Implementing a Multi-Attribute Decision-Making-Based Approach to Evaluate Small Electric Vertical Takeoff and Landing Fixed-Wing Drones with Mission Efficiency

Abstract

Evaluating the mission efficiency of various drone configurations under complex, multi-source, and multi-dimensional requirements remains a significant challenge. This study aimed to develop a comprehensive decision support system (DSS) that employs mission efficiency evaluation, probabilistic hesitant fuzzy sets (PHFs), and multi-attribute decision-making (MADM) methods to assess and optimize drone design. In the proposed method, mission efficiency is defined as a composite measure of the flight performance, adaptability, and economic viability required to complete a mission. By designing a “demand–capability–design” mapping approach, this system effectively resolves multi-attribute conflicts in the decision-making process. To demonstrate the proposed approach, a set of small electric vertical takeoff and landing fixed-wing (e-VTOLFW) drones are compared and ranked based on their mission efficiency. The impacts of different mission requirements on drone evaluation are also discussed. The results demonstrate that this model resolves the traditional issue of unclear information flow in drone design. By improving the evaluation criteria, it enhances informed decision making and the robustness of evaluation results in drone design assessments. Additionally, the model is generalizable and can be widely applied to similar fields such as “demand–product design”, improving the understanding and optimization of product performance.

1. Introduction

Assessing mission efficiency is a significant contemporary challenge [1]. With the growing urgency in military systems for mission efficiency solutions, there is an escalating demand to comprehensively define and accurately assess mission efficiency. The literature offers diverse interpretations of mission efficiency. For instance, ref. [2] defines it as the probability of successfully completing a specified mission. In model-based systems engineering (MBSE), there is a lack of clear mapping between system design and model libraries [3], as shown in Figure 1. Mission efficiency assessment, as a method to associate “top-level demand–design models” effectively addresses the ambiguous connection between system design and model libraries [4,5].

Mission efficiency is a critical concern in the aerospace manufacturing industry as it directly impacts the success or failure of the final product. Trade-offs are involved in the design of aerospace products, but the constraints of design parameters are often insufficiently considered [6]. Previous models have not adequately taken into account important design parameters such as maximum range, flight speed, payload efficiency, and economic factors [7].

In the current context, the assessment of mission efficiency has become crucial in the conceptual design of drones in the aerospace industry [6]. The selection of drone configurations is vital for rapid design processes. Given the widespread use of small drones in conflicts such as that in Ukraine, there is an increasing need for the swift development of drones tailored to specific missions. Therefore, an evaluation method that considers drone design standards while fully addressing mission requirements is necessary. Multiple-attribute decision making (MADM) can effectively reduce subjective bias and maintain manageability as the number of decision alternatives increases [8,9].

Table 1. Characteristics of different decision-making assessment methods [10,11,12,13,14,15,16,17,18,19]. (Abbreviations are in Appendix A.)

Algorithm	User Consideration	Complexity	Flexibility	Reliability	Extendibility
AHP [10,11,12,14]	High	Medium	High	High	Medium
	Merits: Flexibility in problem structuring, handles qualitative and quantitative data, and straightforward comparisons
	Demerits: Requires many comparisons for complex problems, and subjective judgments can lead to inconsistency
ANP [11,12]	High	High	High	Medium	Medium
	Merits: Handles internal and external dependencies in a network and flexible for complex decisions
	Demerits: Complex computation and requires expertise for model construction and analysis
SAW [13,15]	Medium	Low	Medium	Medium	Low
	Merits: Simple and easy to use, and effective for ranking and selection among limited alternatives
	Demerits: Limited to problems with clear criteria and may not handle interdependent relationships well
TOPSIS [14,15]	Medium	Medium	Medium	High	High
	Merits: Effective in various applications and provides a solution close to the ideal case
	Demerits: Can be sensitive to the normalization method used and may be biased if weights are not assigned properly
VIKOR [15,16,17]	Medium	Medium	Medium	High	High
	Merits: Provides a compromise solution and is useful for conflicting criteria
	Demerits: Can be complex to interpret and may become complex with many criteria or alternatives
ELECTRE [15]	High	High	Medium	Medium	Medium
	Merits: Handles conflicting criteria without commensurability and effective for complex decisions
	Demerits: Complex methodology, requires expert knowledge, and computationally intensive for large alternatives
PROMETHEE [16,18]	High	Medium	High	Medium	Medium
	Merits: Handles quantitative and qualitative data, and supports decisions under uncertainty
	Demerits: Complex application and substantial effort to determine preferences and weights
EDAS [19]	Medium	Medium	Medium	High	High
	Merits: Useful in performance measurement and productivity analysis, and no need for a priori weights
	Demerits: The accuracy of rankings heavily relies on the accurate assignment of weights

shows a comparison of the notable features of the different available decision assessment options.

