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Assessment of Near-Earth Asteroid Deflection Techniques via Spherical Fuzzy Sets

Assessment of Near-Earth Asteroid Deflection Techniques via Spherical Fuzzy Sets Hindawi Advances in Astronomy Volume 2021, Article ID 6678056, 12 pages https://doi.org/10.1155/2021/6678056 Research Article Assessment of Near-Earth Asteroid Deflection Techniques via Spherical Fuzzy Sets M. Ferna ´ ndez-Martı´nez and J. M. Sa ´ nchez-Lozano University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720–Santiago De La Ribera, Regio´n De Murcia, Spain ´ ´ Correspondence should be addressed to M. Fernandez-Martınez; manuel.fernandez-martinez@cud.upct.es Received 23 October 2020; Revised 22 January 2021; Accepted 18 February 2021; Published 8 March 2021 Academic Editor: Maria Gritsevich Copyright © 2021 M. Ferna´ndez-Mart´ınez and J. M. Sa´nchez-Lozano. .is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Extensions of fuzzy sets to broader contexts constitute one of the leading areas of research in the context of problems in artificial intelligence. .eir aim is to address decision-making problems in the real world whenever obtaining accurate and sufficient data is not a straightforward task. In this way, spherical fuzzy sets were recently introduced as a step beyond to modelize such problems more precisely on the basis of the human nature, thus expanding the space of membership levels, which are defined under imprecise circumstances. .e main goal in this study is to apply the spherical fuzzy set version of Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), a well-established multicriteria decision-making approach, in the context of planetary defense. As of the extraction of knowledge from a group of experts in the field of near-Earth asteroids, they rated four deflection technologies of asteroids (kinetic impactor, ion beam deflection, enhanced gravity tractor, and laser ablation) that had been previously assessed by means of the classical theory of fuzzy series. .is way, a comparative study was carried out whose most significant results are the kinetic impactor being the most suitable alternative and the spherical fuzzy set version of the TOPSIS approach behaves more sensitively than the TOPSIS procedure for triangular fuzzy sets with regard to the information provided by our group of experts. object broke up at an altitude of 30km, thus injuring more 1. Introduction than 1500 people [2, 5, 6]. Meteoroids reach the Earth mainly as small rocks and fragile .e latter are clear examples of fatal consequences that aggregates which appear as a consequence of the decay of impact of these kinds of objects with the Earth may result in. asteroids and comets. In this way, the tiny dusts that arrive at .e impact of a single massive cosmic body might leave a fairly the Earth each day amounts to a mean of approximately 100 largecraterinthesurfaceoftheEarthorinduceatsunamiinthe tons [1]. caseitcollideswiththesurfaceoftheocean,thuscontributingto Although it is true that larger objects will unlikely reach overall risk [7–10]. Furthermore, air blasts derived from de- Earth’s orbit, a potential impactor may dramatically affect struction of meteoritic bodies in Earth’s atmosphere may the life and the climate in our planet. For instance, a 10km provoke not only falls of clouds of fragments but also major wide impactor led to the so-called Cretaceous-Paleocene injuries on Earth’s surface even if the object does not make a extinction that happened 64 million years ago [2, 3]. In 1908, touchdown. Indeed, such explosions may cause from glass the fragmentation at a low altitude (5–10km) of an asteroid breakage (for air blasts greater than 15m/s) to extreme dis- with an estimated diameter d ∈ [30,50]m destroyed around tortions in steel structures of bridges or buildings which may 2000km of woodland [4], what was known as the Tunguska lead to their collapse (for air blasts greater than 200m/s) [11] event. Recently, in 2013, a bolide with an estimated weight of and ([12], Table 1). .erefore, such consequences do not only 12000 tons and an estimated diameter of 19m entered depend onthe sizeof the impactor butalsoon otherparameters Earth’s atmosphere at a relative velocity of 19km/s. .e such as entry angle, velocity, density, and shape [12]. 2 Advances in Astronomy carried out in regard to eight criteria (build time, level of Near-Earth objects (NEOs) are asteroids or comets whose perihelion distance is less than 1.3AU (about 195 maturity of a NEA deflection technology, asteroid rotation, asteroid structure, asteroid composition, asteroid shape, million km). Short period comets with orbital periods less than 200 years are known as near-Earth comets (NECs) active deflection duration, and mission risk). Such a decision whose orbits lie far away from the Earth. In this way, re- problem was addressed throughout a combination of fuzzy searchers are mainly focused on the tracking of the near- logic and multicriteria decision-making (MCDM, hereafter) Earth asteroids (NEAs) [2, 13]. approaches. .ree key reasons that support the study of NEAs are, We recall that a MCDM problem consists of looking for namely, planetary defense, scientific knowledge (e.g., deepen the best choice from a set of alternatives by a set of criteria, and to deal with, all that information is arranged into the so- our solar system origins), and mining. In this regard, the current study is allocated to planetary defense. called decision matrix [20]. A wide collection of MCDM algorithms, such as ELECTRE, OWA, VIKOR, ANP, and, In particular, the so-called potentially hazardous aster- oids are NEAs whose minimum orbit interception distance PROMETHEE, can be found in the literature. Despite MCDM methodologies had been previously with respect to Earth’s orbit is less than 0.05AU. .ey are also characterized by an estimated size greater than 140m in applied to address several issues regarding NEAs ([20, 21]), diameterand anabsolutemagnitude notgreaterthan22.0,as they were first combined with fuzzy logic recently in [19], well. Since they are able to closely approach Earth’s orbit, where four NEA deflection technologies (described in small perturbations regarding their orbits may place them Section 3.2) were rated with respect to a set consisting of on a collision path to the Earth. As such, to perform an eight criteria (Section 3.3). .e involvement of fuzzy logic exhaustive tracking of these asteroids is recommended therein was mainly motivated by the existence of qualitative criteria whose values are difficult to be specified or mea- [14, 15]. Some efforts have already been made with the aim of sured. In this way, linguistic labels that were associated with triangular fuzzy numbers, as well as (the valuable knowledge redirecting a NEA out of a risky trajectory with the Earth’, at least at a first glance. Indeed, in 2013, the NASA introduced from) the judgments provided by an international board consisting of great standing scientists in the field of NEAs, the Asteroid Redirect Mission (ARM) with a double pur- pose, namely, redirect small (i.e., less than 8m diameter) were used to quantify the level of importance of such criteria. asteroids and extract small (less than 4m) boulders from the .is way, the level of importance of each criterion was surface of a wide asteroid and place it on a distant retrograde calculated, thus leading to solve the decision problem that orbitaroundthemoon[16]..elater couldbe understoodas had been posed by means of the MCDM approach named a preliminary test of the current technical capacity to deliver Technique for Order of Preference by Similarity to Ideal and catch an object to a safe environment, despite that Solution (TOPSIS). Zadeh was the first tointroduce thatapproach to manage mission did not actually consist of redirecting a small PHA. However, that technological and scientific project was uncertainty and ambiguity when there exist attributes that are hard to be quantified [22]. In fuzzy logic, the level of cancelled in 2017 [17]. Notwithstanding, several encouraging approaches for membership of an element to a series is determined by a real NEA redirection have been posed, even though the tech- number lying in the interval [0,1], thus leading to a fuzzy nology underlying them has not yet completely developed. series. Since that pioneer work, fuzzy series have been ap- In this regard, we would like to highlight the Double As- plied not only to deal with decision problems in a wide range teroid Redirection Test (DART) that will be the first in situ of contexts (e.g., [23]) but also to contribute new viewpoints exhibition of a kinetic impactor to deflect an asteroid in in that field including intuitionistic fuzzy sets [24], Py- space. .is was formerly applied to the satellite of the binary thagorean fuzzy sets [25–28], neutrosophic fuzzy sets [29], NEA (65803) Didymos [18]. or picture fuzzy sets [30, 31]. Extensions of fuzzy sets to broader contexts constitute Redirecting an asteroid should be distinguished from deflecting it. In fact, deflecting an asteroid consists of one of the leading areas of research in the context of problems in computational intelligence. .eir aim is to modifying its trajectory to avert a potential impact with the Earth. As such, nuclear blast and kinetic impactor are ex- address decision-making problems in the real world amples of deflection techniques. On the other hand, the aim whenever obtaining accurate and sufficient data is not a of redirecting an object is to induce a controlled change in its straightforward task. In this way, spherical fuzzy sets were orbit with a further purpose, as it is the case of laser ablation/ recently introduced as a step beyond the picture fuzzy sets to sublimation, tugboat, mass driver, and ion beam ([2] and modelize MCDM problems more precisely on the basis of references therein). However, in this study, we shall un- the human nature, thus expanding the space of membership levels, which are defined under imprecise circumstances derstand a deflection technology as that one being able of deflecting or redirecting an asteroid. [32]. Even though they were introduced recently [33], it is worth mentioning that several operators, distance measures, It is worth pointing out that, in [19], an assessment involving four deflection technologies for asteroids smaller and even some applications have already been contributed [34–38]. than 250 meters in diameter (a range of sizes that covers most of the impactors with the Earth that occur in timescales Being encouraged by such novel contributions, we up to 100000 years), i.e., kinetic impactor, ion beam de- wondered to what extent ranking of alternatives would vary flection, enhanced gravity tractor, and laser ablation, was if either fuzzy sets are considered to deal with a MCDM Advances in Astronomy 3 problem or the most recent extensions of fuzzy sets (such as been widely applied including the Gaussian, the PI (or spherical fuzzy sets) are used with the same purpose. Fol- trapezoidal), and the LAMBDA (or triangular) ones. lowing the above, this study addresses that question by Several extensions of ordinary fuzzy sets have appeared applying the spherical fuzzy version of the TOPSIS approach in the literature (e.g., [33] for a chronological tracking of to a decision problem that had previously been posed in the them). Among them, we would like to highlight those context