the most recent and trend topic; also, there are no sufficient datasets that can be used for reliable results. However, the proposed FO-MPA approach has an advantage in performance compared to other works. Also, all other works do not give further statistics about their model's complexity and the number of  www.nature.com/scientificreports/ featurset produced, unlike, our approach which extracts the most informative features (130 and 86 features for dataset 1 and dataset 2) that imply faster computation time and, accordingly, lower resource consumption. Compared to 59 which is one of the most recent published works on X-ray COVID-19, a combination between You Only Look Once (YOLO) which is basically a real time object detection system and DarkNet as a classifier was proposed. They achieved 98.08 % and 96.51 % of accuracy and F-Score, respectively compared to our approach with 98.77 % and 98.2% for accuracy and F-Score, respectively. While no feature selection was applied to select best features or to reduce model complexity.

Discussion
The proposed IMF approach successfully achieves two important targets, selecting small feature numbers with high accuracy. Therefore, reducing the size of the feature from about 51 K as extracted by deep neural networks (Inception) to be 128.5 and 86 in dataset 1 and dataset 2, respectively, after applying FO-MPA algorithm while increasing the general performance can be considered as a good achievement as a machine learning goal. Besides, the used statistical operations improve the performance of the FO-MPA algorithm because it supports the algorithm in selecting only the most important and relevant features. It also contributes to minimizing resource consumption which consequently, reduces the processing time.
In addition, the good results achieved by the FO-MPA against other algorithms can be seen as an advantage of FO-MPA, where a balancing between exploration and exploitation stages and escaping from local optima were achieved. As a result, the obtained outcomes outperformed previous works in terms of the model's general performance measure.
Furthermore, using few hundreds of images to build then train Inception is considered challenging because deep neural networks need large images numbers to work efficiently and produce efficient features. However, the proposed IMF approach achieved the best results among the compared algorithms in least time. One of the main disadvantages of our approach is that it's built basically within two different environments. The first one is based on Python, where the deep neural network architecture (Inception) was built and the feature extraction part was performed. The second one is based on Matlab, where the feature selection part (FO-MPA algorithm) was performed. So, there might be sometimes some conflict issues regarding the features vector file types or issues related to storage capacity and file transferring.

Conclusion
Computational image analysis techniques play a vital role in disease treatment and diagnosis. Taking into consideration the current spread of COVID-19, we believe that these techniques can be applied as a computer-aided tool for diagnosing this virus. Therefore, in this paper, we propose a hybrid classification approach of COVID-19. It based on using a deep convolutional neural network (Inception) for extracting features from COVID-19 images, then filtering the resulting features using Marine Predators Algorithm (MPA), enhanced by fractionalorder calculus(FO).
The proposed IMF approach is employed to select only relevant and eliminate unnecessary features. Extensive evaluation experiments had been carried out with a collection of two public X-ray images datasets. Extensive comparisons had been implemented to compare the FO-MPA with several feature selection algorithms, including SMA, HHO, HGSO, WOA, SCA, bGWO, SGA, BPSO, besides the classic MPA. The results showed that the proposed approach showed better performances in both classification accuracy and the number of extracted features that positively affect resource consumption and storage efficiency. The results are the best achieved compared to other CNN architectures and all published works in the same datasets.
According to the promising results of the proposed model, that combines CNN as a feature extractor and FO-MPA as a feature selector could be useful and might be successful in being applied in other image classification tasks.