F. Pollet et al. designed a multidisciplinary optimization framework, but it faces implementation challenges due to its complexity [20]. Dionysios N. Markatos developed a MADM model for selecting sustainable aircraft, but it assumes equal importance for all criteria, ignoring variability in weight [21]. Pei Chi et al. utilized the Department of Defense Architecture Framework (DoDAF) to design an evaluation model for drone autonomy capabilities based on the AHP [22]. Yash Chitale et al. proposed a method for the rapid prototyping of drones, but further research is needed to improve its generality and scalability [23]. Slavica Dožić et al. conducted a sensitivity analysis of AHP and ESM in aircraft selection, revealing that AHP outperforms ESM [24]. Furthermore, they proposed a decision-making process under uncertainty; however, this approach relies on subjective judgment to determine the criteria weights, which may introduce bias and variability [25]. Giuseppe Bruno and his team developed a hybrid AHP-FST model for aircraft evaluation, but its complexity may pose challenges for practitioners unfamiliar with these methods [26]. In the aerospace field, the evaluation of multiple alternatives has increasingly relied on MADM methods.

Table 2. Application of the MADM methodology in aerospace [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. (Abbreviations are in Appendix A.)

Author	Objective	Theme	Methods Used
Ma, J. (2022) [28]	Selection of aircraft to meet training needs	Ranking	BWM, Fuzzy TOPSIS, and AHP
Rasaizadi, A. et al. (2021) [29]	Investigation of aircraft to meet air transportation needs	Evaluation	AHP, SAW, TOPSIS, and ELECTRE
Markatos, D. et al. (2023) [30]	Selection of sustainable materials	Evaluation	AHP and WSM
Liu, H. et al. (2020) [31]	Cockpit displaying ergonomic evaluation	Evaluation	IFGOWGA
Dahooie, J.H. et al. (2021) [32]	Selecting an appropriate forecasting method for aircraft engine	Ranking	SWARA and MUTLIMOORA
Xiong, S.H. et al. (2021) [33]	Selection of green airport plans	Evaluation	MAGDM
AlKheder, S. et al. (2022) [34]	Airport runway material selection	Evaluation	Fuzzy AHP
Canumalla, R. et al. (2024) [35]	Landing gear material evaluation	Evaluation	PCA and HC
Zhang, Y. et al. (2023) [36]	Evaluation of airline business operations	Evaluation	CPT and TOPSIS
Deveci, M. et al. (2022) [37]	Evaluation of airline route schedules	Evaluation	BWM and TOPSIS
Kaya, S.K. et al. (2020) [38]	A sustainable airport design	Ranking	QFD
Ahmad, F. et al. (2023) [39]	Selection methodology for an aircraft skin	Ranking	Ashby
Liou, J.J.H. et al. (2024) [40]	Exploring the impact of pandemic measures on airport performance	Evaluation	DEMATEL
Todorov, V.T. et al. (2022) [41]	Conceptual design	Evaluation	Fuzzy AHP
Khan, S.A. et al. (2024) [42]	Selection of dissimilar joining materials	Ranking	QFD and TOPSIS
Gül, A.Y. et al. (2024) [43]	Drone selection for forest and fire detection	Evaluation	TOPSIS and CODAS

shows the MADM methods applied in this field over recent years.

The performance of small UAVs in recent international regional conflicts has garnered significant attention from scholars and aerospace companies. Studies based on actual usage have demonstrated that endurance, payload efficiency, and portability are the key design criteria for small drones [44,45,46]. These criteria are often conflicting in overall design. For instance, simplifying the actuator design may reduce the drone’s complexity and cost of the drone, but it can result in an underactuated system, resulting in flight instability. This is a significant issue for small electric vertical takeoff and landing fixed-wing (e-VTOLFW) drones. The MADM methods, which are effective in resolving multi-attribute conflicts, are widely used in product design decision making [47]. Therefore, to reconcile the conflicting design attributes in the drone design process, we introduced mission efficiency evaluation and MADM methods.

This study addresses the challenge of unclear information flow between top-level demand and model libraries by leveraging shared characteristics, such as capability indicators (attributes). This approach establishes a “demand–design” mapping relationship, thereby clarifying complex decision-making processes and achieving designs that meet specified requirements. Additionally, to address the traditional issue in UAV design of not fully considering the interdependencies between multi-dimensional attributes, a PHFs-MADM model was developed. This model enables the comprehensive evaluation of UAV designs from the perspective of mission effectiveness. This study specifically focuses on e-VTOLFW drones, a current research hotspot, and evaluates various configurations, providing a novel design approach for this type of UAV.