of planetary defense [19]. generalizations of fuzzy sets with a three-dimensional .e structure of this study is as follows. Section 2 membership function. containsthebasicson fuzzysetsand fuzzylogic (Section2.1), An intuitionistic fuzzy set is one of the form spherical fuzzy sets and their operators (Section 2.2), and A � ⟨μ (u), ] (u)⟩: u ∈ U , where μ : U ⟶ [0,1] is the 􏽮 􏽯 􏽥 􏽥 􏽥 A A A also includes a description regarding the generalization of membership function that quantifies the degree of mem- the TOPSIS approach to the context of spherical fuzzy sets bership of each element u to A, and ] : U ⟶ [0,1] is the (Section 2.3). .e assumptions of our study are summarized nonmembership function. .ey satisfy that μ (u)+ in Section 3.1. Next, we recall the four NEA deflection ] (u) ∈ [0,1] for all u ∈ U. In addition, π (u) � 􏽥 􏽥 A A technologies to be evaluated in this study (Section 3.2) to- 1 − μ (u) − ] (u) is defined as the degree of hesitancy of u 􏽥 􏽥 A A gether with the selected criteria (Section 3.3). Furthermore, to A. However, in real-life applications, it may happen that, some comments on the board of experts who provided us for a certain alternative satisfying a criterion, the sum of the valuable information to determine the weights of the criteria squares of the membership and nonmembership functions are provided in Section 3.4. Our results and discussion are stands not greater than 1 with their sum being greater than 1. provided in Section 4, whereas two analyses of sensitivity are With the aim to avoid the experts modifying their prefer- carried out and further discussed in Section 5. Finally, the ences, the second type intuitionistic fuzzy sets were intro- main conclusions of this study are presented in Section 6. duced by Atanassov in [41]. .ey are the form A � (μ (u), ] (u)): u ∈ U , where its membership func- 􏽮 􏽯 􏽥 􏽥 A A tion, μ : U ⟶ [0,1], and its nonmembership function, 2 2 ] : U ⟶ [0,1], satisfy that μ (u) + ] (u) ∈ [0,1] for all 2. Methodology 􏽥 􏽥 􏽥 A A u ∈ U. In addition, the degree of hesitancy of each u ∈ U 2.1. On Fuzzy Sets and Fuzzy Logic. Fuzzy logic constitutes an with respect to A is given by the following expression: alternative to classical logic to deal with decision making by 1/2 introducing some degree of vagueness to assess situations or 2 2 (1) π (u) � 􏼒1 − μ (u) − ] (u)􏼓 . 􏽥 􏽥 􏽥 objects. A A A In 1965, membership functions and fuzzy sets were mathematically introduced to model the level of incertitude We would like also to point out that further general- and ambiguity in regard to human thinking [22]. In this way, izations of the TOPSIS approach under fuzziness have been the domain of a membership function turns into the unit proposed in the literature on the context of interval-valued interval [0,1] rather than the set {0,1}. As such, in the spherical fuzzy sets ([42]). context of the classical logic, the membership of an element to a set is completely determined, whereas in the fuzzy logic, 2.2. Spherical Fuzzy Sets and Operators. Going beyond, in such a membership could be measured gradually. ([33], Definition 3), the spherical fuzzy sets were first in- .e application of fuzzy logic to real-life contexts results troduced to allow the hesitancy of a decision maker be especially appropriate when the rules of membership of a defined independently ofher/hisdegrees ofmembershipand given element to a certain class cannot be stated clearly [39]. nonmembership in regard to an alternative with respect to a In fact, the category itself may depend on the context. criterion. .eir definition, which appears in this section, In fuzzy logic, the level of membership of an element to a consists of using the Euclidean distance on a spherical class is quantified by a real number that belongs to the volume rather than measuring arc distances on the surface of interval [0,1]. In this way, if the membership level of an a sphere, as it was proposed in [43, 44]. element to a certain set is close to 1, then it is more likely that Next, we recall how todefinethem. Let U be a universe of such an element belongs to that class. On the contrary, if that discourse. A spherical fuzzy set (SFS, hereafter) of U is a set degree of membership is close to 0, then it is more unlikely of the form that it belongs to that set. Let A⊆U, where U refers to a universe of discourse. A membership function can be defined as a rule of association, A � 􏼚⟨μ (u), ] (u), π (u)⟩: u ∈ U􏼛, (2) 􏽥 􏽥 􏽥 A A A S S S μ : U ⟶ [0,1], that maps every x ∈ U to its degree of membership to A, μ (x) ∈ [0,1]. Hence, the concept of a where μ , ] ,and π : U ⟶ [0,1] are the functions that 􏽥 􏽥 􏽥 A A A S S S membership function can be further extended to a quali- quantify the degree of membership, nonmembership, and tative setting by means of linguistic labels and variables that hesitancy of each u ∈ U to the SFS A , respectively. .ey are more accurate than crisp numbers in such contexts [40]. 2 2 2 satisfy that μ (u) + ] (u) + π (u) ∈ [0,1] for all u ∈ U. 􏽥 􏽥 􏽥 Reciprocally, each function μ: U ⟶ [0,1] allows defining a A A A S S S Let ε � ⟨μ (u), ] (u), π (u)⟩: u ∈ U be a spherical membership function that is associated to a certain fuzzy set, 􏽥 􏽥 􏽥 A A A S S S fuzzy number (SFN, hereafter). .e product of ε by a scalar thus depending on the context it is applied to and the concept it represents. In this way, several functions have λ>0 was defined as follows: 4 Advances in Astronomy 1/2 1/2 λ λ λ ⎨ ⎬ ⎧ ⎫ 2 λ 2 2 2 (3) λ · ε � 􏼪􏼠1 − 􏼒1 − μ (u)􏼓 􏼡 , ] (u), 􏼠􏼒1 − μ (u)􏼓 − 􏼒1 − μ (u) − π (u)􏼓 􏼡 􏼫: u ∈ U , ⎩ 􏽥 􏽥 􏽥 􏽥 􏽥 ⎭ A A A A S S S S S whereas the λ− power of ε is given by 1/2 1/2 λ λ λ ⎧ ⎨ ⎫ ⎬ λ λ 2 2 2 2 ε � 􏼪μ (u), 􏼠1 − 􏼒1 − ] (u)􏼓 􏼡 , 􏼠􏼒1 − ] (u)􏼓 − 􏼒1 − ] (u) − π (u)􏼓 􏼡 􏼫: u ∈ U . (4) S ⎩ 􏽥 􏽥 􏽥 􏽥 􏽥 ⎭ A A A A A S S S S ([33], Definition 5). We also refer the reader to ([33], recall the definitions of spherical weighted arithmetic mean Definition 6) for some properties regarding products of SFS (SWAM, ([33], Definition 7)) and spherical weighted geo- by scalars (with respect to ⊕) and powers of SFSs (with metric mean (SWGM, [33], Definition 8)) operators with respect to the operator ⊗.) respect to a normalized list ofweights thatwill be usedinthis On the other hand, let ω � (ω , ω , . . . , ω ) be a nor- study. Let 􏼈ε : i � 1, . . . , n􏼉 be a finite list of triangular fuzzy 1 2 n i malized list of weights, i.e., ω ∈ [0,1] for all i � 1,2, . . . , n numbers, where ε � ⟨μ (u), ] (u), π (u)⟩ for each i i 􏽥 􏽥 􏽥 A A A S S S i i i with 􏽐 ω � 1. It is worth mentioning that several oper- i � 1, . . . , n. .en, i�1 i ators for SFNs were introduced in ([33], Section 3). Next, we SWAM ε , . . . , ε 􏼁 ≔ 􏽘 ω ε � ω ε + · · · + ω ε ω 1 n i i 1 1 n n i�1 ω 1/2 ω ω 1/2 n n n n i i i ⎧ ⎨ ⎫ ⎬ 2 ω 2 2 2 ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ � 1 − 􏽙 1 − μ (u) , 􏽙 ] (u), 􏽙 1 − μ (u) − 􏽙 1 − μ (u) − π (u) : u ∈ U , 􏼪 􏼠 􏼡 􏼠 􏼡 􏼠 􏼡 􏼫 􏽥 􏽥 􏽥 􏽥 ⎩ ⎭ A 􏽥 A A A S A S S S i i i i i�1 i�1 i�1 i�1 ω ω ω i 1 SWGM ε , . . . , ε 􏼁 ≔ 􏽘 ε � ε + · · · + ε ω 1 n i 1 n i�1 1/2 1/2 ω ω ω n n i n i n i ⎧ ⎨ ⎫ ⎬ ω 2 2 2 2 i ⎡ ⎢ ⎡ ⎢ ⎣ ⎤ ⎦ ⎣ ⎤ ⎦ � 􏼪􏽙 μ (u), 1 − 􏽙 􏼠1 − ] (u)􏼡 , 􏽙 􏼠1 − ] (u)􏼡 − 􏽙 􏼠1 − ] (u) − π (u)􏼡 􏼫: u ∈ U . ⎩ 􏽥 􏽥 􏽥 􏽥 ⎭ 􏽥 A A A A S S S S S i i i i i�1 i�1 i�1 i�1 (5) Finally, for a SFN, ε � ⟨μ (u), ] (u), π (u)⟩: u ∈ U, decision matrix constitutes the starting point to apply the 􏽥 􏽥 􏽥 A A A S S S recall that its score was defined in the following terms ([33], SFS TOPSIS approach, which includes the following stages. Definition 9): Step 1: the evaluation matrices of alternatives and 2 2 criteria have to be filled in by the decision makers. With Score(ε) � 􏼒μ (u) − π (u)􏼓 − 􏼒] (u) − π (u)􏼓 . 􏽥 􏽥 􏽥 􏽥 A A A A this aim, the linguistic labels that appear in Table 1 S S S S should be used. (6) Step 2: the judgments of the decision makers have to be aggregated by means of the SWAM (respectively, the SWGM) operator as defined above. Specifically, 2.3. 2e SFS TOPSIS. Interestingly, some extensions of fuzzy Step 2.1: the individual valuations of the decision sets have led to new versions of the TOPSIS approach (e.g., makers in regard to the relative importance of each [33] for a literature review concerning them). In this study, criterion have to be combined to obtain the weights of we shall apply the SFS version of the TOPSIS approach (SFS the criteria. TOPSIS, hereafter) that was first introduced ([33], Section 5) and already applied in the literature (e.g., [45, 46]). Step 2.2: construction of the aggregated spherical fuzzy First, recall that a MCDM problem can be expressed by a decision matrix by taking into account the judgments decision matrix whose entries contain the evaluation of the of the decision makers. In fact, let us denote the evaluation of the alternative X alternatives with respect to each criterion. .us, first, let with respect to the m ≥2, X � X , X , . . . , X be a finite set of alternatives, criterion C by C (X ) � (μ , ] , π ) for all 􏼈 􏼉 j j i ij ij ij 1 2 m i � 1, . . . , m and all j � 1, . . . , n. Hence, let C � 􏼈C , C , . . . , C 􏼉 be a discrete set of criteria, and ω � 1 2 n (ω , ω , . . . , ω ) be a normalized list of weights, i.e., D ≔ (C (X )) be the decision matrix of a SFS j i m×n 1 2 n MCDM problem. ω ∈ [0,1] for all i � 1,2, . . . , n and 􏽐 ω � 1. .en, that i i�1 i Advances in Astronomy 5 Table 1: Linguistic terms and their associated linguistic labels and SFNs (μ, ], π). Linguistic term Label μ ] π Absolutely more importance AMI 0.9 0.1 0.1 Very high importance VHI 0.8 0.2 0.2 High importance HI 0.7 0.3 0.3 Slightly more importance SMI 0.6 0.4 0.4 Equally importance EI 0.5 0.5 0.5 Slightly low importance SLI 0.4 0.6 0.4 Low importance LI 0.3 0.7 0.3 Very low importance VLI 0.2 0.8 0.2 Absolutely low importance ALI 0.1 0.9 0.1 Step 3: construction of the aggregated weighted function (equation (6)). To tackle with, use 2 2 spherical fuzzy decision matrix. Once the alternatives Score (C (X )) � (μ − π ) − (] − π ) . j iω ijω ijω ijω ijω have been ranked and the weights of the criteria de- Step 5: calculation of both the Spherical Fuzzy Negative termined, calculate D � (C (X )) , where j iω m×n Ideal Solution (SF-NIS), denoted by X , and the C (X ) � (μ , ] , π ) for all i � 1, . . . , m and all Spherical Fuzzy Positive Ideal Solution (SF-PIS), j iω ijω ijω ijω j � 1, . . . , n. Notice that ([33], equation (14)) is applied denoted by X , throughout the following expressions, in this step. respectively: Step 4: defuzzification of the aggregated weighted spherical fuzzy decision matrix is by applying the score X ≔ 􏽮⟨C , min􏽮Score 􏼐 C X 􏼁􏼑 : i � 1, . . . , m􏽯⟩ : j � 1, . . . , n􏽯 j j iω − − − � 􏽮⟨C , 􏼐μ , ] , π 􏼑⟩ : j � 1, . . . , n􏽯, j j j (7) X ≔ 􏽮⟨C , max􏽮Score 􏼐 C X 􏼁􏼑 : i � 1, . . . , m􏽯⟩ : j � 1, . . . , n􏽯 j j iω ∗ ∗ ∗ � 􏽮⟨C , 􏼐μ , ] , π 􏼑⟩ : j � 1, . . . , n􏽯. j j j Step 6:calculation ofthe normalized Euclidean distance (respectively, the SF-PIS) for all i � 1, . . . , m by means ([47]) of each alternative X with respect to the SF-NIS of the next expressions: 􏽶������������������������������������ � 1 2 2 2 − − − − D X , X � 􏽘 μ − μ + ] − ] + π − π , 􏼁 􏼔􏼐 􏼑 􏼐 􏼑 􏼐 􏼑 􏼕 i ij i ij i ij i 2n j�1 (8) 􏽶������������������������������������ � 1 2 2 2 ∗ ∗ ∗ ∗ D X , X 􏼁 � 􏽘􏼔􏼐μ − μ 􏼑 + 􏼐] − ] 􏼑 + 􏼐π − π 􏼑 􏼕, i ij i ij i ij i 2n j�1 where C (X ) � (μ , ] , π ) for all i � 1, . . . , m and all Step 8: calculation of the closeness ratio as provided in j i ij ij ij j � 1, . . . , n. ([33], equation (37)), thus taking the absolute value of the expression suggested in [48], namely, Step 7: calculation of the minimum distance with re- ∗ − spect to the SF-PIS as well as the maximum distance D X , X 􏼁 D X , X 􏼁 i i ξ X􏼁 � − , forall i � 1, . . . , m. i + − with respect to the SF-NIS, i.e., D X , X 􏼁 D X , X 􏼁 min i max i (10) ∗ ∗ D X , X � min D X , X : i � 1, . . . , m , 􏼁 􏼈 􏼁 􏼉 min i i (9) Step 9: list the alternatives by increasing the order of − − D X , X 􏼁 � max􏼈D X , X 􏼁 : i � 1, . . . , m􏼉. max i i their corresponding closeness ratios. In this way, the 6 Advances in Astronomy optimal alternative is the one that appears rated in the board a spacecraft (namedthe “shepherd”) that points a first position of that ranking. highly collimated high-velocity ion beam at NEA. Si- multaneously, a secondary thruster points in the op- posite direction to maintain a uniform distance from 3. Assessment of the NEA the asteroid [49, 53]. In this way, a hovering distance of Deflection Technologies twice the diameter of the targeted asteroid allows leaving the gravitational force of NEA negligible [54]. 