Determination of multi-UAVs formation shape: Using a requirement satisfaction and spherical fuzzy ANP based TOPSIS approach

Abstract

Multiple unmanned aerial vehicles (multi-UAVs) formation shape refers to the geometric shape when multi-UAVs fly in formation and describes their relative positions. It plays a necessary role in multi-UAVs collaboration to improve performance, avoid collision, and provide reference for control. This study aims to determine the most appropriate multi-UAVs formation shape in a specific mission to meet different and even conflicting requirements. The proposed approach introduces requirement satisfaction and spherical fuzzy analytic network process (SFANP) to improve the technique for order preference by similarity to ideal solution (TOPSIS). First, multi-UAVs capability criteria and their evaluation models are constructed. Next, performance data are transformed into requirement satisfaction of capability and unified into a same scale. Qualitative judgments are made and quantified based on spherical fuzzy sets and nonlinear transformation functions are developed for benefit, cost, and interval metrics. Then, SFANP is used to handle interrelationships among criteria and determine their global weights, which takes decision vagueness and hesitancy into account and extends decision-makers’ preference domain onto a spherical surface. Finally, alternative formation shapes are ranked by their distances to the positive and negative ideal solution according to the TOPSIS. Furthermore, a case study of 9 UAVs performing a search-attack mission is set up to illustrate the proposed approach, and a comparative analysis is conducted to verify the applicability and credibility.

Keywords

Multi-UAVs formation shape requirement satisfaction spherical fuzzy sets analytic network process TOPSIS

1 Introduction

Unmanned aerial vehicles (UAVs) are gaining in popularity for their advantages of low risk, low cost, and long endurance. Multiple UAVs (multi-UAVs) have been widely deployed as a cooperative formation to perform complicated military and civilian missions, such as target tracking [1], search and rescue [2], data collection [3], and search-attack [4]. This formation typically has a geometric shape to improve performance and avoid collisions [5]. For example, a V-shaped formation can reduce energy consumption during flight [6] and a straight-line formation can be used to navigate through valleys or obstacles [7, 8]. A clearly defined formation shape also describes relative positions of each UAV and is the assumption and goal of multi-UAVs formation control studies. Square formation [9], circular formation [10], and triangular formation [11, 12] have been taken as target states to verify the efficiency and stability of formation keeping methods. [13] conducted simulations of reconfiguration from triangular formation to V-shaped or diamond formation after loss of UAVs. Formation shapes have also been adopted in field flight. [14] conducted 21-UAV flight experiments and performed formation maintaining and reconfiguration with triangular, V-shaped, and linear patterns. [15] took linear, circular, and triangular formations in a 10-UAV flight test for cooperative trajectory planning.

Although researchers have proposed such a variety of formation shapes, rare studies have been conducted on how to choose the most appropriate formation shape for multi-UAVs under different or even conflicting requirements, which plays a fundamental role in maximizing overall mission effectiveness. As determination of multi-UAVs formation shape is affected by many factors, such as tracking target performance [16], aerodynamic efficiency [17], and power consumption [18], it is a typical multi-criteria decision-making (MCDM) problem which needs to evaluate alternatives considering multiple criteria. Among various MCDM approaches, the technique for order preference by similarity to ideal solution (TOPSIS) is widely used because of its simple and rational concept that the optimal alternative is the closest to the positive ideal solution and simultaneously the farthest from the negative ideal solution [19]. TOPSIS is also easy to implement and has no strict limitations on data distribution or size. Despite its simplicity and computational efficiency, some issues remain in traditional TOPSIS when scaling the relative performance of each alternative formation shape under multiple criteria:

Classical TOPSIS requires all performance values of criteria to be numerical [20]. However, when determining the formation shape of multi-UAVs, some evaluations may not be expressed quantitatively and rely on the experience and knowledge of experts to make judgments. Therefore, subjective and objective evaluations should be discussed under the same scale.

In classical TOPSIS, only benefit and cost criteria are considered, which is obviously insufficient to capture all numerical metrics in multi-UAVs missions. Moreover, TOPSIS usually applies a linear min-max normalization to the initial data to make criteria in different units comparable [21]. However, this normalization does not handle outliers well and fails to address mission requirements.

Assigning reliable weights to each criterion is a prerequisite for obtaining unbiased ranking results through TOPSIS. However, the criteria measuring different multi-UAVs formation shapes interact with each other. For example, formations with poor communications will have difficulty conducting coordinated search and coordinated attack; formations that can successfully jam enemy radars will have better survivability. These net-like interdependencies create difficulties in determining the weights of the criteria.

The user’s need for a complex system, such as multi-UAVs, to achieve desired objectives is often expressed as multiple capability requirements. Thus, requirement satisfaction of the capability describes the extent to which multi-UAVs accomplish their mission. This satisfaction reveals the gap between the inherent characteristics and expectations, offering a perspective for unifying subjective and objective evaluations and allowing more types of metrics to be considered. Moreover, the characteristics with crisp values are usually transformed into requirement satisfaction non-linearly. This enables a better handling of data with different distributions and ranges, and emphasizes the relationship between alternative formation shapes and mission requirements.

To rationally and reliably assign relative weights of interdependent criteria, combining with the analytic network process (ANP) is a common improvement to TOPSIS in the literature [22 –24]. The ANP characterizes interdependent criteria as a network of influences, and determines their global weights through pairwise comparisons and matrix calculations. Although the ANP is naturally suited to deal with net-like criteria structure and has the advantages of simplicity and flexibility, it relies heavily on subjective judgments that experts may not be able to rate precisely in some cases due to their experience and knowledge limitations. There is some literature that applies fuzzy set theory to capture uncertainty in ANP and integrates it with TOPSIS [25 –27]. However, these fuzzy extensions usually only represent experts’ opinions as a simple “yes” or “no”. In practical judgment, experts may also abstain or hesitate which is beyond the scope of these fuzzy representations [28].

First proposed in [29], spherical fuzzy sets (SFSs) represent experts’ opinions with the degree of membership, non-membership, and hesitancy, which extends the preference domain to a three-dimensional spherical surface and is more suitable for practical decision-making when abstention or hesitancy is involved [30]. SFSs provide a better way to handle uncertainties in ANP, and can also be used to represent qualitative judgments when evaluating requirement satisfaction for some characteristics of formation shape.

In this paper, a requirement satisfaction and spherical fuzzy ANP (SFANP) based TOPSIS approach for multi-UAVs formation shape determination is developed. First, capability criteria and corresponding evaluations of multi-UAVs are constructed. Second, requirement satisfaction of capability is introduced to unify quantitative metrics and qualitative judgments on the same scale. Qualitative judgments are made and quantified based on spherical fuzzy sets. Interval criteria are brought into consideration for other quantitative metrics, and nonlinear transformation functions are designed to replace min-max normalization. Third, we deploy SFANP to weight criteria with interrelationships, which considers uncertainty and hesitancy of decision-makers. Alternative formation shapes are then ranked by TOPSIS. Furthermore, a case study of a search-attack mission with comparative analysis is conducted to verify the applicability of the proposed approach.

The rest part of this paper is organized as follows. Section 2 gives a literature review. Section 3 presents preliminary definitions of spherical fuzzy sets and related operations. Section 4 describes the proposed approach. Section 5 conducts a case study on multi-UAVs performing search-attack mission. Conclusions and prospects for future research are given in Section 6.

2 Literature review

Most of existing studies investigating the influence of different formation shapes on multi-UAVs performance are limited to a single indicator. [16] evaluated the performance of following, triangular, and circular formations on tracking malicious UAVs. [17] performed computational fluid dynamics simulations to assess the total drag for echelon, V-shaped, diamond, and half-diamond formation flight of blended-wing UAVs. [18] compared power consumption between V-shaped formation and circular formation considering aerodynamic efficiency and beamforming communication. [31] explored the effect of column, front, echelon, V-shaped, and diamond formation of quadcopter drones on the energy consumptions.