The significance of this research lies in proposing a decision support system (DSS) capable of managing complex decision-making scenarios, which is essential for UAV design, optimization, and evaluation. By introducing a method to capture expert uncertainty, this study integrates mission effectiveness, PHFs, and MADM, thereby bridging existing gaps in quantitative evaluation models.

The remainder of this paper is organized as follows: Section 2 details the methodology, including the DSS architecture and the development of the PHFs-MADM model. Section 3 presents a numerical case study to demonstrate the effectiveness of the proposed approach. Section 4 provides a detailed analysis and explanation of the case study and its results, and Section 5 concludes with a summary of the research contributions and suggestions for future work.

2. Methodology

2.1. Definition of Mission Efficiency

In the realm of drone design, mission efficiency is a critical measure, directly influencing the effectiveness and practicality of drones in operational environments. Mission efficiency, in this context, is multifaceted, encompassing not only flight performance but also adaptability to various mission scenarios and cost-effectiveness. Traditional methodologies have often overlooked these comprehensive aspects in early design stages, leading to a lack of holistic efficiency assessment. Therefore, this paper redefines mission efficiency for drone design as follows: a drone’s mission efficiency is deemed high when it exhibits superior flight performance, demonstrates robust adaptability to diverse operational conditions, and maintains competitive economic viability in comparison with existing alternatives in the market.

2.2. Decision Support System

DSSs focus primarily on enhancing decision-making effectiveness [48]. Inspired by the work of Fikar C. et al. [49,50], we categorize the DSS architecture into two main components: the front end and the back end. The front end primarily functions to analyze requirements, propose criteria, and provide reliable data for the back-end decision algorithms.

Initially, mission scenarios or actual needs are identified through methods such as market research and computer simulation. Subsequently, the expert system deconstructs the required capabilities. Suitable solutions are then selected from the model library using a capability-matching model. For capabilities or criteria that are difficult to quantify, methods such as fuzzy linguistic variables and expert scoring are utilized to gather preliminary evaluations from various experts, as shown in Figure 2.

The back end’s main function involves processing the data output from the front end and generating corresponding evaluation values for different solutions. This process is elaborated in Section 2.4.

2.3. Mission Scenarios and Performance Indicators

DoDAF is widely used in the design of complex systems [51,52,53]. To systematically develop a mission efficiency evaluation system, DoDAF 2.0 was employed to construct the framework from both the mission and capability viewpoints. This approach aligns capability requirements with mission needs, thereby establishing a comprehensive set of mission efficiency evaluation metrics, as shown in Figure 3.

2.3.1. Identifying Criteria for Mission-Based Assessment of Drones

The DoDAF framework for complex system modeling involves a multi-view hierarchical structure; this is similar to AHP analysis [51]. Additionally, the evaluation of complex systems often requires the consideration of multiple criteria. AHP effectively assists decision-makers in balancing these criteria, particularly when subjective judgment is involved [10]. Therefore, when establishing the evaluation criteria for drones, we employed the AHP methodology, as shown in Figure 4. In the first layer of the AHP process, the overall objective is to select a versatile small e-VTOLFW drone conceptual design solution. The second layer describes the scenarios in which the drone needs to be applied, and the third layer introduces the criteria set forth for selecting the drone. The fourth layer presents the evaluation attributes associated with the criteria set, and the fifth layer displays the alternative drone structural solutions being compared.

2.3.2. Mission Scenario

Different design configurations of drones exhibit distinct flight performances. Therefore, drone design assessments vary with mission requirements, influencing final evaluation rankings. Small drones are commonly applied in typical mission scenarios such as urban patrols, environmental mapping, and reconnaissance. Analyzing public safety and military mission requirements, three typical drone scenarios emerge.

• Scenario 1 (Urban Patrol)

In the field of public safety, drones are commonly used for daily patrols in urban areas. Typically, they require a take-off and landing area of less than 10 m² and are equipped with a visible light payload. They possess wide-area flight capabilities.

2.

Scenario 2 (Environmental Mapping)

Environmental mapping is another common application area for small drones. They typically require a take-off and landing area under 8 m² and are outfitted with dual payloads, including visible light for standard imaging and infrared for thermal detection, along with multispectral imaging devices, facilitating comprehensive environmental mapping. Furthermore, to ensure comprehensive area surveying, drones should have hovering capabilities.

3.

Scenario 3 (Wide-Area Reconnaissance)

In the military domain, small drones have become essential for individual combatants due to their light weight, portability, and ease of operation. In complex terrains and combat zones, drones assist combat personnel in situational awareness through wide-area reconnaissance and fire guidance. Additionally, in signal-absent areas, they are equipped with relay communication capabilities. To fulfill these requirements, they typically need a take-off and landing area under 5 m² and carry three types of payloads—visible light, infrared, and strike payloads—ensuring effective wide-area reconnaissance.