3.1. Assumptions of Our Study. We would like to highlight Interestingly, the IBD rendezvous spacecraft may be that the primary goal in the current study is to perform a sent to NEA beforehand, which allows decreasing the fuzzy MCDM analysis with the aim to assess the following uncertainty in regard to the orbit of the asteroid. .is NEA deflection technologies: kinetic impactor (KI), en- could be understood as an advantage of the IBD with hanced gravity tractor (EGT), ion-beam deflection (IBD), respect to the KI approach. Moreover, IBD permits an and laser ablation (LA). Such alternatives will be evaluated accurate retargeting of the impact point at the asteroid, with respect to the 8 criteria that have been described in which becomes especially useful in regard to large Section 3.3. Furthermore, to deal with that task, the infor- asteroids that may be deflected only a few Earth radii mation provided by a group of experts (Section 3.4) will (except if a nuclear blast is utilized). Nevertheless, a allow us to calculate the aggregated relative importance of a satisfactory level of autonomy regarding the hovering given alternative for each criterion in terms of linguistic of the shepherd has not yet been reached. In addition, a labels that are identified with SFNs (Section 4). greater accuracy concerning the pointing of the beam .e deflection of an asteroid consists of accelerating the still lacks [52]. object just enough in such a way it crosses Earth’s orbit by a Alternative A : enhanced gravity tractor (EGT). .e minimum distance from the point the NEA would have gravity tractor (GT) consists of a spaceship that hovers crossed it providing that it had not been deflected. over a targeted NEA being aimed at redirecting its .e assumptions of our study that were disclosed to the trajectory by taking advantage of the gravitational at- group of experts were as follows. First, we intend to conduct traction between the asteroid and the spacecraft. Note a (nonnuclear) primary deflection greater than or equal to that the GT constitutes a trim/observer approach itself twice Earth’ radii (excluding the KI) on a threatening NEA [54]. In the case of the enhanced gravity tractor (EGT), with an estimated diameter lower than or equal to 250m. the hovering spacecraft increases its mass by removing Also, the warning time was assumed to range between 5 and some rocks or regolith from the targeted NEA. .at 30 years. amount of mass is calculated in such a way that its We would also like to point out that the assignment of a thrusters at full power and in the general direction of threatening asteroid to one of the four orbital groups the NEA do not increase the distance between the (Apollos, Atens, Atiras, or Amors) has not been specifically asteroid and the spaceship. In fact, a uniform separa- considered in the current analysis. Alternatively, and re- tion distance between the spacecraft and the targeted garding the orbital dynamics of NEAs, they have assumed asteroid has to be preserved, so the thrusters slowly those assumptions that can be found in [49] and ([50], impulse the whole system in the opposite direction of equation (7)). the asteroid (to reduce the velocity of the NEA) or in .e alternatives described in ([19], Section 3.1) and the the actual direction of the object (thus improving its criteria appeared in ([19], Section 3.2) are also considered velocity) [49, 55]. throughout this study. Along the next two sections, we summarize them for the sake of completeness. Alternative A : laser ablation (LA). .e energy from the combined effects of a set of phase locked laser am- plifiers is continuously impinged on NEA, thus ejecting 3.2. Description of the Alternatives for NEA Deflection some material away from its surface and having an effect on the velocity of the targeted asteroid Alternative A : the kinetic impactor (KI) consists of [49, 54, 56]. placing a spaceship on a trajectory to crash a NEA. .is way, both the momentum and the velocity of the tar- geted asteroid would be modified [49]. It is worth 3.3. 2e Selected Attributes. In this study, all the following mentioning that it is already possible to impact an as- criteria described will be evaluated by means of scales of teroid at a high velocity as NASA’s Deep Impact mission importance that are given in terms of SFNs. With this aim, it reported in 2005 [51]. According to the space science used the information provided by our group of experts. community, one of the advantages of the KI deflection technology lies in its immediate effect as well as the high Attribute C : build time. .is criterion, Tb, could be level of momentum that may be delivered to the targeted understood in the following terms: asteroid. However, there is still a nonnegligible level of uncertainty regarding the amount of momentum that is Tb � requiredwarningtime − Tr − T, (11) effectively delivered to the NEA [52]. Alternative A : the technology under the ion beam where the required warning time is the timeframe from deflection (IBD) mainly consists of an ion thruster on the discovery of the threat to the predicted date of Advances in Astronomy 7 collision, Tr is the rendezvous time, and T is defined for quantified separately from the TRL to identify those each NEA deflection alternative as follows: specific risks that may appear when applying each NEA deflection technique. It is worth mentioning that a scale T + T , if thealternativeiseitherEGTorIBD, based on the Goddard risk matrix has been proposed to ⎧ ⎪ 1 2 address the risk assessment ([54, 57–59]). T , inthecaseof LA, T � 3.4. Our Group of Experts. A group of 10 researchers whose expertise areas include NEA deflection technologies com- 1 ΔX , if thealternativeisKI. pleted the questionnaires sent by the authors, thus providing 3 ΔV some valuable information in regard to the alternatives and (12) criteria involved in our study. .eir affiliations were as follows: Langley Research Center and Jet Propulsion Lab- In equation (12), T denotes the active deflection time, oratory of the National Aeronautics and Space Adminis- T is the coasting time, ΔX denotes the required de- tration (three experts), Planetary Defence Office and Galileo flection distance (in m), and ΔV is the achievable ve- Mission of the European Space Agency (two experts), In- locity change (in m/s). It should be highlighted that the stitute of Space Sciences at the Spanish National Research build time does not include the time each technology Council, Institute for Aerospace Studies at the University of needs to achieve the TRL 6 [49]. Observe that the build Toronto, Department of Physics Applied to Aeronautical time is especially important when the warning time is Engineering of the Polytechnic University in Madrid, De- short which, in turn, may be produced by a significant partment of Mathematics and SpaceDyS at the University of uncertainty concerning the probability of impact of the Pisa, and Laboratory of Applied Physics at the Johns asteroid with the Earth. Hopkins University. Attribute C : duration of the active deflection. It is the time needed to achieve a deflection of the targeted 4. Results and Discussion asteroid of at least twice Earth’ radius (except in the case of the KI). As stated above, the scale of importance appearing in Table 1 ([33]), which identifies a set of linguistic labels with their Attribute C : asteroid rotation. As it was suggested by corresponding SFNs, is considered to assess the criteria and our group of experts, it is unlikely to tackle with a fast the alternatives involved in the current study. To deal with, rotator for objects with estimated diameters ranging the information provided by our advisory board was used. In 150–240m. this way, Table 2 provides the weights of the criteria de- Attribute C : asteroid composition. It is worth noting scribed in terms of SFNs via the SWAM operator. that the efficiency of several NEA deflection approaches .e following order of preference regarding our set of may strongly depend on this criterion. For instance, LA criteria holds from the results appeared in Table 2: may not work appropriately when being applied on metallic surfaces since the heat produced may be C > C > C > C > C > C > C > C . (13) 1 2 8 4 7 5 3 6 conducted away. According to equation (13), C (build time) appears Attribute C : asteroid structure. .is is related to the ranked in the first position being followed by C (active porosity and the internal structure of the object instead deflection duration), C (mission risk), C (asteroid com- 8 4 of the surface material structure of the asteroid or its position), C (level of maturity), and C (asteroid structure). 