Depending on the type of characteristics, the degree of requirement satisfaction can be obtained through either experts’ opinions or crisp values. In [32, 33], experts’ assessments were quantified based on fuzzy sets, and the satisfaction degrees of capability requirements were represented through membership functions. In [34], basic uncertain linguistic information was proposed to express performance assessment of experts, and the value function of prospect theory was introduced to calculate the satisfaction level of each requirement. Regarding characteristics or attributes with crisp values, [35] has pointed out that they do not translate linearly into requirement satisfaction. A rule-based transforming method was applied in [36], but it only generated discrete satisfaction grades. [37] used a continuous sigmoid function to model user satisfaction for routers. In [38], a sigmoid-like satisfaction function was constructed based on unimodal and multi-peak distributions. Although sigmoid-like function is an effective way to describe the relationship between numerical characteristic and requirement satisfaction, its expression is rather complex and has a step at the boundary.

The combination of ANP and TOPSIS has been employed to solve many MCDM problems, where the ANP was often extended with different fuzzy sets. Figure 1 shows the commonly used fuzzy representations for handling uncertainty in subjective comparisons in studies combining fuzzy ANP and TOPSIS from 2014 to 2023, and Table 1 summarizes some of them. Although these publications have considered uncertainty of experts’ opinions when dealing with interdependent criteria through ANP, most of them were conducted on a one-dimensional preference domain. Researchers are trying to extend preference domain to meet practical needs, but very few have handled the degree of hesitation.

Fig. 1

The annual numbers of studies based on fuzzy ANP and TOPSIS from 2014 to 2023.

Table 1

A summary of some studies based on fuzzy ANP and TOPSIS

Study	MCDM problem	Fuzzy extension of ANP	Preference domain
[39]	Optimization of user experience in mobile application design	Triangular fuzzy numbers	One-dimensional
[26]	Collection centers selection	Triangular fuzzy numbers	One-dimensional
[27]	Healthcare device security assessment	Triangular fuzzy numbers	One-dimensional
[40]	Evaluation of renewable energy resources	Trapezoidal fuzzy numbers	One-dimensional
[41]	Selection of the best store plan alternative	Interval type-2 fuzzy sets	Two-dimensional
[42]	Evaluation for the innovation capacity of financial institutions	Interval type-2 fuzzy sets	Two-dimensional
[25]	Location selection for a freight village	Interval-valued intuitionistic fuzzy sets	Two-dimensional
[43]	Real-time location systems selection	Interval-valued intuitionistic fuzzy sets	Two-dimensional

From the studies reviewed above, the contribution of this paper can be clearly stated as follows:

An approach for determining the formation shape of multi-UAVs considering multiple factors is proposed, and relevant criteria and evaluations are modeled.

Based on SFSs and newly designed non-linear transformation functions, the requirement satisfactions for various types of characteristics are obtained and make different evaluations comparable.

SFSs are introduced to ANP to handle subjective comparisons involving decision hesitancy and to weight interdependent criteria in TOPSIS.

3 Preliminaries

In this section, we introduce definitions of SFSs and some related operations.

Definition 1. A spherical fuzzy set ${\tilde{A}}_{S}$ of the universe of discourse U is given by $\begin{matrix} {\tilde{A}}_{S} = {(u, (μ_{{\tilde{A}}_{S}} (u), v_{{\tilde{A}}_{S}} (u), π_{{\tilde{A}}_{S}} (u))) | u \in U} \end{matrix}$ (1) where $μ_{{\tilde{A}}_{S}} : U \to [0, 1]$ , $ν_{{\tilde{A}}_{S}} : U \to [0, 1]$ $π_{{\tilde{A}}_{S}} : U \to [0, 1]$ and $0 ⩽ μ_{{\tilde{A}}_{S}}^{2} (u) + v_{{\tilde{A}}_{S}}^{2} (u) + π_{{\tilde{A}}_{S}}^{2} (u) ⩽ 1 \forall u \in U$ .

For each u, $μ_{{\tilde{A}}_{S}} (u)$ , $v_{{\tilde{A}}_{S}} (u)$ , and $π_{{\tilde{A}}_{S}} (u)$ represent the degree of membership, non-membership, and hesitancy of u to ${\tilde{A}}_{S}$ , respectively. Note ${\tilde{A}}_{S} = (μ_{{\tilde{A}}_{S}}, v_{{\tilde{A}}_{S}}, π_{{\tilde{A}}_{S}})$ and ${\tilde{B}}_{S} = (μ_{{\tilde{B}}_{S}}, v_{{\tilde{B}}_{S}}, π_{{\tilde{B}}_{S}})$ as two SFSs, and some basic arithmetic operations can be defined as follows.

Definition 2. Addition; ${\tilde{A}}_{S} \oplus {\tilde{B}}_{S} = {\begin{matrix} {(μ_{{\tilde{A}}_{S}}^{2} + μ_{{\tilde{B}}_{S}}^{2} - μ_{{\tilde{A}}_{S}}^{2} μ_{{\tilde{B}}_{S}}^{2})}^{\frac{1}{2}}, v_{{\tilde{A}}_{S}} v_{{\tilde{B}}_{S}}, \\ {((1 - μ_{{\tilde{B}}_{S}}^{2}) π_{{\tilde{A}}_{S}}^{2} + (1 - μ_{{\tilde{A}}_{S}}^{2}) π_{{\tilde{B}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2} π_{{\tilde{B}}_{S}}^{2})}^{\frac{1}{2}} \end{matrix}}$ (2)

Multiplication; ${\tilde{A}}_{S} \otimes {\tilde{B}}_{S} = {\begin{matrix} μ_{{\tilde{A}}_{S}} μ_{{\tilde{B}}_{S}}, {(v_{{\tilde{A}}_{S}}^{2} + v_{{\tilde{B}}_{S}}^{2} - v_{{\tilde{A}}_{S}}^{2} v_{{\tilde{B}}_{S}}^{2})}^{\frac{1}{2}}, \\ {((1 - v_{{\tilde{B}}_{S}}^{2}) π_{{\tilde{A}}_{S}}^{2} + (1 - v_{{\tilde{A}}_{S}}^{2}) π_{{\tilde{B}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2} π_{{\tilde{B}}_{S}}^{2})}^{\frac{1}{2}} \end{matrix}}$ (3)

Table 2

Scoring criteria for qualitative judgment

Satisfaction level	Positive membership μ_sc	Negative membership v_sc	Hesitancy π_sc
Dissatisfied	[0.0, 0.2)	[0.8, 1.0)	[0.0, 0.2)
Hardly satisfied	[0.2, 0.4)	[0.6, 0.8)	[0.0, 0.3)
Basically satisfied	[0.4, 0.6)	[0.4, 0.6)	[0.0, 0.4)
Almost satisfied	[0.6, 0.8)	[0.2, 0.4)	[0.0, 0.3)
Satisfied	[0.8, 1.0)	[0.0, 0.2)	[0.0, 0.2)

Multiplication by a scalar; λ > 0 $λ \cdot {\tilde{A}}_{S} = {\begin{matrix} {(1 - {(1 - μ_{{\tilde{A}}_{S}}^{2})}^{λ})}^{\frac{1}{2}} r v_{{\tilde{A}}_{S}}^{λ} r \\ {({(1 - μ_{{\tilde{A}}_{S}}^{2})}^{λ} - {(1 - μ_{{\tilde{A}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2})}^{λ})}^{\frac{1}{2}} \end{matrix}}$ (4)

λ. Power of ${\tilde{A}}_{S}$ ; λ > 0 ${\tilde{A}}_{S}^{λ} = {\begin{matrix} μ_{{\tilde{A}}_{S}}^{λ} r {(1 - {(1 - v_{{\tilde{A}}_{S}}^{2})}^{λ})}^{\frac{1}{2}} r \\ {({(1 - v_{{\tilde{A}}_{S}}^{2})}^{λ} - {(1 - v_{{\tilde{A}}_{S}}^{2} - π_{{\tilde{A}}_{S}}^{2})}^{λ})}^{\frac{1}{2}} \end{matrix}}$ (5)