7 5 friability. It should be pointed out that KI is sensitive to .en, C (asteroid rotation) and C (asteroid shape) are 3 6 the internal structure of the object and its porosity, found with a same level of importance. In this regard, a good which mayaffect the momentum transfer. Also, itcould reference to identify a set of linguistic labels with their influence the ability of EGTto collect material from the corresponding SFNs can be found in [33]. NEA surface. When applying the triangular fuzzy set (TFS, hereafter) Attribute C : asteroid shape. A great variety of irregular 6 version of the analytic hierarchy process (AHP) approach contours may appear in targeted NEAs. ([19]), the next order of preference was found for our set of Attribute C : level of maturity of a deflection tech- criteria by means of the valuable information provided by nology or technological readiness level (TRL). .is is a the group of experts: standardized scale suggested by NASA to evaluate the C > C > C > C > C > C > C > C . (14) 1 2 7 4 5 8 6 3 current level of development of a technology in regard to a desired maturity level for that approach. In this Hence, from both equations (13) and (14), it holds that study, targeted maturity means a redirection technol- the criteria C (level of maturity) and C (asteroid com- 7 4 ogy for asteroids that is ready to be proved in space at position) interchange their relative level of importance from the next level, which is equivalent to TRL 6 ([54]). the SWAM operator to the TFS version of AHP. Specifically, Attribute C : mission risk. It takes into account the it holds that C appears as a more important criterion than possibility of a technological failure or an unsuccessful C when applying SFS TOPSIS. It is also worth pointing out result regarding the asteroid deflection mission. .is is that the attribute C (mission risk) has been assigned a 8 8 Advances in Astronomy Table 2: Weights of the criteria in terms of SFNs. Weights Spherical fuzzy numbers Criteria μ ] π C (build time) 0.8 0.2 0.2 C (active deflection duration) 0.6 0.5 0.4 C (asteroid rotation) 0.4 0.6 0.4 C (asteroid composition) 0.6 0.4 0.4 C (asteroid structure) 0.5 0.5 0.4 C (asteroid shape) 0.4 0.6 0.4 C (level of maturity) 0.6 0.4 0.3 C (mission risk) 0.6 0.5 0.3 greater level of importance when applying the SWAM op- alternatives through the SFS TOPSIS procedure, we can use erator than TFS AHP. either the SWAM operator or the SWGM operator, which .e next step was to generate a new decision matrix constitutes an advantage of SFSs over TFSs to deal with fuzzy (Table 3) that contains the assessment of the alternatives for series. In fact, applying geometric mean to generate the such criteria from the judgments provided by the experts via aggregated matrix of decision (by scales of importance linguistic labels defined in terms of SFNs (Table 1). through TFS) could provoke that the TOPSIS algorithm may From that decision matrix and taking into account the not be executed. In this section, we compare the ranking of alternatives weightsofthe criteria (obtained bythe SWAM operator), the SFS TOPSIS methodology was applied to rank the alter- providedbytheSFSTOPSISapproachandtheSWAMoperator natives of our case of study. In this way, Table 4 displays a (Section 4) vs. the one provided by the SFS TOPSIS approach comparison between the rankings provided by the SFS when the SWAM operator is applied. In both cases, the weights TOPSIS approach vs. the one obtained by means of TFS of the criteria were calculated according to the information TOPSIS methodology. provided by our group of experts. We found that both rankings .e SFS TOPSIS-based ranking in Table 4 shows that the of alternatives were found to be the same (Table 5), which alternatives LA and EGTdo interchange their positions with suggests that the choice of the SWGM (respectively, the respect to their TFS TOPSIS rankings. .is could be due to SWAM) operator does not influence the ranking positions of the alternatives. Only slight deviations were found in regard to the greater SFS TOPSIS relative importance that has been assigned to the criterion C (asteroid composition) to the the absolute values of the differences between the closeness ratios of pairs of consecutive alternatives. As such, the choice of detriment of C (level of maturity). In fact, a greater valu- ation for that criterion (Table3) placesLA with respect to C , one of such operators to the detriment of the other would thus being followed by KI, IBD, and EGT. mainly depend on the computational cost required to carry out Similarly, since mission risk (criterion C ) for both al- the corresponding calculations. However, in this case, the ternatives LA and EGT is greater than the one for both KI computational cost is similar for both operators. and IBD, and it was the 3rd most important criterion according to SWAM operator (equation (13)), both LA and EGT become closer from KI and IBD in the SFS TOPSIS 5.2. On the Dependence of the SFS TOPSIS Approach on the ranking. Judgments from the Group of Experts. Next, we highlight the influence of the information provided by the group of ex- 5. Sensitivity Analyses perts over our SFS TOPSIS rankings of alternatives. With this aim, a pair of SFS TOPSIS rankings was obtained (one Two sensitivity analyses have been carried out in this section per each operator, SWAM and SWGM) by assuming that the with the aim to validate the robustness of the results pro- weights of all the criteria are the same. First, as shown in vided in Section 4. In fact, the first one consists of carrying Table 6, it holds that the positions of the four alternatives out the SFS TOPSIS calculations by means of the SWGM involved in the present study were found to be the same in operator and taking into account the weights of the criteria both SFS TOPSIS-based rankings. However, such SFS as provided by the judgments from the group of experts TOPSIS-based rankings differ from the one provided in [19], (Section 5.1), whereas the second sensitivity analysis repeats where TFS TOPSIS was used to assess these four NEA the SFS TOPSIS calculations by both operators, SWAM and deflection technologies. In fact, the use of one of such op- SWGM, but assuming that the weights of all the criteria are erators (SWAM or SWGM) may lead to some changes in the same. Two interesting facts follow from the results regard to the rankings of alternatives as provided by the SFS provided by each sensitivity analysis. TOPSIS approachwith respect tothe rankingsof alternatives provided by the TFS TOPSIS procedure. 5.1. On the Effect of the SWGM Operator. Recall that Specifically, observe that KI keeps the first position in all arithmetic mean is used to aggregate the valuations provided such SFS TOPSIS-based rankings. On the other hand, LA is ranked in 3rd position in both SFS TOPSIS rankings when by the experts to generate the decision matrix of the TOPSIS approach (e.g., [19]). However, when ranking the the weights of the criteria are calculated from the group of Advances in Astronomy 9 Table 3: Assessment of the alternatives for criteria C − C in terms of SFNs as provided by the SFS TOPSIS approach via the SWAM 1 8 operator. Criteria C C C C C C C C 1 2 3 4 5 6 7 8 Alternatives μ ] π μ ] π μ ] π μ ] π μ ] π μ ] π μ ] π μ ] π A (KI) 0.6 0.4 0.3 0.7 0.3 0.3 0.5 0.5 0.3 0.6 0.4 0.3 0.6 0.4 0.3 0.6 0.4 0.3 0.7 0.3 0.3 0.6 0.4 0.4 A (IBD) 0.5 0.5 0.4 0.4 0.6 0.4 0.5 0.5 0.3 0.5 0.5 0.4 0.3 0.7 0.3 0.5 0.5 0.4 0.5 0.5 0.4 0.6 0.4 0.3 A (EGT) 0.4 0.6 0.4 0.3 0.7 0.3 0.4 0.6 0.3 0.4 0.6 0.3 0.5 0.6 0.3 0.4 0.6 0.4 0.4 0.6 0.3 0.7 0.4 0.3 A (LA) 0.4 0.6 0.3 0.5 0.6 0.4 0.6 0.4 0.3 0.7 0.3 0.3 0.5 0.5 0.4 0.5 0.5 0.3 0.4 0.7 0.2 0.7 0.4 0.3 Table 4: Comparison of rankings of alternatives between the SFSTOPSIS (SWAM operator) approach to TFS TOPSISprocedure ([19]). .e weights of the criteria were obtained from the information provided by the group of experts. SFS TOPSIS (SWAM) ranking TFS TOPSIS ranking Alternative Closeness ratio Ranking R Ranking A (KI) 0.00 1 3.07 1 A (IBD) 5.39 2 1.85 2 A (EGT) 7.82 4 1.16 3 A (LA) 5.99 3 0.74 4 Table 5: SFS TOPSIS (SWGM) ranking of alternatives as described in the first analysis of sensitivity. .e weights of the criteria were chosen to be those calculated from the information provided by the group of experts. .e “Diff.” column contains the differences between the closeness ratios (in absolute value) from pairs of alternatives. SFS TOPSIS (SWGM) ranking of alternatives (criteria weights from experts) Alternative Closeness ratio Rank Diff. A (KI) 0.00 1 – A (IBD) 3.24 2 3.24 A (EGT) 4.81 4 1.57 A (LA) 3.86 3 0.95 Table 6: SFS-based rankings of alternatives for both operators, SWAM and SWGM, under the assumption that the all the criteria are equally weighted (Section 5.2) vs. TFS-based ranking of alternatives for equally weighted criteria ([19]). SFS rankings (equally weighted criteria) TFS ranking (equally weighted criteria) Alternative SWAM rank SWGM rank Rank A (KI) 1 1 1 A (IBD) 3 3 2 A (EGT) 4 4 3 A (LA) 2 2 4 experts, though it appears ranked in 4th position in the TFS 6. Conclusions TOPSIS ranking for equally weighted criteria. However, it In this section, we summarize the main conclusions to be occupies 2nd position in both SFS TOPSIS rankings for highlighted from the study carried out. equally weighted criteria. .e next ranked alternatives are First of all, it is worth pointing out that the KI alternative IBD and EGT (notice that such a consecutive order for such is consolidated as the best choice for active NEA deflection alternatives coincides with the one that appears in the TFS purposes. In fact, the results thrown by the SFS TOPSIS TOPSIS-based rankings). methodology coincide with all those presented in [19] when .is analysis of sensitivity highlights that, unlike the TFS it was applied to the TFS TOPSIS approach with the same TOPSIS procedure, the weights of the criteria should be purpose. assigned carefully when applying a SFS TOPSIS approach However, this study highlights the fact that a SFS TOPSIS- since variations regarding the weights of the criteria may based ranking of alternatives may vary widely when a sensitivity induce changes of positions among the ranked alternatives. 10 Advances in Astronomy analysis is carried out. Specifically, we showed that the ranking acknowledges the support of Grants TIN2017-86647-P and of alternatives as provided by the SFS TOPSIS approach when 19882-GERM-15 from the Spanish Ministry of Economy ´ ´ taking into account the information from the group of experts and Competitiveness (MINECO) and Fundacion Seneca becomes quite different from the SFS TOPSIS ranking of al- (Region ´ de Murcia), respectively. .is work could have not ternatives we obtained provided that all the weights of the been carried out without the generous collaboration of criteria are assumed to be the same. In other words, this study experts from the following institutions: Langley Research reveals a nonnegligible dependence of the SFS TOPSIS results Center and Jet Propulsion Laboratory of the National from the judgments that could be provided by the group of Aeronautics and Space Administration (NASA), Planetary experts with the aim of ranking a set of alternatives. Defence Office and Galileo Mission of the European Space On the other hand, we would like to mention that the use Agency (ESA), Laboratory of Applied Physics at the Johns of either the SWAM operator or the SWGM operator is Hopkins University, Institute for Aerospace Studies at the indifferent when carrying out SFS TOPSIS calculations. In University of Toronto, Institute of Space Sciences at the fact, only slight differences between the absolute value of the Spanish National Research Council, Department of Physics closeness ratios from pairs of consecutive alternatives were Applied to Aeronautical Engineering of the Polytechnic found with a similar computational cost. .is fact could be University in Madrid, and the Department of Mathematics understood as an advantage of SFS TOPSIS approach to the and SpaceDyS at the University of Pisa. .e survey results detriment of TFS TOPSIS. In fact, the latter only uses are based on the expert opinions of the participating indi- arithmetic mean in contexts where it is necessary to utilize viduals and do not necessarily reflect the official positions of the lowest level of the standard TFS scale of importance. their parent institutions. .e authors would also like to Note that a consistency analysis regarding the judgments express their gratitude to the editor and anonymous re- provided by the group of experts cannot be carried out viewers whose suggestions, comments, and remarks have through the SWAM (respectively, the SWGM) operator. allowed them to enhance the quality of this paper. Notwithstanding, recently, it has been contributed in [60] a SFS version of the AHP methodology, which encourages us References to calculate the weights of the criteria throughout that novel approach as a future research task. Also, we would like to [1] D. E. 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Assessment of Near-Earth Asteroid Deflection Techniques via Spherical Fuzzy Sets