Definition 3. For the SFSs ${\tilde{A}}_{S}$ and ${\tilde{B}}_{S}$ the followings are valid under the condition λ r λ₁ r λ₂ > 0

i. ${\tilde{A}}_{S} \oplus {\tilde{B}}_{S} = {\tilde{B}}_{S} \oplus {\tilde{A}}_{S} (6)$

ii. ${\tilde{A}}_{S} \otimes {\tilde{B}}_{S} = {\tilde{B}}_{S} \otimes {\tilde{A}}_{S} (7)$

iii. $λ ({\tilde{A}}_{S} \oplus {\tilde{B}}_{S}) = λ {\tilde{A}}_{S} \oplus λ {\tilde{B}}_{S} (8)$

iv. $λ_{1} {\tilde{A}}_{S} \oplus λ_{2} {\tilde{A}}_{S} = (λ_{1} + λ_{2}) {\tilde{A}}_{S} (9)$

v. ${({\tilde{A}}_{S} \otimes {\tilde{B}}_{S})}^{λ} = {\tilde{A}}_{S}^{λ} \otimes {\tilde{B}}_{S}^{λ} (10)$

vi. ${\tilde{A}}_{S}^{λ_{1}} \otimes {\tilde{A}}_{S}^{λ_{2}} = {\tilde{A}}_{S}^{λ_{1} + λ_{2}} (11)$

Definition 4. Spherical Weighted Geometric Mean (SWGM) with respect to, w = (w₁, w₂, ⋯ , w_n); w_i ∈ 0, 1 ]; $\sum_{i = 1}^{n} w_{i} = 1$ is defined as $\begin{matrix} {SWGM}_{w} ({\tilde{A}}_{S 1}, {\tilde{A}}_{S 2}, \dots, {\tilde{A}}_{Sn}) = \prod_{i = 1}^{n} {\tilde{A}}_{Si}^{w_{i}} \\ = {\begin{matrix} \prod_{i = 1}^{n} μ_{{\tilde{A}}_{Si}}^{w_{i}}, {[1 - \prod_{i = 1}^{n} {(1 - v_{{\tilde{A}}_{Si}}^{2})}^{w_{i}}]}^{\frac{1}{2}}, \\ {[\prod_{i = 1}^{n} {(1 - v_{{\tilde{A}}_{Si}}^{2})}^{w_{i}} - \prod_{i = 1}^{n} {(1 - v_{{\tilde{A}}_{Si}}^{2} - π_{{\tilde{A}}_{Si}}^{2})}^{w_{i}}]}^{\frac{1}{2}} \end{matrix}} \end{matrix}$ (12)

Definition 5. The defuzzification of spherical fuzzy (DSF) is calculated as follows [44]: $DSF ({\tilde{A}}_{S}) = \sqrt{| 100 \times [{(3 μ_{{\tilde{A}}_{S}} - \frac{π_{{\tilde{A}}_{S}}}{2})}^{2} - {(\frac{v_{{\tilde{A}}_{S}}}{2} - π_{{\tilde{A}}_{S}})}^{2}] |}$ (13)

4 The proposed approach

In this section, the proposed approach for multi-UAVs formation shape determination is presented. Figure 2 provides an overview of the methodology. First, capability criteria and corresponding evaluations are constructed based on mission requirements. Then, performance data are transformed into requirement satisfaction of capability. Meanwhile, spherical fuzzy ANP is applied to obtain weights of different sub-capabilities. Finally different formation shapes are ranked by TOPSIS.

Fig. 2

Overview of the proposed approach.

4.1 Constructing capability criteria and evaluations

The mission performance of multi-UAVs is usually judged by the satisfaction of capability requirements [45, 46]. The degree of this satisfaction can be evaluated based on existing schemes. In order to determine the optimal formation shape, the capability criteria and corresponding evaluations should first be constructed. Based on literature and common missions, we model 5 main capabilities of multi-UAVs: reconnaissance capability (C1), maneuverability (C2), strike capability (C3), defense capability (C4), and communication capability (C5). Due to the complexity of multi-UAVs missions, these required capabilities are further divided into 13 sub-capabilities as shown in Fig. 3. For each sub-capability criterion, characteristics of different formation shapes are evaluated through qualitative judgment or quantitative metric, which are later translated into requirement satisfaction of capability.

Fig. 3

Capability and sub-capability criteria for multi-UAVs.

Detailed capability criteria and corresponding evaluations are modeled below.

4.1.1 Reconnaissance capability (C1)

Reconnaissance capability refers to the ability of multi-UAVs to perform area reconnaissance, battlefield surveillance, and other scouting tasks. After detecting targets, multi-UAVs need to locate and track them.

Search sub-capability (SC11). The search sub-capability is measured by the total search area covered, which reflects the ability of multi-UAVs to detect possible targets. The value S_search can be calculated as $S_{search} = \cup_{i = 1}^{n} S_{i}$ (14) where S_i denotes the search area covered by the i-th UAV in the formation.

Target locating sub-capability (SC12). The target locating sub-capability of multi-UAVs is measured with co-locating accuracy. According to [47], the two-dimensional co-locating accuracy can be expressed through Geometric Dilution of Precision (GDOP), which can be expressed as $GDOP = \sqrt{σ_{x}^{2} + σ_{y}^{2}}$ (15) where σ_x, σ_y is the standard deviation of the locating error in x, y directions, respectively, and locating errors conform to a random uniform distribution.

Target tracking sub-capability (SC13). Target tracking is an essential part before decisions. Since the effectiveness of target tracking is related to the subsequent operations taken by multi-UAVs, the target tracking sub-capability is evaluated by experts’ opinions.

4.1.2 Maneuverability (C2)

Flying in formation can have an impact on UAV aerodynamics, as well as its safety and the ability to pass through certain environments. Maneuverability reflects the ability of multi-UAVs to carry long-range flight and avoid unexpected collisions or obstacles.

Long-range flying sub-capability (SC21). When flying in formation, the leading UAV will create upwash, which will change the drag and lift of the following UAV as shown in Fig. 4.

Fig. 4

View of two close UAVs from above.

This affects the energy consumption of the following UAV and thus changes the available mission time. We take the average rate of change of the lift-to-drag ratio to evaluate the long-range flying sub-capability of multi-UAVs. Since all UAVs fly in the same plane, the equation of change in drag and lift with two UAVs from [48] can be expressed as

$Δ C_{D_{W}} = \frac{2 C_{L_{L}} C_{L_{W}}}{π^{3} A_{R}} {\begin{matrix} \ln \frac{x'^{2} + μ^{2}}{{(x' - \frac{π}{4})}^{2} + μ^{2}} - \ln \frac{{(x' + \frac{π}{4})}^{2} + μ^{2}}{x'^{2} + μ^{2}} \end{matrix}},$ (16) $Δ C_{L_{W}} = \frac{2 a_{W} C_{L_{L}}}{π^{3} A_{R}} {\begin{matrix} \ln \frac{x'^{2} + μ^{2}}{{(x' - \frac{π}{4})}^{2} + μ^{2}} - \ln \frac{{(x' + \frac{π}{4})}^{2} + μ^{2}}{x'^{2} + μ^{2}} \end{matrix}}$ (17) where ΔC_{D
_W}, ΔC_{L
_W} is the nondimensional coefficient of drag and lift increment, A_R is the aspect ratio of the wing, C_{L
_L} is the lift coefficient of the leading UAV, C_{L
_W} is the lift coefficient of the following UAV, $x' = \bar{x} / b$ is the nondimensional formation geometry parameter, μ² is the correction term considering physical viscosity effects, and a_W is the lift curve slope of the wing.

Collision avoidance sub-capability (SC22). The ratio of the desired distance to the safety distance between UAVs determines the difficulty of collision avoidance control in an emergency. Thus, we use this ratio as the initial data to measure the collision avoidance sub-capability.

Obstacle avoidance sub-capability (SC23). When multi-UAVs encounter obstacles, the safety width of the formation determines whether the current formation can pass through. The smaller the safety width, the more scenes multi-UAVs can pass through obstacles without reconfiguration. The formation safety width can be calculated as $W_{safe} = x_{\max} - x_{\min} + 2 d_{safe},$ (18) where x_max and x_min are the x-axis coordinates of the leftmost and rightmost UAVs in the formation, and d_safe indicates the minimum safety distance.

4.1.3 Strike capability (C3)

In missions, multi-UAVs are often used as weapons and electronic warfare platforms to conduct firepower strikes or electronic jamming against ground and maritime targets, aiming to make them inoperable.

Firepower strike sub-capability (SC31). The firepower strike sub-capability shows the ability of multi-UAVs to use weapons to destroy the target. In order to measure the firepower strike sub-capability uniformly, we calculate the ratio of the formation strike effect of multi-UAVs to the individual strike effect as the initial value. The strike effect of UAVs in formation can be expressed as $Q_{fs, i} = {\begin{matrix} \ln (1 + \frac{r_{\max} P_{W} \frac{n_{\max}}{35} \frac{ω_{\max}}{20} e^{\frac{δ_{T, i}}{δ_{\max}} - 1}}{r_{T, i}}) & if r_{\min} ⩽ r_{T, i} ⩽ r_{\max} and | δ_{T, i} | ⩽ δ_{\max} \\ 0 & otherwise \end{matrix}$ (19) where Q_fs,i is the firepower strike effect of the i-th UAV in the formation, r_max, r_min is the maximum and minimum engagement range of the weapon, r_T,i is the distance between the i-th UAV and the target, P_W is the kill probability of weapon, n_max is the maximum available overload, ω_max is the maximum tracking angular velocity of the weapon guidance system, δ_max is the maximum off-axis launch angle of the weapon, and δ_T,i is the angle between the velocity of the i-th UAV and the line of sight of the target and the UAV.

Electronic jamming sub-capability (SC32). Electronic jamming is an important tool against radar-type targets. The electronic jamming sub-capability is usually measured by the jamming to signal ratio (J/S). Based on [49], the total J/S for multi-UAVs can be simplified as $J / S = (\sum_{i = 1}^{n} \frac{k_{J, i} G_{{rJ}_{i}}}{r_{T, i}^{2}}) / \frac{k_{S} σ (θ_{el}, θ_{az})}{\min r_{T, i}^{4}}$ (20) where r_T,i is the distance between the i-th UAV and the target, k_J,i is an UAV-related coefficient, G_{rJ
_i} is the radar antenna gain in the direction of the i-th UAV, k_S is a target radar-related coefficient, and σ (·) is the radar cross section of the UAV closest to the target radar which varies with the elevation angle θ_el and azimuth angle θ_az.

4.1.4 Defense capability (C4)

Multi-UAVs are often deployed in environments filled with anti-aircraft fire and electronic jamming. As a result, on the one hand, multi-UAVs need to avoid detection by satellites or radars as much as possible and keep the formation intact. On the other hand, multi-UAVs must be able to quickly reform into a specific shape to continue the mission after a loss or leaving the original positions.

Survivability (SC41). Survivability reflects whether the multi-UAVs are easy to be found or attacked in a threat environment. Since evaluating survivability of different formation shapes requires a combination of experience and knowledge from different fields, this evaluation is made jointly by qualitative judgment of multiple experts.

Formation generating sub-capability (SC42). When multi-UAVs avoid attacks or are disturbed by environmental factors and leave their original positions, generating the desired formation is an important means to defend and maintain the ability to complete missions. We use the method proposed in [50] to generate the specified formation from random positions, and take the time to form the formation as the evaluation of this sub-capability.

4.1.5 Communication capability (C5)

Communication is the basis for multi-UAVs to receive commands, act in coordination, and transmit intelligence. The communication capability is further evaluated by data transmission, communication security, and link maintenance.

Data transmission sub-capability (SC51). Data transmission between UAVs is the foundation of cooperation. The information can be transmitted through satellite relays or a network between UAVs. Since the latter has a better transmission performance than the former, we take the average network communication distance between UAVs to measure the data transmission sub-capability. If no communication network can be established between the UAVs, this distance is represented by zero.

Communication security sub-capability (SC52). Communication security ensures that UAVs transmit information without interference or cyber attack. When UAVs communicate over the network, the more hops in the transmission route, the less secure the link. Thus, the value of communication security sub-capability can be expressed as ${com}_{\sec} = {\begin{matrix} η_{dl}^{{\bar{n}}_{hop}} & if {\bar{n}}_{hop} > 0 \\ η_{sat} & if {\bar{n}}_{hop} = 0 \end{matrix},$ (21) where com_sec denotes the communication security of the formation, ${\bar{n}}_{hop}$ is the average number of hops between UAVs when communicating over the network and ${\bar{n}}_{hop} = 0$ means UAVs communicate with satellite relays, η_dl and η_sat are security coefficients of the UAV data link and the satellite link, respectively.

Link maintenance sub-capability (SC53). The ability to maintain communication links is an important factor affecting the available time of multi-UAVs. The link maintenance sub-capability can be evaluated using ${com}_{power} = {\begin{matrix} {\bar{n}}_{hop} \ln (1 + p_{dl}) & if {\bar{n}}_{hop} > 0 \\ \ln (1 + p_{sat}) & if {\bar{n}}_{hop} = 0 \end{matrix},$ (22) where com_power is the average power consumption of communication, p_dl is the power cost to maintain the data links between UAVs, p_sat is the power cost to maintain the data link between UAV and satellite relay.

4.2 Transforming data into requirement satisfaction of capability

As described in Section 3.1, characteristics data of different formation shapes are collected through qualitative judgments and quantitative metrics. In order to apply the TOPSIS method, it is necessary to unify these data of different types to the same comparable scale. Here, we first translate experts’ opinions into crisp values based on SFSs and then transform all data into requirement satisfaction of capability on the interval [0, 1].

4.2.1 Quantifying qualitative judgment

For sub-capabilities that are difficult to measure by precise numerical data, we take a multi-experts qualitative judgment to evaluate based on SFSs. First, the experts make a judgment on whether the formation shape can satisfy the sub-capability requirement based on their knowledge and experience. This judgment is expressed by the positive membership μ_sc, the negative membership v_sc, and the hesitancy of this judgment π_sc. Inspired by [51], the values of μ_sc, v_sc, and π_sc are limited in ranges to unify the scoring criteria, and $μ_{sc}^{2} + v_{sc}^{2} + π_{sc}^{2} ⩽ 1$ . The scoring criteria are shown in Table 2. Let ${\tilde{e}}_{k} = (μ_{sck}, v_{sck}, π_{sck})$ be the judgment of the k-th expert and k = 1, 2, ⋯ , N_e. The weight of the k-th expert’s opinion is denoted by ɛ_k. The judgments on a formation shape are synthesized using the SWGM defined in Equation (12), where the result is denoted as $\bar{\tilde{e}} = {\tilde{e}}_{1}^{ɛ_{1}} \otimes {\tilde{e}}_{2}^{ɛ_{2}} \otimes \dots \otimes {\tilde{e}}_{N_{e}}^{ɛ_{N_{e}}}$ (23)

Then $\bar{\tilde{e}}$ is defuzzified by Equation (13).

4.2.2 Calculating requirement satisfaction of capability

After quantifying qualitative judgments, all capability criteria are represented with crisp values. However, the units of these values are not identical. To unify these data, requirement satisfaction of capability is introduced. For each criterion, the basic value and the ideal value are used to describe decision-makers’ requirement for the corresponding characteristic. As mentioned in [37] and [38], the increase of requirement satisfaction is not always proportional to the improvement of performance. At the beginning, the satisfaction increases linearly as the performance improves; when the decision-makers’ expectation is approached, the rise in satisfaction becomes slow.

The literature has used sigmoid-like functions to simplify this nonlinear relationship, but the sigmoid function asymptotically approaches 1, resulting in discontinuity at the basic and the ideal value boundaries. For initial performance data p and corresponding satisfaction s (p), three continuous transformation functions are designed considering benefit metrics, cost metrics, and interval metrics.

(1) Benefit metric

For a sub-capability with a benefit metric, the satisfaction is positively correlated with the data value. If the performance data is greater than the ideal value, it is considered that the sub-capability requirement is fully satisfied; if the performance data is less than the basic value, it is considered that the sub-capability requirement is barely satisfied. The satisfaction can be calculated as $s (p) = {\begin{matrix} 0, & p < p_{basic} \\ 1 - \frac{1}{1 + {(\frac{p_{ideal} - p_{basic}}{p - p_{basic}} - 1)}^{- 2}}, & p_{basic} ⩽ p ⩽ p_{ideal} \\ 1, & p > p_{ideal} \end{matrix}$ (24) where p_basic, p_ideal are the basic value and ideal value for the metric. Note that the quantified qualitative judgment is also transformed as a benefit metric.