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Hindawi Advances in Astronomy Volume 2021, Article ID 6678056, 12 pages https://doi.org/10.1155/2021/6678056 Research Article Assessment of Near-Earth Asteroid Deflection Techniques via Spherical Fuzzy Sets M. Ferna ´ ndez-Martı´nez and J. M. Sa ´ nchez-Lozano University Centre of Defence at the Spanish Air Force Academy, MDE-UPCT, 30720–Santiago De La Ribera, Regio´n De Murcia, Spain ´ ´ Correspondence should be addressed to M. Fernandez-Martınez; manuel.fernandez-martinez@cud.upct.es Received 23 October 2020; Revised 22 January 2021; Accepted 18 February 2021; Published 8 March 2021 Academic Editor: Maria Gritsevich Copyright © 2021 M. Ferna´ndez-Mart´ınez and J. M. Sa´nchez-Lozano. .is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Extensions of fuzzy sets to broader contexts constitute one of the leading areas of research in the context of problems in artificial intelligence. .eir aim is to address decision-making problems in the real world whenever obtaining accurate and sufficient data is not a straightforward task. In this way, spherical fuzzy sets were recently introduced as a step beyond to modelize such problems more precisely on the basis of the human nature, thus expanding the space of membership levels, which are defined under imprecise circumstances. .e main goal in this study is to apply the spherical fuzzy set version of Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), a well-established multicriteria decision-making approach, in the context of planetary defense. As of the extraction of knowledge from a group of experts in the field of near-Earth asteroids, they rated four deflection technologies of asteroids (kinetic impactor, ion beam deflection, enhanced gravity tractor, and laser ablation) that had been previously assessed by means of the classical theory of fuzzy series. .is way, a comparative study was carried out whose most significant results are the kinetic impactor being the most suitable alternative and the spherical fuzzy set version of the TOPSIS approach behaves more sensitively than the TOPSIS procedure for triangular fuzzy sets with regard to the information provided by our group of experts. object broke up at an altitude of 30km, thus injuring more 1. Introduction than 1500 people [2, 5, 6]. Meteoroids reach the Earth mainly as small rocks and fragile .e latter are clear examples of fatal consequences that aggregates which appear as a consequence of the decay of impact of these kinds of objects with the Earth may result in. asteroids and comets. In this way, the tiny dusts that arrive at .e impact of a single massive cosmic body might leave a fairly the Earth each day amounts to a mean of approximately 100 largecraterinthesurfaceoftheEarthorinduceatsunamiinthe tons [1]. caseitcollideswiththesurfaceoftheocean,thuscontributingto Although it is true that larger objects will unlikely reach overall risk [7–10]. Furthermore, air blasts derived from de- Earth’s orbit, a potential impactor may dramatically affect struction of meteoritic bodies in Earth’s atmosphere may the life and the climate in our planet. For instance, a 10km provoke not only falls of clouds of fragments but also major wide impactor led to the so-called Cretaceous-Paleocene injuries on Earth’s surface even if the object does not make a extinction that happened 64 million years ago [2, 3]. In 1908, touchdown. Indeed, such explosions may cause from glass the fragmentation at a low altitude (5–10km) of an asteroid breakage (for air blasts greater than 15m/s) to extreme dis- with an estimated diameter d ∈ [30,50]m destroyed around tortions in steel structures of bridges or buildings which may 2000km of woodland [4], what was known as the Tunguska lead to their collapse (for air blasts greater than 200m/s) [11] event. Recently, in 2013, a bolide with an estimated weight of and ([12], Table 1). .erefore, such consequences do not only 12000 tons and an estimated diameter of 19m entered depend onthe sizeof the impactor butalsoon otherparameters Earth’s atmosphere at a relative velocity of 19km/s. .e such as entry angle, velocity, density, and shape [12]. 2 Advances in Astronomy carried out in regard to eight criteria (build time, level of Near-Earth objects (NEOs) are asteroids or comets whose perihelion distance is less than 1.3AU (about 195 maturity of a NEA deflection technology, asteroid rotation, asteroid structure, asteroid composition, asteroid shape, million km). Short period comets with orbital periods less than 200 years are known as near-Earth comets (NECs) active deflection duration, and mission risk). Such a decision whose orbits lie far away from the Earth. In this way, re- problem was addressed throughout a combination of fuzzy searchers are mainly focused on the tracking of the near- logic and multicriteria decision-making (MCDM, hereafter) Earth asteroids (NEAs) [2, 13]. approaches. .ree key reasons that support the study of NEAs are, We recall that a MCDM problem consists of looking for namely, planetary defense, scientific knowledge (e.g., deepen the best choice from a set of alternatives by a set of criteria, and to deal with, all that information is arranged into the so- our solar system origins), and mining. In this regard, the current study is allocated to planetary defense. called decision matrix [20]. A wide collection of MCDM algorithms, such as ELECTRE, OWA, VIKOR, ANP, and, In particular, the so-called potentially hazardous aster- oids are NEAs whose minimum orbit interception distance PROMETHEE, can be found in the literature. Despite MCDM methodologies had been previously with respect to Earth’s orbit is less than 0.05AU. .ey are also characterized by an estimated size greater than 140m in applied to address several issues regarding NEAs ([20, 21]), diameterand anabsolutemagnitude notgreaterthan22.0,as they were first combined with fuzzy logic recently in [19], well. Since they are able to closely approach Earth’s orbit, where four NEA deflection technologies (described in small perturbations regarding their orbits may place them Section 3.2) were rated with respect to a set consisting of on a collision path to the Earth. As such, to perform an eight criteria (Section 3.3). .e involvement of fuzzy logic exhaustive tracking of these asteroids is recommended therein was mainly motivated by the existence of qualitative criteria whose values are difficult to be specified or mea- [14, 15]. Some efforts have already been made with the aim of sured. In this way, linguistic labels that were associated with triangular fuzzy numbers, as well as (the valuable knowledge redirecting a NEA out of a risky trajectory with the Earth’, at least at a first glance. Indeed, in 2013, the NASA introduced from) the judgments provided by an international board consisting of great standing scientists in the field of NEAs, the Asteroid Redirect Mission (ARM) with a double pur- pose, namely, redirect small (i.e., less than 8m diameter) were used to quantify the level of importance of such criteria. asteroids and extract small (less than 4m) boulders from the .is way, the level of importance of each criterion was surface of a wide asteroid and place it on a distant retrograde calculated, thus leading to solve the decision problem that orbitaroundthemoon[16]..elater couldbe understoodas had been posed by means of the MCDM approach named a preliminary test of the current technical capacity to deliver Technique for Order of Preference by Similarity to Ideal and catch an object to a safe environment, despite that Solution (TOPSIS). Zadeh was the first tointroduce thatapproach to manage mission did not actually consist of redirecting a small PHA. However, that technological and scientific project was uncertainty and ambiguity when there exist attributes that are hard to be quantified [22]. In fuzzy logic, the level of cancelled in 2017 [17]. Notwithstanding, several encouraging approaches for membership of an element to a series is determined by a real NEA redirection have been posed, even though the tech- number lying in the interval [0,1], thus leading to a fuzzy nology underlying them has not yet completely developed. series. Since that pioneer work, fuzzy series have been ap- In this regard, we would like to highlight the Double As- plied not only to deal with decision problems in a wide range teroid Redirection Test (DART) that will be the first in situ of contexts (e.g., [23]) but also to contribute new viewpoints exhibition of a kinetic impactor to deflect an asteroid in in that field including intuitionistic fuzzy sets [24], Py- space. .is was formerly applied to the satellite of the binary thagorean fuzzy sets [25–28], neutrosophic fuzzy sets [29], NEA (65803) Didymos [18]. or picture fuzzy sets [30, 31]. Extensions of fuzzy sets to broader contexts constitute Redirecting an asteroid should be distinguished from deflecting it. In fact, deflecting an asteroid consists of one of the leading areas of research in the context of problems in computational intelligence. .eir aim is to modifying its trajectory to avert a potential impact with the Earth. As such, nuclear blast and kinetic impactor are ex- address decision-making problems in the real world amples of deflection techniques. On the other hand, the aim whenever obtaining accurate and sufficient data is not a of redirecting an object is to induce a controlled change in its straightforward task. In this way, spherical fuzzy sets were orbit with a further purpose, as it is the case of laser ablation/ recently introduced as a step beyond the picture fuzzy sets to sublimation, tugboat, mass driver, and ion beam ([2] and modelize MCDM problems more precisely on the basis of references therein). However, in this study, we shall un- the human nature, thus expanding the space of membership levels, which are defined under imprecise circumstances derstand a deflection technology as that one being able of deflecting or redirecting an asteroid. [32]. Even though they were introduced recently [33], it is worth mentioning that several operators, distance measures, It is worth pointing out that, in [19], an assessment involving four deflection technologies for asteroids smaller and even some applications have already been contributed [34–38]. than 250 meters in diameter (a range of sizes that covers most of the impactors with the Earth that occur in timescales Being encouraged by such novel contributions, we up to 100000 years), i.e., kinetic impactor, ion beam de- wondered to what extent ranking of alternatives would vary flection, enhanced gravity tractor, and laser ablation, was if either fuzzy sets are considered to deal with a MCDM Advances in Astronomy 3 problem or the most recent extensions of fuzzy sets (such as been widely applied including the Gaussian, the PI (or spherical fuzzy sets) are used with the same purpose. Fol- trapezoidal), and the LAMBDA (or triangular) ones. lowing the above, this study addresses that question by Several extensions of ordinary fuzzy sets have appeared applying the spherical fuzzy version of the TOPSIS approach in the literature (e.g., [33] for a chronological tracking of to a decision problem that had previously been posed in the them). Among them, we would like to highlight those context of planetary defense [19]. generalizations of fuzzy sets with a three-dimensional .e structure of this study is as follows. Section 2 membership function. containsthebasicson fuzzysetsand fuzzylogic (Section2.1), An intuitionistic fuzzy set is one of the form spherical fuzzy sets and their operators (Section 2.2), and A � ⟨μ (u), ] (u)⟩: u ∈ U , where μ : U ⟶ [0,1] is the 􏽮 􏽯 􏽥 􏽥 􏽥 A A A also includes a description regarding the generalization of membership function that quantifies the degree of mem- the TOPSIS approach to the context of spherical fuzzy sets bership of each element u to A, and ] : U ⟶ [0,1] is the (Section 2.3). .e assumptions of our study are summarized nonmembership function. .ey satisfy that μ (u)+ in Section 3.1. Next, we recall the four NEA deflection ] (u) ∈ [0,1] for all u ∈ U. In addition, π (u) � 􏽥 􏽥 A A technologies to be evaluated in this study (Section 3.2) to- 1 − μ (u) − ] (u) is defined as the degree of hesitancy of u 􏽥 􏽥 A A gether with the selected criteria (Section 3.3). Furthermore, to A. However, in real-life applications, it may happen that, some comments on the board of experts who provided us for a certain alternative satisfying a criterion, the sum of the valuable information to determine the weights of the criteria squares of the membership and nonmembership functions are provided in Section 3.4. Our results and discussion are stands not greater than 1 with their sum being greater than 1. provided in Section 4, whereas two analyses of sensitivity are With the aim to avoid the experts modifying their prefer- carried out and further discussed in Section 5. Finally, the ences, the second type intuitionistic fuzzy sets were intro- main conclusions of this study are presented in Section 6. duced by Atanassov in [41]. .ey are the form A � (μ (u), ] (u)): u ∈ U , where its membership func- 􏽮 􏽯 􏽥 􏽥 A A tion, μ : U ⟶ [0,1], and its nonmembership function, 2 2 ] : U ⟶ [0,1], satisfy that μ (u) + ] (u) ∈ [0,1] for all 2. Methodology 􏽥 􏽥 􏽥 A A u ∈ U. In addition, the degree of hesitancy of each u ∈ U 2.1. On Fuzzy Sets and Fuzzy Logic. Fuzzy logic constitutes an with respect to A is given by the following expression: alternative to classical logic to deal with decision making by 1/2 introducing some degree of vagueness to assess situations or 2 2 (1) π (u) � 􏼒1 − μ (u) − ] (u)􏼓 . 􏽥 􏽥 􏽥 objects. A A A In 1965, membership functions and fuzzy sets were mathematically introduced to model the level of incertitude We would like also to point out that further general- and ambiguity in regard to human thinking [22]. In this way, izations of the TOPSIS approach under fuzziness have been the domain of a membership function turns into the unit proposed in the literature on the context of interval-valued interval [0,1] rather than the set {0,1}. As such, in the spherical fuzzy sets ([42]). context of the classical logic, the membership of an element to a set is completely determined, whereas in the fuzzy logic, 2.2. Spherical Fuzzy Sets and Operators. Going beyond, in such a membership could be measured gradually. ([33], Definition 3), the spherical fuzzy sets were first in- .e application of fuzzy logic to real-life contexts results troduced to allow the hesitancy of a decision maker be especially appropriate when the rules of membership of a defined independently ofher/hisdegrees ofmembershipand given element to a certain class cannot be stated clearly [39]. nonmembership in regard to an alternative with respect to a In fact, the category itself may depend on the context. criterion. .eir definition, which appears in this section, In fuzzy logic, the level of membership of an element to a consists of using the Euclidean distance on a spherical class is quantified by a real number that belongs to the volume rather than measuring arc distances on the surface of interval [0,1]. In this way, if the membership level of an a sphere, as it was proposed in [43, 44]. element to a certain set is close to 1, then it is more likely that Next, we recall how todefinethem. Let U be a universe of such an element belongs to that class. On the contrary, if that discourse. A spherical fuzzy set (SFS, hereafter) of U is a set degree of membership is close to 0, then it is more unlikely of the form that it belongs to that set. Let A⊆U, where U refers to a universe of discourse. A membership function can be defined as a rule of association, A � 􏼚⟨μ (u), ] (u), π (u)⟩: u ∈ U􏼛, (2) 􏽥 􏽥 􏽥 A A A S S S μ : U ⟶ [0,1], that maps every x ∈ U to its degree of membership to A, μ (x) ∈ [0,1]. Hence, the concept of a where μ , ] ,and π : U ⟶ [0,1] are the functions that 􏽥 􏽥 􏽥 A A A S S S membership function can be further extended to a quali- quantify the degree of membership, nonmembership, and tative setting by means of linguistic labels and variables that hesitancy of each u ∈ U to the SFS A , respectively. .ey are more accurate than crisp numbers in such contexts [40]. 2 2 2 satisfy that μ (u) + ] (u) + π (u) ∈ [0,1] for all u ∈ U. 􏽥 􏽥 􏽥 Reciprocally, each function μ: U ⟶ [0,1] allows defining a A A A S S S Let ε � ⟨μ (u), ] (u), π (u)⟩: u ∈ U be a spherical membership function that is associated to a certain fuzzy set, 􏽥 􏽥 􏽥 A A A S S S fuzzy number (SFN, hereafter). .e product of ε by a scalar thus depending on the context it is applied to and the concept it represents. In this way, several functions have λ>0 was defined as follows: 4 Advances in Astronomy 1/2 1/2 λ λ λ ⎨ ⎬ ⎧ ⎫ 2 λ 2 2 2 (3) λ · ε � 􏼪􏼠1 − 􏼒1 − μ (u)􏼓 􏼡 , ] (u), 􏼠􏼒1 − μ (u)􏼓 − 􏼒1 − μ (u) − π (u)􏼓 􏼡 􏼫: u ∈ U , ⎩ 􏽥 􏽥 􏽥 􏽥 􏽥 ⎭ A A A A S S S S S whereas the λ− power of ε is given by 1/2 1/2 λ λ λ ⎧ ⎨ ⎫ ⎬ λ λ 2 2 2 2 ε � 􏼪μ (u), 􏼠1 − 􏼒1 − ] (u)􏼓 􏼡 , 􏼠􏼒1 − ] (u)􏼓 − 􏼒1 − ] (u) − π (u)􏼓 􏼡 􏼫: u ∈ U . (4) S ⎩ 􏽥 􏽥 􏽥 􏽥 􏽥 ⎭ A A A A A S S S S ([33], Definition 5). We also refer the reader to ([33], recall the definitions of spherical weighted arithmetic mean Definition 6) for some properties regarding products of SFS (SWAM, ([33], Definition 7)) and spherical weighted geo- by scalars (with respect to ⊕) and powers of SFSs (with metric mean (SWGM, [33], Definition 8)) operators with respect to the operator ⊗.) respect to a normalized list ofweights thatwill be usedinthis On the other hand, let ω � (ω , ω , . . . , ω ) be a nor- study. Let 􏼈ε : i � 1, . . . , n􏼉 be a finite list of triangular fuzzy 1 2 n i malized list of weights, i.e., ω ∈ [0,1] for all i � 1,2, . . . , n numbers, where ε � ⟨μ (u), ] (u), π (u)⟩ for each i i 􏽥 􏽥 􏽥 A A A S S S i i i with 􏽐 ω � 1. It is worth mentioning that several oper- i � 1, . . . , n. .en, i�1 i ators for SFNs were introduced in ([33], Section 3). Next, we SWAM ε , . . . , ε 􏼁 ≔ 􏽘 ω ε � ω ε + · · · + ω ε ω 1 n i i 1 1 n n i�1 ω 1/2 ω ω 1/2 n n n n i i i ⎧ ⎨ ⎫ ⎬ 2 ω 2 2 2 ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ � 1 − 􏽙 1 − μ (u) , 􏽙 ] (u), 􏽙 1 − μ (u) − 􏽙 1 − μ (u) − π (u) : u ∈ U , 􏼪 􏼠 􏼡 􏼠 􏼡 􏼠 􏼡 􏼫 􏽥 􏽥 􏽥 􏽥 ⎩ ⎭ A 􏽥 A A A S A S S S i i i i i�1 i�1 i�1 i�1 ω ω ω i 1 SWGM ε , . . . , ε 􏼁 ≔ 􏽘 ε � ε + · · · + ε ω 1 n i 1 n i�1 1/2 1/2 ω ω ω n n i n i n i ⎧ ⎨ ⎫ ⎬ ω 2 2 2 2 i ⎡ ⎢ ⎡ ⎢ ⎣ ⎤ ⎦ ⎣ ⎤ ⎦ � 􏼪􏽙 μ (u), 1 − 􏽙 􏼠1 − ] (u)􏼡 , 􏽙 􏼠1 − ] (u)􏼡 − 􏽙 􏼠1 − ] (u) − π (u)􏼡 􏼫: u ∈ U . ⎩ 􏽥 􏽥 􏽥 􏽥 ⎭ 􏽥 A A A A S S S S S i i i i i�1 i�1 i�1 i�1 (5) Finally, for a SFN, ε � ⟨μ (u), ] (u), π (u)⟩: u ∈ U, decision matrix constitutes the starting point to apply the 􏽥 􏽥 􏽥 A A A S S S recall that its score was defined in the following terms ([33], SFS TOPSIS approach, which includes the following stages. Definition 9): Step 1: the evaluation matrices of alternatives and 2 2 criteria have to be filled in by the decision makers. With Score(ε) � 􏼒μ (u) − π (u)􏼓 − 􏼒] (u) − π (u)􏼓 . 􏽥 􏽥 􏽥 􏽥 A A A A this aim, the linguistic labels that appear in Table 1 S S S S should be used. (6) Step 2: the judgments of the decision makers have to be aggregated by means of the SWAM (respectively, the SWGM) operator as defined above. Specifically, 2.3. 2e SFS TOPSIS. Interestingly, some extensions of fuzzy Step 2.1: the individual valuations of the decision sets have led to new versions of the TOPSIS approach (e.g., makers in regard to the relative importance of each [33] for a literature review concerning them). In this study, criterion have to be combined to obtain the weights of we shall apply the SFS version of the TOPSIS approach (SFS the criteria. TOPSIS, hereafter) that was first introduced ([33], Section 5) and already applied in the literature (e.g., [45, 46]). Step 2.2: construction of the aggregated spherical fuzzy First, recall that a MCDM problem can be expressed by a decision matrix by taking into account the judgments decision matrix whose entries contain the evaluation of the of the decision makers. In fact, let us denote the evaluation of the alternative X alternatives with respect to each criterion. .us, first, let with respect to the m ≥2, X � X , X , . . . , X be a finite set of alternatives, criterion C by C (X ) � (μ , ] , π ) for all 􏼈 􏼉 j j i ij ij ij 1 2 m i � 1, . . . , m and all j � 1, . . . , n. Hence, let C � 􏼈C , C , . . . , C 􏼉 be a discrete set of criteria, and ω � 1 2 n (ω , ω , . . . , ω ) be a normalized list of weights, i.e., D ≔ (C (X )) be the decision matrix of a SFS j i m×n 1 2 n MCDM problem. ω ∈ [0,1] for all i � 1,2, . . . , n and 􏽐 ω � 1. .en, that i i�1 i Advances in Astronomy 5 Table 1: Linguistic terms and their associated linguistic labels and SFNs (μ, ], π). Linguistic term Label μ ] π Absolutely more importance AMI 0.9 0.1 0.1 Very high importance VHI 0.8 0.2 0.2 High importance HI 0.7 0.3 0.3 Slightly more importance SMI 0.6 0.4 0.4 Equally importance EI 0.5 0.5 0.5 Slightly low importance SLI 0.4 0.6 0.4 Low importance LI 0.3 0.7 0.3 Very low importance VLI 0.2 0.8 0.2 Absolutely low importance ALI 0.1 0.9 0.1 Step 3: construction of the aggregated weighted function (equation (6)). To tackle with, use 2 2 spherical fuzzy decision matrix. Once the alternatives Score (C (X )) � (μ − π ) − (] − π ) . j iω ijω ijω ijω ijω have been ranked and the weights of the criteria de- Step 5: calculation of both the Spherical Fuzzy Negative termined, calculate D � (C (X )) , where j iω m×n Ideal Solution (SF-NIS), denoted by X , and the C (X ) � (μ , ] , π ) for all i � 1, . . . , m and all Spherical Fuzzy Positive Ideal Solution (SF-PIS), j iω ijω ijω ijω j � 1, . . . , n. Notice that ([33], equation (14)) is applied denoted by X , throughout the following expressions, in this step. respectively: Step 4: defuzzification of the aggregated weighted spherical fuzzy decision matrix is by applying the score X ≔ 􏽮⟨C , min􏽮Score 􏼐 C X 􏼁􏼑 : i � 1, . . . , m􏽯⟩ : j � 1, . . . , n􏽯 j j iω − − − � 􏽮⟨C , 􏼐μ , ] , π 􏼑⟩ : j � 1, . . . , n􏽯, j j j (7) X ≔ 􏽮⟨C , max􏽮Score 􏼐 C X 􏼁􏼑 : i � 1, . . . , m􏽯⟩ : j � 1, . . . , n􏽯 j j iω ∗ ∗ ∗ � 􏽮⟨C , 􏼐μ , ] , π 􏼑⟩ : j � 1, . . . , n􏽯. j j j Step 6:calculation ofthe normalized Euclidean distance (respectively, the SF-PIS) for all i � 1, . . . , m by means ([47]) of each alternative X with respect to the SF-NIS of the next expressions: 􏽶������������������������������������ � 1 2 2 2 − − − − D X , X � 􏽘 μ − μ + ] − ] + π − π , 􏼁 􏼔􏼐 􏼑 􏼐 􏼑 􏼐 􏼑 􏼕 i ij i ij i ij i 2n j�1 (8) 􏽶������������������������������������ � 1 2 2 2 ∗ ∗ ∗ ∗ D X , X 􏼁 � 􏽘􏼔􏼐μ − μ 􏼑 + 􏼐] − ] 􏼑 + 􏼐π − π 􏼑 􏼕, i ij i ij i ij i 2n j�1 where C (X ) � (μ , ] , π ) for all i � 1, . . . , m and all Step 8: calculation of the closeness ratio as provided in j i ij ij ij j � 1, . . . , n. ([33], equation (37)), thus taking the absolute value of the expression suggested in [48], namely, Step 7: calculation of the minimum distance with re- ∗ − spect to the SF-PIS as well as the maximum distance D X , X 􏼁 D X , X 􏼁 i i ξ X􏼁 � − , forall i � 1, . . . , m. i + − with respect to the SF-NIS, i.e., D X , X 􏼁 D X , X 􏼁 min i max i (10) ∗ ∗ D X , X � min D X , X : i � 1, . . . , m , 􏼁 􏼈 􏼁 􏼉 min i i (9) Step 9: list the alternatives by increasing the order of − − D X , X 􏼁 � max􏼈D X , X 􏼁 : i � 1, . . . , m􏼉. max i i their corresponding closeness ratios. In this way, the 6 Advances in Astronomy optimal alternative is the one that appears rated in the board a spacecraft (namedthe “shepherd”) that points a first position of that ranking. highly collimated high-velocity ion beam at NEA. Si- multaneously, a secondary thruster points in the op- posite direction to maintain a uniform distance from 3. Assessment of the NEA the asteroid [49, 53]. In this way, a hovering distance of Deflection Technologies twice the diameter of the targeted asteroid allows leaving the gravitational force of NEA negligible [54]. 