(2) Cost metric

Similarly, the requirement satisfaction of the sub-capability with the cost metric is negatively correlated with the data, which is $s (p) = {\begin{matrix} 1, & p < p_{ideal} \\ 1 - \frac{1}{1 + {(\frac{p_{basic} - p_{ideal}}{p_{basic} - p} - 1)}^{- 2}}, & p_{ideal} ⩽ p ⩽ p_{basic} \\ 0, & p > p_{basic} \end{matrix}$ (25)

(3) Interval metric

For the sub-capability with the interval metric, both the ideal value and the basic value are in the form of an interval. Like the above two metrics, the satisfaction is transformed as follows: $s (p) = {\begin{matrix} 0, & p < p_{basicl} \\ 1 - \frac{1}{1 + {(\frac{p_{ideall} - p_{basicl}}{p - p_{basicl}} - 1)}^{- 2}}, & p_{basicl} ⩽ p ⩽ p_{ideall} \\ 1, & p_{ideall} < p < p_{idealh} \\ 1 - \frac{1}{1 + {(\frac{p_{basich} - p_{idealh}}{p_{basich} - p} - 1)}^{- 2}}, & p_{idealh} ⩽ p ⩽ p_{basich} \\ 0, & p > p_{basich} \end{matrix},$ (26) where [p_basicl, p_basich] and [p_ideall, p_idealh] represent the basic value interval and the ideal value interval, respectively. Visual representations of these transformations are given in Fig. 5.

Fig. 5

Satisfaction transformation functions of sub-capabilities with (a) benefit metric, (b) cost metric, and (c) interval metric.

4.3 Weighting criteria based on SFANP

Step 1: Construct linguistic pairwise comparison matrices. Each decision-maker constructs linguistic pairwise comparison matrices using the terms listed in Table 3. For each linguistic matrix, the consistency is checked as in traditional ANP by converting linguistic terms to corresponding consistency scales. If the calculated consistency is greater than 0.1, the matrix will be asked to be adjusted.

Table 3
Linguistic terms and their corresponding spherical fuzzy numbers

Linguistic Terms Abbreviations Consistency Scale Spherical Fuzzy Numbers

Equally important EI 1 (0.5, 0.5, 0.4)

Moderately more important MMI 3 (0.6, 0.4, 0.3)

Strongly more important SMI 5 (0.7, 0.3, 0.2)

Very Strongly more Important VSMI 7 (0.8, 0.2, 0.1)

Absolutely more important AMI 9 (0.9, 0.1, 0.0)

Moderately less important MLI 1/3 (0.4, 0.6, 0.3)

Strongly less important SLI 1/5 (0.3, 0.7, 0.2)

Very Strongly less Important VSLI 1/7 (0.2, 0.8, 0.1)

Absolutely less important ALI 1/9 (0.1, 0.9, 0.0)

Linguistic Terms	Abbreviations	Consistency Scale	Spherical Fuzzy Numbers
Equally important	EI	1	(0.5, 0.5, 0.4)
Moderately more important	MMI	3	(0.6, 0.4, 0.3)
Strongly more important	SMI	5	(0.7, 0.3, 0.2)
Very Strongly more Important	VSMI	7	(0.8, 0.2, 0.1)
Absolutely more important	AMI	9	(0.9, 0.1, 0.0)
Moderately less important	MLI	1/3	(0.4, 0.6, 0.3)
Strongly less important	SLI	1/5	(0.3, 0.7, 0.2)
Very Strongly less Important	VSLI	1/7	(0.2, 0.8, 0.1)
Absolutely less important	ALI	1/9	(0.1, 0.9, 0.0)

Step 2: Synthesis ratings from decision-makers. In this step, linguistic terms in pairwise comparison matrices are converted into spherical fuzzy numbers based on Table 3. Then with respect to each criterion, pairwise comparisons from different decision-makers are synthesized using SWGM.

Step 3: Determine spherical fuzzy local weights. For a synthesized spherical fuzzy comparison matrix $\tilde{M} = {({\tilde{m}}_{ij})}_{n_{M} \times n_{M}}$ , spherical fuzzy local weights are determined as follows: ${\tilde{w}}_{i} = \frac{1}{n_{M}} ({\tilde{m}}_{i 1} \oplus {\tilde{m}}_{i 2} \oplus \dots \oplus {\tilde{m}}_{{in}_{M}})$ (27)

Step 4: Construct the cluster matrix and the unweighted supermatrix. The spherical fuzzy local weights are defuzzified by Equation and normalized for each spherical fuzzy comparison matrix. Defuzzified local weights in a same capability cluster construct a matrix block w_ij and all blocks construct the unweighted supermatrix W = (w_ij) _5×5. The cluster matrix A = (a_ij) _5×5 is constructed in the same way while a_ij is a numerical value.

Step 5: Calculate the weighted supermatrix. The weighted supermatrix $\bar{W} = {({\bar{w}}_{ij})}_{5 \times 5}$ is calculated by ${\bar{w}}_{ij} = a_{ij} w_{ij}$ (28)

and normalized by column.

Step 6: Calculate the limited supermatrix and obtain global weights. The limit weighted supermatrix is calculated by $\bar{W^{\infty}} = \lim_{t \to \infty} \bar{W^{t}}$ (29)

According to this limit weighted supermatrix, global weights of sub-capabilities under certain mission requirements can be obtained through columns.

4.4 Ranking alternatives

After transforming data to the same scale and obtaining the weights of criteria, the final priorities of different formation shapes can be ranked based on TOPSIS. Assume that the number of sub-capability criteria is n_sc. Note that the satisfaction of the i-th formation for the j-th sub-capability is s_ij, and the weight of the j-th sub-capability criteria is noted as ω_j. The positive ideal solution Z⁺ and the negative ideal solution Z^- are identified as ${\begin{matrix} Z^{+} = {z_{1}^{+}, z_{2}^{+}, \dots, z_{n_{sc}}^{+}} = {\max_{i} s_{ij} | j = 1, 2, \dots, n_{sc}} \\ Z^{-} = {z_{1}^{-}, z_{2}^{-}, \dots, z_{n_{sc}}^{-}} = {\min_{i} s_{ij} | j = 1, 2, \dots, n_{sc}} \end{matrix}$ (30)

For the i-th alternative, its distances to the positive and negative ideal solutions with weights determined by SFANP can be calculated as ${\begin{matrix} Δ_{i}^{+} = \sqrt{\sum_{j = 1}^{n_{sc}} ω_{j} {(s_{ij} - z_{j}^{+})}^{2}} \\ Δ_{i}^{-} = \sqrt{\sum_{j = 1}^{n_{sc}} ω_{j} {(s_{ij} - z_{j}^{-})}^{2}} \end{matrix}$ (31)

The final priority of the i-th multi-UAVs formation shape under the mission requirement is obtained as $Δ_{i} = Δ_{i}^{-} / (Δ_{i}^{+} + Δ_{i}^{-}) .$ (32)

5 Case study

A typical application for multi-UAVs is the search and attack of ground or maritime targets. In this process, multi-UAVs collaboratively find and locate targets and coordinate the strike on the targets. In this section, a search-attack mission is used as a case study to illustrate the proposed approach. Based on constructed capability criteria and evaluations, the performances of six alternative formation shapes are collected and transformed into requirement satisfactions of capability. Three decision-makers are involved to weight the capability criteria using SFANP. The final priorities are given utilizing the TOPSIS method. To validate the proposed approach, some extensions to the TOPSIS method are compared.

5.1 Definition of alternatives

Based on the literature mentioned in Section 1, this case study takes triangular, V-shaped, horizontal linear, longitudinal linear, circular, and diamond formations with 9 UAVs as alternative shapes. For convenience of description, these alternatives are numbered from F1 to F6. The shapes of these alternatives are shown in Fig. 6. Formation-related parameters include: d₁ = 10b, d_2x = πb/4, d_2y = 2b, d₃ = 10b, d₄ = 10b, d₅ = 100b, and d_6x = d_6y = 8b. UAV parameters are displayed in Table 4.

Fig. 6

Shape of alternative (a) F1, triangular formation, (b) F2, V-shaped formation, (c) F3, horizontal linear formation, (d) F4, longitudinal linear formation, (e) F5, circular formation, and (f) F6, diamond formation.

Table 4

Parameters of the UAV

Parameter	Symbol	Value	Parameter	Symbol	Value
Cruise speed	V	20 m/s	Kill probability	P _W	0.5
Mass	m	30 kg	Maximum overload	n _max	35
Wingspan	b	4 m	Maximum tracking angular velocity	ω_max	15°
Aspect ratio	A _R	8	Jamming coefficient	k _J	1
Lift coefficient	C _L	0.5	Safety distance	d _safe	8 m
Drag coefficient	C _D	0.0358	Flying height	H	800 m
Lift curve slope	a _W	0.089	Communication range	r _com	100 m
Search range	r _search	3000 m	Link security coefficient with UAV	η _dl	0.95
Maximum engagement range	r _max	2000 m	Link security coefficient with satellite	η _sat	0.85
Minimum engagement range	r _min	20 m	Link power cost with UAV	p _dl	5
Maximum off-axis launch angle	δ _max	30	Link power cost with satellite	p _sat	20

5.2 Data collection and transformation

Among the multi-UAVs sub-capabilities presented in Section 3, the target tracking sub-capability (SC13) and survivability (SC41) are evaluated by qualitative judgments. According to the scoring criteria, 5 experts are invited to evaluate SC13 and SC41, and the experts have the same weight. The original evaluation matrices and synthesis results are shown in Table 5 and Table 6.

Table 5
Evaluation matrix and synthesis result of the target tracking sub-capability (SC13)

Expert 1 Expert 2 Expert 3 Expert 4 Expert 5 Synthesis Result Defuzzified

F1 (0.28, 0.61, 0.17) (0.87, 0.05, 0.11) (0.38, 0.60, 0.26) (0.52, 0.42, 0.33) (0.48, 0.42, 0.15) (0.47, 0.48, 0.22) 12.9939

F2 (0.12, 0.82, 0.14) (0.28, 0.61, 0.17) (0.31, 0.70, 0.07) (0.50, 0.58, 0.25) (0.38, 0.72, 0.06) (0.29, 0.70, 0.14) 7.6292

F3 (0.88, 0.07, 0.19) (0.23, 0.62, 0.03) (0.77, 0.31, 0.27) (0.62, 0.21, 0.07) (0.76, 0.28, 0.06) (0.59, 0.36, 0.14) 17.0421

F4 (0.23, 0.62, 0.27) (0.53, 0.54, 0.21) (0.29, 0.69, 0.02) (0.04, 0.86, 0.03) (0.33, 0.64, 0.28) (0.22, 0.70, 0.18) 5.3010

F5 (0.92, 0.18, 0.02) (0.82, 0.02, 0.16) (0.93, 0.09, 0.11) (0.75, 0.37, 0.28) (0.87, 0.05, 0.09) (0.86, 0.19, 0.16) 24.8299

F6 (0.65, 0.35, 0.15) (0.80, 0.10, 0.18) (0.62, 0.21, 0.07) (0.56, 0.48, 0.20) (0.75, 0.37, 0.28) (0.67, 0.33, 0.19) 19.1480

	Expert 1	Expert 2	Expert 3	Expert 4	Expert 5	Synthesis Result	Defuzzified
F1	(0.28, 0.61, 0.17)	(0.87, 0.05, 0.11)	(0.38, 0.60, 0.26)	(0.52, 0.42, 0.33)	(0.48, 0.42, 0.15)	(0.47, 0.48, 0.22)	12.9939
F2	(0.12, 0.82, 0.14)	(0.28, 0.61, 0.17)	(0.31, 0.70, 0.07)	(0.50, 0.58, 0.25)	(0.38, 0.72, 0.06)	(0.29, 0.70, 0.14)	7.6292
F3	(0.88, 0.07, 0.19)	(0.23, 0.62, 0.03)	(0.77, 0.31, 0.27)	(0.62, 0.21, 0.07)	(0.76, 0.28, 0.06)	(0.59, 0.36, 0.14)	17.0421
F4	(0.23, 0.62, 0.27)	(0.53, 0.54, 0.21)	(0.29, 0.69, 0.02)	(0.04, 0.86, 0.03)	(0.33, 0.64, 0.28)	(0.22, 0.70, 0.18)	5.3010
F5	(0.92, 0.18, 0.02)	(0.82, 0.02, 0.16)	(0.93, 0.09, 0.11)	(0.75, 0.37, 0.28)	(0.87, 0.05, 0.09)	(0.86, 0.19, 0.16)	24.8299
F6	(0.65, 0.35, 0.15)	(0.80, 0.10, 0.18)	(0.62, 0.21, 0.07)	(0.56, 0.48, 0.20)	(0.75, 0.37, 0.28)	(0.67, 0.33, 0.19)	19.1480

Table 6

Evaluation matrix and synthesis result of the survivability (SC41)

	Expert 1	Expert 2	Expert 3	Expert 4	Expert 5	Synthesis Result	Defuzzified
F1	(0.85, 0.11, 0.1)	(0.3, 0.61, 0.06)	(0.74, 0.23, 0.02)	(0.84, 0.11, 0.05)	(0.65, 0.34, 0.21)	(0.63, 0.35, 0.11)	18.4785
F2	(0.17, 0.84, 0.16)	(0.82, 0.12, 0.16)	(0.3, 0.65, 0.13)	(0.79, 0.37, 0.06)	(0.91, 0.18, 0.17)	(0.50, 0.57, 0.16)	14.0489
F3	(0.49, 0.47, 0.27)	(0.67, 0.31, 0.08)	(0.44, 0.58, 0.34)	(0.5, 0.5, 0.16)	(0.25, 0.75, 0.15)	(0.45, 0.56, 0.22)	12.3022
F4	(0.56, 0.56, 0.24)	(0.69, 0.37, 0.18)	(0.9, 0.19, 0.16)	(0.39, 0.64, 0.27)	(0.41, 0.58, 0.37)	(0.56, 0.51, 0.27)	15.4815
F5	(0.9, 0.07, 0.05)	(0.26, 0.72, 0.23)	(0.87, 0.15, 0.05)	(0.26, 0.67, 0.14)	(0.1, 0.99, 0.13)	(0.35, 0.81, 0.34)	8.7833
F6	(0.62, 0.37, 0.02)	(0.43, 0.41, 0.16)	(0.41, 0.53, 0.01)	(0.18, 0.89, 0.09)	(0.04, 0.89, 0.15)	(0.24, 0.73, 0.13)	6.0953

Other sub-capabilities are evaluated following models built in Section 3.1. The performance data of different formation shapes are shown in Table 7.

Table 7

The performance data of alternative formation shapes

	SC11	SC12	SC13	SC21	SC22	SC23	SC31	SC32	SC41	SC42	SC51	SC52	SC53
Evaluation	BM	CM	QJ	BM	BM	CM	BM	IM	QJ	CM	IM	BM	CM
type
F1	29356294	3267873	12.99	–0.07	5.00	136.00	9.74	0.67	18.48	70.41	64.97	0.94	1.79
F2	28548771	13871974	7.63	0.22	0.79	41.13	9.08	2.20	14.05	53.16	16.32	0.95	1.51
F3	30185988	21216966	17.04	0.00	10.00	336.00	9.94	0.46	12.30	181.10	118.52	0.92	2.78
F4	30185988	6468145	5.30	–0.56	10.00	16.00	10.43	0.35	15.48	158.98	118.52	0.92	2.78
F5	36111424	395627	24.83	0.00	34.20	803.85	1.58	0.13	8.78	298.32	0.00	0.85	3.00
F6	29368217	2572153	19.15	–0.16	5.66	144.00	9.47	0.52	6.10	67.03	67.36	0.94	1.83

BM: Benefit metric; CM: Cost metric; IM: Interval metric; QJ: Qualitative judgement.

For each sub-capability criterion, the basic value and ideal value of the corresponding evaluation are displayed in Table 8. The basic value and the ideal value for qualitative judgment are DSF((0.8, 0.2, 0.1)) and DSF((0.4, 0.6, 0.2)), respectively. Following statement in Section 3.2, the performance data are transformed into requirement satisfactions of capability, and the results are shown in Fig. 7.