3.1. Assumptions of Our Study. We would like to highlight Interestingly, the IBD rendezvous spacecraft may be that the primary goal in the current study is to perform a sent to NEA beforehand, which allows decreasing the fuzzy MCDM analysis with the aim to assess the following uncertainty in regard to the orbit of the asteroid. .is NEA deflection technologies: kinetic impactor (KI), en- could be understood as an advantage of the IBD with hanced gravity tractor (EGT), ion-beam deflection (IBD), respect to the KI approach. Moreover, IBD permits an and laser ablation (LA). Such alternatives will be evaluated accurate retargeting of the impact point at the asteroid, with respect to the 8 criteria that have been described in which becomes especially useful in regard to large Section 3.3. Furthermore, to deal with that task, the infor- asteroids that may be deflected only a few Earth radii mation provided by a group of experts (Section 3.4) will (except if a nuclear blast is utilized). Nevertheless, a allow us to calculate the aggregated relative importance of a satisfactory level of autonomy regarding the hovering given alternative for each criterion in terms of linguistic of the shepherd has not yet been reached. In addition, a labels that are identified with SFNs (Section 4). greater accuracy concerning the pointing of the beam .e deflection of an asteroid consists of accelerating the still lacks [52]. object just enough in such a way it crosses Earth’s orbit by a Alternative A : enhanced gravity tractor (EGT). .e minimum distance from the point the NEA would have gravity tractor (GT) consists of a spaceship that hovers crossed it providing that it had not been deflected. over a targeted NEA being aimed at redirecting its .e assumptions of our study that were disclosed to the trajectory by taking advantage of the gravitational at- group of experts were as follows. First, we intend to conduct traction between the asteroid and the spacecraft. Note a (nonnuclear) primary deflection greater than or equal to that the GT constitutes a trim/observer approach itself twice Earth’ radii (excluding the KI) on a threatening NEA [54]. In the case of the enhanced gravity tractor (EGT), with an estimated diameter lower than or equal to 250m. the hovering spacecraft increases its mass by removing Also, the warning time was assumed to range between 5 and some rocks or regolith from the targeted NEA. .at 30 years. amount of mass is calculated in such a way that its We would also like to point out that the assignment of a thrusters at full power and in the general direction of threatening asteroid to one of the four orbital groups the NEA do not increase the distance between the (Apollos, Atens, Atiras, or Amors) has not been specifically asteroid and the spaceship. In fact, a uniform separa- considered in the current analysis. Alternatively, and re- tion distance between the spacecraft and the targeted garding the orbital dynamics of NEAs, they have assumed asteroid has to be preserved, so the thrusters slowly those assumptions that can be found in [49] and ([50], impulse the whole system in the opposite direction of equation (7)). the asteroid (to reduce the velocity of the NEA) or in .e alternatives described in ([19], Section 3.1) and the the actual direction of the object (thus improving its criteria appeared in ([19], Section 3.2) are also considered velocity) [49, 55]. throughout this study. Along the next two sections, we summarize them for the sake of completeness. Alternative A : laser ablation (LA). .e energy from the combined effects of a set of phase locked laser am- plifiers is continuously impinged on NEA, thus ejecting 3.2. Description of the Alternatives for NEA Deflection some material away from its surface and having an effect on the velocity of the targeted asteroid Alternative A : the kinetic impactor (KI) consists of [49, 54, 56]. placing a spaceship on a trajectory to crash a NEA. .is way, both the momentum and the velocity of the tar- geted asteroid would be modified [49]. It is worth 3.3. 2e Selected Attributes. In this study, all the following mentioning that it is already possible to impact an as- criteria described will be evaluated by means of scales of teroid at a high velocity as NASA’s Deep Impact mission importance that are given in terms of SFNs. With this aim, it reported in 2005 [51]. According to the space science used the information provided by our group of experts. community, one of the advantages of the KI deflection technology lies in its immediate effect as well as the high Attribute C : build time. .is criterion, Tb, could be level of momentum that may be delivered to the targeted understood in the following terms: asteroid. However, there is still a nonnegligible level of uncertainty regarding the amount of momentum that is Tb � requiredwarningtime − Tr − T, (11) effectively delivered to the NEA [52]. Alternative A : the technology under the ion beam where the required warning time is the timeframe from deflection (IBD) mainly consists of an ion thruster on the discovery of the threat to the predicted date of Advances in Astronomy 7 collision, Tr is the rendezvous time, and T is defined for quantified separately from the TRL to identify those each NEA deflection alternative as follows: specific risks that may appear when applying each NEA deflection technique. It is worth mentioning that a scale T + T , if thealternativeiseitherEGTorIBD, based on the Goddard risk matrix has been proposed to ⎧ ⎪ 1 2 address the risk assessment ([54, 57–59]). T , inthecaseof LA, T � 3.4. Our Group of Experts. A group of 10 researchers whose expertise areas include NEA deflection technologies com- 1 ΔX , if thealternativeisKI. pleted the questionnaires sent by the authors, thus providing 3 ΔV some valuable information in regard to the alternatives and (12) criteria involved in our study. .eir affiliations were as follows: Langley Research Center and Jet Propulsion Lab- In equation (12), T denotes the active deflection time, oratory of the National Aeronautics and Space Adminis- T is the coasting time, ΔX denotes the required de- tration (three experts), Planetary Defence Office and Galileo flection distance (in m), and ΔV is the achievable ve- Mission of the European Space Agency (two experts), In- locity change (in m/s). It should be highlighted that the stitute of Space Sciences at the Spanish National Research build time does not include the time each technology Council, Institute for Aerospace Studies at the University of needs to achieve the TRL 6 [49]. Observe that the build Toronto, Department of Physics Applied to Aeronautical time is especially important when the warning time is Engineering of the Polytechnic University in Madrid, De- short which, in turn, may be produced by a significant partment of Mathematics and SpaceDyS at the University of uncertainty concerning the probability of impact of the Pisa, and Laboratory of Applied Physics at the Johns asteroid with the Earth. Hopkins University. Attribute C : duration of the active deflection. It is the time needed to achieve a deflection of the targeted 4. Results and Discussion asteroid of at least twice Earth’ radius (except in the case of the KI). As stated above, the scale of importance appearing in Table 1 ([33]), which identifies a set of linguistic labels with their Attribute C : asteroid rotation. As it was suggested by corresponding SFNs, is considered to assess the criteria and our group of experts, it is unlikely to tackle with a fast the alternatives involved in the current study. To deal with, rotator for objects with estimated diameters ranging the information provided by our advisory board was used. In 150–240m. this way, Table 2 provides the weights of the criteria de- Attribute C : asteroid composition. It is worth noting scribed in terms of SFNs via the SWAM operator. that the efficiency of several NEA deflection approaches .e following order of preference regarding our set of may strongly depend on this criterion. For instance, LA criteria holds from the results appeared in Table 2: may not work appropriately when being applied on metallic surfaces since the heat produced may be C > C > C > C > C > C > C > C . (13) 1 2 8 4 7 5 3 6 conducted away. According to equation (13), C (build time) appears Attribute C : asteroid structure. .is is related to the ranked in the first position being followed by C (active porosity and the internal structure of the object instead deflection duration), C (mission risk), C (asteroid com- 8 4 of the surface material structure of the asteroid or its position), C (level of maturity), and C (asteroid structure). 7 5 friability. It should be pointed out that KI is sensitive to .en, C (asteroid rotation) and C (asteroid shape) are 3 6 the internal structure of the object and its porosity, found with a same level of importance. In this regard, a good which mayaffect the momentum transfer. Also, itcould reference to identify a set of linguistic labels with their influence the ability of EGTto collect material from the corresponding SFNs can be found in [33]. NEA surface. When applying the triangular fuzzy set (TFS, hereafter) Attribute C : asteroid shape. A great variety of irregular 6 version of the analytic hierarchy process (AHP) approach contours may appear in targeted NEAs. ([19]), the next order of preference was found for our set of Attribute C : level of maturity of a deflection tech- criteria by means of the valuable information provided by nology or technological readiness level (TRL). .is is a the group of experts: standardized scale suggested by NASA to evaluate the C > C > C > C > C > C > C > C . (14) 1 2 7 4 5 8 6 3 current level of development of a technology in regard to a desired maturity level for that approach. In this Hence, from both equations (13) and (14), it holds that study, targeted maturity means a redirection technol- the criteria C (level of maturity) and C (asteroid com- 7 4 ogy for asteroids that is ready to be proved in space at position) interchange their relative level of importance from the next level, which is equivalent to TRL 6 ([54]). the SWAM operator to the TFS version of AHP. Specifically, Attribute C : mission risk. It takes into account the it holds that C appears as a more important criterion than possibility of a technological failure or an unsuccessful C when applying SFS TOPSIS. It is also worth pointing out result regarding