Fig. 7

Requirement satisfaction of capability of each formation shape.

Table 8

Basic value and ideal value of each sub-capability evaluation

	SC11	SC12	SC13	SC21	SC22	SC23	SC31	SC32	SC41	SC42	SC51	SC52	SC53
Basic value	2.8e7	1e8	10.95	–0.05	2.00	300	6	[0, 2.00]	10.95	200	[0, 150]	0.80	3.00
Ideal value	3.2e7	1e6	23.50	0.25	12.00	50	12	[0.60, 1.50]	23.50	50	[10, 100]	0.95	1.00

5.3 Weighting criteria

Based on decision-makers’ opinions, the interdependencies between the sub-capability criteria for the search-attack mission are displayed in Table 9.

Table 9
Interdependencies between sub-capability criteria for search-attack mission

SC11 SC12 SC13 SC21 SC22 SC23 SC31 SC32 SC41 SC42 SC51 SC52 SC53

SC11 ↗ ↗ ↗ ↗ ↗

SC12 ↗

SC13 ↗ ↗ ↗

SC21 ↗ ↗ ↗

SC22

SC23 ↗

SC31 ↗ ↗ ↗ ↗ ↗

SC32 ↗ ↗ ↗

SC41 ↗ ↗ ↗ ↗

SC42 ↗ ↗ ↗

SC51

SC52 ↗ ↗ ↗ ↗

SC53

Based on the linguistic terms defined in Table 3, pairwise comparisons are conducted. After converting linguistic matrices to spherical fuzzy number forms, pairwise comparisons are synthesized. Then spherical fuzzy local weights are determined following Equation (27). Taking the sub-capabilities in C5 as an example, the synthesized results with respect to SC31 are shown in Table 10, and the corresponding local weights are shown in Table 11. Similarly, the local weights for sub-capabilities in each cluster with respect to different criteria can be obtained and used to construct the cluster matrix A and the unweighted supermatrix W, which are shown in Table 12 and Table 13, respectively. Following Equation (28), the matrix blocks in the unweighted supermatrix are multiplied by the elements in the cluster matrix. The result is normalized by columns and the weighted supermatrix can be obtained as shown in Table 14. Then following Equation (29), the weighted supermatrix is multiplied by itself and normalized by columns until each column converge to the same value. This stabilized matrix is the limit weighted supermatrix as shown in Table 15. Among the columns, any non-zero column stands for the global weights of sub-capabilities. The global weights of the sub-capabilities for the search-attack mission are noted in Table 16.

Table 10

Synthesized pairwise comparisons for sub-capabilities in C5 with respect to SC31

With respect to SC31		SC51	SC52	SC53
Decision-maker 1	SC51	(0.5, 0.4, 0.4)	(0.7, 0.3, 0.2)	(0.8, 0.2, 0.1)
	SC52	(0.3, 0.7, 0.2)	(0.5, 0.4, 0.4)	(0.6, 0.4, 0.3)
	SC53	(0.2, 0.8, 0.1)	(0.4, 0.6, 0.3)	(0.5, 0.4, 0.4)
Decision-maker 2	SC51	(0.5, 0.4, 0.4)	(0.6, 0.4, 0.3)	(0.7, 0.3, 0.2)
	SC52	(0.4, 0.6, 0.3)	(0.5, 0.4, 0.4)	(0.5, 0.4, 0.4)
	SC53	(0.3, 0.7, 0.2)	(0.5, 0.4, 0.4)	(0.5, 0.4, 0.4)
Decision-maker 3	SC51	(0.5, 0.4, 0.4)	(0.6, 0.4, 0.3)	(0.6, 0.4, 0.3)
	SC52	(0.4, 0.6, 0.3)	(0.5, 0.4, 0.4)	(0.5, 0.4, 0.4)
	SC53	(0.4, 0.6, 0.3)	(0.5, 0.4, 0.4)	(0.5, 0.4, 0.4)
Synthesized	SC51	(0.5000, 0.4000, 0.4000)	(0.6316, 0.3705, 0.2733)	(0.6952, 0.3131, 0.2221)
	SC52	(0.3634, 0.6377, 0.2673)	(0.5000, 0.4000, 0.4000)	(0.5313, 0.4000, 0.3707)
	SC53	(0.2884, 0.7143, 0.2022)	(0.4642, 0.4825, 0.3656)	(0.5000, 0.4000, 0.4000)

Table 11

Local weights for sub-capabilities in C5 with respect to SC31

With respect to SC31	Spherical fuzzy local weights	Defuzzified	Normalized
SC51	(0.6203, 0.3593, 0.2996)	17.0681	0.4204
SC52	(0.4734, 0.4673, 0.3570)	12.3540	0.3043
SC53	(0.4312, 0.5166, 0.3444)	11.1816	0.2754

Table 12

The cluster matrix for search-attack mission

	C1	C2	C3	C4	C5
C1	0.0000	0.3942	0.3332	0.4614	0.0000
C2	0.3320	0.2523	0.0000	0.0000	0.6147
C3	0.0000	0.0000	0.0000	0.0000	0.3853
C4	0.3895	0.3534	0.3837	0.0000	0.0000
C5	0.2785	0.0000	0.2831	0.5386	0.0000

Table 13

The unweighted supermatrix for search-attack mission

		C1			C2			C3		C4		C5
		SC11	SC12	SC13	SC21	SC22	SC23	SC31	SC32	SC41	SC42	SC51	SC52	SC53
C1	SC11	0.0000	0.0000	0.0000	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	SC12	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000
	SC13	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
C2	SC21	0.0000	0.0000	0.5929	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	SC22	0.0000	0.0000	0.0000	0.4435	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.5750	0.0000
	SC23	1.0000	0.0000	0.4071	0.5565	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.4250	0.0000
C3	SC31	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.4621	0.0000
	SC32	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.5379	0.0000
C4	SC41	0.4622	0.0000	0.0000	1.0000	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	SC42	0.5378	1.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
C5	SC51	0.3837	0.0000	0.0000	0.0000	0.0000	0.0000	0.4204	0.2749	0.4363	0.4031	0.0000	0.0000	0.0000
	SC52	0.2703	0.0000	0.0000	0.0000	0.0000	0.0000	0.3043	0.3795	0.2458	0.2538	0.0000	0.0000	0.0000
	SC53	0.3461	0.0000	0.0000	0.0000	0.0000	0.0000	0.2754	0.3456	0.3180	0.3431	0.0000	0.0000	0.0000

As shown in the results above, the search sub-capability (SC11) is weighted as the most important criterion. This is due to the fact that targets detection is a necessary precondition for striking in the search-attack mission. The sub-capabilities in communication capability (C5) cluster also have higher priorities, which is consistent with the general perception that communication plays an important role in multi-UAVs’ collaboration. The weights of the obstacle avoidance sub-capability (SC23) and formation generating sub-capability (SC42) reflect decision-makers’ focus on formation flexibility. The relatively low priorities of firepower strike sub-capability (SC31) and electronic jamming sub-capability (SC32) indicate that decision-makers prefer to use multi-UAVs as a situational awareness tool, leaving strike requirements to other platforms with greater lethality.

5.4 Ranking formation shapes

For alternative formation shapes, their final priorities are obtained through Equation (30) to Equation (32). As shown in Table 17, the order is F1 > F6 > F2 > F4 > F3 > F5, and the triangular formation shape is the optimal alternative for the search-attack mission in this case.

Table 14
The weighted supermatrix for search-attack mission

C1 C2 C3 C4 C5

SC11 SC12 SC13 SC21 SC22 SC23 SC31 SC32 SC41 SC42 SC51 SC52 SC53

C1 SC11 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

SC12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.4614 0.0000 0.0000 0.0000 0.0000

SC13 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3332 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

C2 SC21 0.0000 0.0000 0.2729 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

SC22 0.0000 0.0000 0.0000 0.1848 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.3534 0.0000

SC23 0.3320 0.0000 0.1873 0.2318 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2612 0.0000

C3 SC31 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1781 0.0000

SC32 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2073 0.0000

C4 SC41 0.1800 0.0000 0.0000 0.5834 0.0000 0.0000 0.3837 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

SC42 0.2095 1.0000 0.5398 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

C5 SC51 0.1068 0.0000 0.0000 0.0000 0.0000 0.0000 0.1190 0.2749 0.2350 0.4031 0.0000 0.0000 0.0000

SC52 0.0753 0.0000 0.0000 0.0000 0.0000 0.0000 0.0861 0.3795 0.1324 0.2538 0.0000 0.0000 0.0000

SC53 0.0964 0.0000 0.0000 0.0000 0.0000 0.0000 0.0780 0.3456 0.1713 0.3431 0.0000 0.0000 0.0000

		C1			C2		C3		C4
C1	SC11	0.0000	0.0000	0.0000	0.0000	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	SC12	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.4614	0.0000	0.0000
	SC13	0.0000	0.0000	0.0000	0.0000	0.0000	0.3332	0.0000	0.0000	0.0000	0.0000
C2	SC21	0.0000	0.0000	0.2729	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	SC22	0.0000	0.0000	0.0000	0.1848	0.0000	0.0000	0.0000	0.0000	0.0000	0.3534
	SC23	0.3320	0.0000	0.1873	0.2318	0.0000	0.0000	0.0000	0.0000	0.0000	0.2612
C3	SC31	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.1781
	SC32	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.2073
C4	SC41	0.1800	0.0000	0.0000	0.5834	0.0000	0.3837	0.0000	0.0000	0.0000	0.0000
	SC42	0.2095	1.0000	0.5398	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
C5	SC51	0.1068	0.0000	0.0000	0.0000	0.0000	0.1190	0.2749	0.2350	0.4031	0.0000
	SC52	0.0753	0.0000	0.0000	0.0000	0.0000	0.0861	0.3795	0.1324	0.2538	0.0000
	SC53	0.0964	0.0000	0.0000	0.0000	0.0000	0.0780	0.3456	0.1713	0.3431	0.0000

Table 15

The limit weighted supermatrix for search-attack mission

		C1			C2			C3		C4		C5
		SC11	SC12	SC13	SC21	SC22	SC23	SC31	SC32	SC41	SC42	SC51	SC52	SC53
C1	SC11	0.1803	0.1803	0.1803	0.1803	0.0000	0.1803	0.1803	0.1803	0.1803	0.1803	0.0000	0.1803	0.0000
	SC12	0.0420	0.0420	0.0420	0.0420	0.0000	0.0420	0.0420	0.0420	0.0420	0.0420	0.0000	0.0420	0.0000
	SC13	0.0115	0.0115	0.0115	0.0115	0.0000	0.0115	0.0115	0.0115	0.0115	0.0115	0.0000	0.0115	0.0000
C2	SC21	0.0045	0.0045	0.0045	0.0045	0.0000	0.0045	0.0045	0.0045	0.0045	0.0045	0.0000	0.0045	0.0000
	SC22	0.0491	0.0491	0.0491	0.0491	0.0000	0.0491	0.0491	0.0491	0.0491	0.0491	0.0000	0.0491	0.0000
	SC23	0.1258	0.1258	0.1258	0.1258	0.0000	0.1258	0.1258	0.1258	0.1258	0.1258	0.0000	0.1258	0.0000
C3	SC31	0.0241	0.0241	0.0241	0.0241	0.0000	0.0241	0.0241	0.0241	0.0241	0.0241	0.0000	0.0241	0.0000
	SC32	0.0281	0.0281	0.0281	0.0281	0.0000	0.0281	0.0281	0.0281	0.0281	0.0281	0.0000	0.0281	0.0000
C4	SC41	0.0636	0.0636	0.0636	0.0636	0.0000	0.0636	0.0636	0.0636	0.0636	0.0636	0.0000	0.0636	0.0000
	SC42	0.1233	0.1233	0.1233	0.1233	0.0000	0.1233	0.1233	0.1233	0.1233	0.1233	0.0000	0.1233	0.0000
C5	SC51	0.1354	0.1354	0.1354	0.1354	0.0000	0.1354	0.1354	0.1354	0.1354	0.1354	0.0000	0.1354	0.0000
	SC52	0.0946	0.0946	0.0946	0.0946	0.0000	0.0946	0.0946	0.0946	0.0946	0.0946	0.0000	0.0946	0.0000
	SC53	0.1177	0.1177	0.1177	0.1177	0.0000	0.1177	0.1177	0.1177	0.1177	0.1177	0.0000	0.1177	0.0000

Table 16

Global weights of sub-capabilities for search-attack mission

	SC11	SC12	SC13	SC21	SC22	SC23	SC31	SC32	SC41	SC42	SC51	SC52	SC53
Global weights	0.1803	0.042	0.0115	0.0045	0.0491	0.1258	0.0241	0.0281	0.0636	0.1233	0.1354	0.0946	0.1177

Table 17

Final ranking of formation shapes for the search-attack mission

	Δ⁺	Δ^-	Δ	Rank
F1	0.4171	0.7263	0.6352	1
F2	0.5347	0.7422	0.5813	3
F3	0.6419	0.5160	0.4457	5
F4	0.5073	0.6228	0.5511	4
F5	0.7855	0.4821	0.3803	6
F6	0.4373	0.6992	0.6153	2

To verify the applicability of the proposed approach, three different TOPSIS approaches have been conducted for comparative analysis as follows:

TOPSIS with analytic hierarchy process (AHP) to determine criteria weights (AHP-TOPSIS), which is applied in [52]. Only hierarchical relationships are considered and 1–9 scales are used.

TOPSIS with ANP determining criteria weights (ANP-TOPSIS), which is applied in [24]. The same 1–9 integer scales are used to construct comparison matrices.

TOPSIS with fuzzy ANP determining criteria weights (FANP-TOPSIS), which is applied in [39]. Fuzzy pairwise comparisons are built based on triangular fuzzy numbers.

All the calculations are based on the same satisfaction data shown in Fig. 7. The evaluated priorities are shown in Fig. 8. ANP-TOPSIS and FANP-TOPSIS report the same ranking result as the proposed approach. This proves that the calculation in this paper is consistent and reliable. The small difference in the priority values indicates that the proposed approach is able to reflect the hesitancy in the real decision-making progress after expanding the preference domain. On the other hand, the AHP-TOPSIS approach reports a ranking that F6 > F1 > F3 > F4 > F2 > F5 without considering interdependencies between criteria. The relatively high priority of F6 possibly suggests that the impact of survivability on other sub-capabilities is ignored, which proves that the proposed approach is more practical and credible.

Fig. 8

Comparison of priorities of alternative formation shapes calculated by different approaches.

6 Conclusions

In this paper, a requirement satisfaction and SFANP based TOPSIS approach is presented to determine multi-UAVs formation shape. First, multi-UAVs capability criteria and corresponding evaluations are constructed. Then, performance data of different formation shapes are transformed into requirement satisfaction of capability. Meanwhile, sub-capability criteria are weighted based on SFANP. Alternative formation shapes are ranked by their distances to positive and negative ideal solutions. A case study in which 9 UAVs perform a search-attack mission is set up to validate the presented approach with a comparative analysis.

The approach proposed in this paper provides a practical framework to evaluate different multi-UAVs formation shapes considering the overall mission effectiveness. The approach has the following strengths:

Multi-UAVs capability criteria and their evaluation models are constructed, which reflect the influence of different formation shapes.

Traditional data normalization is replaced by requirement satisfaction of capability. Qualitative judgments and quantitative metrics are unified to a same comparable scale. Qualitative judgments are made and quantified based on SFSs which can better express uncertain information. Nonlinear transformation functions are developed for benefit, cost, and interval metrics to address expectations of decision-makers.

SFANP is deployed to handle interrelationships among criteria, which takes decision vagueness and hesitancy into account and extends decision-makers’ preference domain.

In terms of future work, it would be more realistic to evaluate different formation shapes with experimental data. Further research can also explore broader capabilities and formation shapes for more complex missions. Additionally, this paper only considers the subjective weights of the criteria, and the credibility of the criteria weights of TOPSIS can be improved by combining objective weights such as information entropy. Other MCDM approaches can be also applied or integrated.

Footnotes

Acknowledgments

This research is supported by the National Natural Science Foundation of China (No. 62073267; No. 61903305), and the Fundamental Research Funds for the Central Universities (No. HXGJXM202214)

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