the asteroid deflection mission. .is is that the attribute C (mission risk) has been assigned a 8 8 Advances in Astronomy Table 2: Weights of the criteria in terms of SFNs. Weights Spherical fuzzy numbers Criteria μ ] π C (build time) 0.8 0.2 0.2 C (active deflection duration) 0.6 0.5 0.4 C (asteroid rotation) 0.4 0.6 0.4 C (asteroid composition) 0.6 0.4 0.4 C (asteroid structure) 0.5 0.5 0.4 C (asteroid shape) 0.4 0.6 0.4 C (level of maturity) 0.6 0.4 0.3 C (mission risk) 0.6 0.5 0.3 greater level of importance when applying the SWAM op- alternatives through the SFS TOPSIS procedure, we can use erator than TFS AHP. either the SWAM operator or the SWGM operator, which .e next step was to generate a new decision matrix constitutes an advantage of SFSs over TFSs to deal with fuzzy (Table 3) that contains the assessment of the alternatives for series. In fact, applying geometric mean to generate the such criteria from the judgments provided by the experts via aggregated matrix of decision (by scales of importance linguistic labels defined in terms of SFNs (Table 1). through TFS) could provoke that the TOPSIS algorithm may From that decision matrix and taking into account the not be executed. In this section, we compare the ranking of alternatives weightsofthe criteria (obtained bythe SWAM operator), the SFS TOPSIS methodology was applied to rank the alter- providedbytheSFSTOPSISapproachandtheSWAMoperator natives of our case of study. In this way, Table 4 displays a (Section 4) vs. the one provided by the SFS TOPSIS approach comparison between the rankings provided by the SFS when the SWAM operator is applied. In both cases, the weights TOPSIS approach vs. the one obtained by means of TFS of the criteria were calculated according to the information TOPSIS methodology. provided by our group of experts. We found that both rankings .e SFS TOPSIS-based ranking in Table 4 shows that the of alternatives were found to be the same (Table 5), which alternatives LA and EGTdo interchange their positions with suggests that the choice of the SWGM (respectively, the respect to their TFS TOPSIS rankings. .is could be due to SWAM) operator does not influence the ranking positions of the alternatives. Only slight deviations were found in regard to the greater SFS TOPSIS relative importance that has been assigned to the criterion C (asteroid composition) to the the absolute values of the differences between the closeness ratios of pairs of consecutive alternatives. As such, the choice of detriment of C (level of maturity). In fact, a greater valu- ation for that criterion (Table3) placesLA with respect to C , one of such operators to the detriment of the other would thus being followed by KI, IBD, and EGT. mainly depend on the computational cost required to carry out Similarly, since mission risk (criterion C ) for both al- the corresponding calculations. However, in this case, the ternatives LA and EGT is greater than the one for both KI computational cost is similar for both operators. and IBD, and it was the 3rd most important criterion according to SWAM operator (equation (13)), both LA and EGT become closer from KI and IBD in the SFS TOPSIS 5.2. On the Dependence of the SFS TOPSIS Approach on the ranking. Judgments from the Group of Experts. Next, we highlight the influence of the information provided by the group of ex- 5. Sensitivity Analyses perts over our SFS TOPSIS rankings of alternatives. With this aim, a pair of SFS TOPSIS rankings was obtained (one Two sensitivity analyses have been carried out in this section per each operator, SWAM and SWGM) by assuming that the with the aim to validate the robustness of the results pro- weights of all the criteria are the same. First, as shown in vided in Section 4. In fact, the first one consists of carrying Table 6, it holds that the positions of the four alternatives out the SFS TOPSIS calculations by means of the SWGM involved in the present study were found to be the same in operator and taking into account the weights of the criteria both SFS TOPSIS-based rankings. However, such SFS as provided by the judgments from the group of experts TOPSIS-based rankings differ from the one provided in [19], (Section 5.1), whereas the second sensitivity analysis repeats where TFS TOPSIS was used to assess these four NEA the SFS TOPSIS calculations by both operators, SWAM and deflection technologies. In fact, the use of one of such op- SWGM, but assuming that the weights of all the criteria are erators (SWAM or SWGM) may lead to some changes in the same. Two interesting facts follow from the results regard to the rankings of alternatives as provided by the SFS provided by each sensitivity analysis. TOPSIS approachwith respect tothe rankingsof alternatives provided by the TFS TOPSIS procedure. 5.1. On the Effect of the SWGM Operator. Recall that Specifically, observe that KI keeps the first position in all arithmetic mean is used to aggregate the valuations provided such SFS TOPSIS-based rankings. On the other hand, LA is ranked in 3rd position in both SFS TOPSIS rankings when by the experts to generate the decision matrix of the TOPSIS approach (e.g., [19]). However, when ranking the the weights of the criteria are calculated from the group of Advances in Astronomy 9 Table 3: Assessment of the alternatives for criteria C − C in terms of SFNs as provided by the SFS TOPSIS approach via the SWAM 1 8 operator. Criteria C C C C C C C C 1 2 3 4 5 6 7 8 Alternatives μ ] π μ ] π μ ] π μ ] π μ ] π μ ] π μ ] π μ ] π A (KI) 0.6 0.4 0.3 0.7 0.3 0.3 0.5 0.5 0.3 0.6 0.4 0.3 0.6 0.4 0.3 0.6 0.4 0.3 0.7 0.3 0.3 0.6 0.4 0.4 A (IBD) 0.5 0.5 0.4 0.4 0.6 0.4 0.5 0.5 0.3 0.5 0.5 0.4 0.3 0.7 0.3 0.5 0.5 0.4 0.5 0.5 0.4 0.6 0.4 0.3 A (EGT) 0.4 0.6 0.4 0.3 0.7 0.3 0.4 0.6 0.3 0.4 0.6 0.3 0.5 0.6 0.3 0.4 0.6 0.4 0.4 0.6 0.3 0.7 0.4 0.3 A (LA) 0.4 0.6 0.3 0.5 0.6 0.4 0.6 0.4 0.3 0.7 0.3 0.3 0.5 0.5 0.4 0.5 0.5 0.3 0.4 0.7 0.2 0.7 0.4 0.3 Table 4: Comparison of rankings of alternatives between the SFSTOPSIS (SWAM operator) approach to TFS TOPSISprocedure ([19]). .e weights of the criteria were obtained from the information provided by the group of experts. SFS TOPSIS (SWAM) ranking TFS TOPSIS ranking Alternative Closeness ratio Ranking R Ranking A (KI) 0.00 1 3.07 1 A (IBD) 5.39 2 1.85 2 A (EGT) 7.82 4 1.16 3 A (LA) 5.99 3 0.74 4 Table 5: SFS TOPSIS (SWGM) ranking of alternatives as described in the first analysis of sensitivity. .e weights of the criteria were chosen to be those calculated from the information provided by the group of experts. .e “Diff.” column contains the differences between the closeness ratios (in absolute value) from pairs of alternatives. SFS TOPSIS (SWGM) ranking of alternatives (criteria weights from experts) Alternative Closeness ratio Rank Diff. A (KI) 0.00 1 – A (IBD) 3.24 2 3.24 A (EGT) 4.81 4 1.57 A (LA) 3.86 3 0.95 Table 6: SFS-based rankings of alternatives for both operators, SWAM and SWGM, under the assumption that the all the criteria are equally weighted (Section 5.2) vs. TFS-based ranking of alternatives for equally weighted criteria ([19]). SFS rankings (equally weighted criteria) TFS ranking (equally weighted criteria) Alternative SWAM rank SWGM rank Rank A (KI) 1 1 1 A (IBD) 3 3 2 A (EGT) 4 4 3 A (LA) 2 2 4 experts, though it appears ranked in 4th position in the TFS 6. Conclusions TOPSIS ranking for equally weighted criteria. However, it In this section, we summarize the main conclusions to be occupies 2nd position in both SFS TOPSIS rankings for highlighted from the study carried out. equally weighted criteria. .e next ranked alternatives are First of all, it is worth pointing out that the KI alternative IBD and EGT (notice that such a consecutive order for such is consolidated as the best choice for active NEA deflection alternatives coincides with the one that appears in the TFS purposes. In fact, the results thrown by the SFS TOPSIS TOPSIS-based rankings). methodology coincide with all those presented in [19] when .is analysis of sensitivity highlights that, unlike the TFS it was applied to the TFS TOPSIS approach with the same TOPSIS procedure, the weights of the criteria should be purpose. assigned carefully when applying a SFS TOPSIS approach However, this study highlights the fact that a SFS TOPSIS- since variations regarding the weights of the criteria may based ranking of alternatives may vary widely when a sensitivity induce changes of positions among the ranked alternatives. 10 Advances in Astronomy analysis is carried out. Specifically, we showed that the ranking acknowledges the support of Grants TIN2017-86647-P and of alternatives as provided by the SFS TOPSIS approach when 19882-GERM-15 from the Spanish Ministry of Economy ´ ´ taking into account the information from the group of experts and Competitiveness (MINECO) and Fundacion Seneca becomes quite different from the SFS TOPSIS ranking of al- (Region ´ de Murcia), respectively. .is work could have not ternatives we obtained provided that all the weights of the been carried out without the generous collaboration of criteria are assumed to be the same. In other words, this study experts from the following institutions: Langley Research reveals a nonnegligible dependence of the SFS TOPSIS results Center and Jet Propulsion Laboratory of the National from the judgments that could be provided by the group of Aeronautics and Space Administration (NASA), Planetary experts with the aim of ranking a set of alternatives. Defence Office and Galileo Mission of the European Space On the other hand, we would like to mention that the use Agency (ESA), Laboratory of Applied Physics at the Johns of either the SWAM operator or the SWGM operator is Hopkins University, Institute for Aerospace Studies at the indifferent when carrying out SFS TOPSIS calculations. In University of Toronto, Institute of Space Sciences at the fact, only slight differences between the absolute value of the Spanish National Research Council, Department of Physics closeness ratios from pairs of consecutive alternatives were Applied to Aeronautical Engineering of the Polytechnic found with a similar computational cost. .is fact could be University in Madrid, and the Department of Mathematics understood as an advantage of SFS TOPSIS approach to the and SpaceDyS at the University of Pisa. .e survey results detriment of TFS TOPSIS. In fact, the latter only uses are based on the expert opinions of the participating indi- arithmetic mean in contexts where it is necessary to utilize viduals and do not necessarily reflect the official positions of the lowest level of the standard TFS scale of importance. their parent institutions. .e authors would also like to Note that a consistency analysis regarding the judgments express their gratitude to the editor and anonymous re- provided by the group of experts cannot be carried out viewers whose suggestions, comments, and remarks have through the SWAM (respectively, the SWGM) operator. allowed them to enhance the quality of this paper. Notwithstanding, recently, it has been contributed in [60] a SFS version of the AHP methodology, which encourages us References to calculate the weights of the criteria throughout that novel approach as a future research task. Also, we would like to [1] D. E. 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Advances in AstronomyHindawi Publishing Corporation

Published: Mar 8, 2021

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