2.3.3. Performance Indicators

In assessing the mission efficiency of drones, the selection of relevant standards and criteria significantly influences the ranking of different drone types. In light of the analysis in Section 2.3.1, the Delphi method was employed [54], utilizing an expert system to categorize the performance indicators into three main groups: fundamental performance, user operability, and economic viability. This classification was designed to provide a comprehensive decision-making analysis for the DSS. This approach is inspired by previous drone evaluation methods [21,29], which have demonstrated the effectiveness of these classifications in assessing complex systems.

• Fundamental Performance

Fundamental performance refers to quantifiable technical parameters. In the proposed approach, this performance is reflected through four indicators: maximum range ($B_1$), payload efficiency ($B_2$), maximum speed ($B_3$), and endurance capability ($B_4$).

2.

User Operability

In contrast with the directly quantifiable metrics of fundamental performance, user operability data are obtained through a combination of subjective assessments and objective measurements. To comprehensively evaluate the user operability of the drone, assessments were conducted focusing on flight maneuverability ($B_5$), flight stability ($B_6$), and portability ($B_7$).

3.

Economic Viability

Economic viability includes drone structural complexity and related costs, including manufacturing and maintenance, collectively termed as economic performance, reflecting overall economic efficiency ($B_8$).

Table 3. Attribute specifications.

Attribute	Instructions
$B_1$	It is defined as the drone’s ability to achieve the longest distance while carrying a mission payload, with no consideration given to communication constraints.
$B_2$	It is defined as the ratio of the drone’s mission payload to its maximum takeoff weight, indicating its payload carriage capacity under strict size and weight limitations.
$B_3$	It is defined as the highest speed achievable by a drone during flight without compromising its structural integrity and safety regulations.
$B_4$	It is defined as the drone’s capacity for sustained operation with a mission payload.
$B_5$	It is defined as a rapid transition between rotor and fixed-wing modes during flight, a minimal turn radius, and executing specific aerial maneuvers.
$B_6$	It is defined as the drone’s capacity to resist external disturbances, such as airflow disruptions and stall resistance, in rotor mode, fixed-wing modes, and during mode transitions.
$B_7$	It is defined as the drone’s capacity to be transported by an individual during mission execution.
$B_8$	It is defined as the total cost associated with the drone’s entire lifespan, with no consideration cost of software design (e.g., controller design)

provides detailed descriptions of each attribute.

2.4. The PHFs-MADM Approach

This section explains how the PHFs-MADM model, as the back end of the DSS, quantifies the front-end information and ultimately derives the evaluation ranking of alternative solutions.

2.4.1. Modeling Framework and Process Content

This study developed a hybrid model for the conceptual design of small e-VTOLFW drones, as depicted in Figure 5. Initially, probabilistic hesitant fuzzy sets (PHFs) and triangular fuzzy numbers (TFNs) were utilized to evaluate the criteria, establishing the study’s indicator system. Expert knowledge and experience in the domain of e-VTOLFW drone design were then gathered through a survey, capturing the influential relationships between criteria and the evaluation scores for various product design alternatives. The PHFs method was employed to determine the criteria’s causal structure and weights. The MADM method was applied to calculate actual evaluation scores for each design alternative, identifying the gaps from the ideal level. The optimal solution was identified as the one with the minimal gap. Additionally, the dispersion of gaps across criteria for each solution was analyzed, enabling the development of improvement strategies based on causal influence relationships. The design and the analysis process of the hybrid model are illustrated in Figure 5.

2.4.2. Fusion and Defuzzification of Fuzzy Information Based on PHFs

• Fuzzy Information Fusion

Fuzzy information fusion using PHFs, built upon the principles of fuzzy sets, effectively addresses uncertainties in the description of complex information [55]. These sets are notable for reflecting the number of experts contributing to each assessment value, thereby preserving crucial information that may influence decision-making processes. The mathematical formulation of these sets is presented in Theorem 1 [56].

Theorem 1.

Let X be a reference set, and function H is defined as a probabilistic hesitant fuzzy set.

$$H=\{<x,h_x(p_x)>|x\in X\}$$

(1)

$$h\left(p\right)=\left\{h_x\right(p_x\left)\right|x=\mathrm{1,2},...,\left|h\right(p\left)\right\}$$

(2)

where $h_x$ is a subset of [0, 1], and $p_x$ is also a subset of [0, 1], for the probability of the corresponding $h_x$, and satisfies ${\sum }_{x=1}^{\left|h\right(p\left)\right|}{p}_x\le 1$, which is called a probabilistic fuzzy hesitant element, abbreviated as $h\left(p\right)$.

For example, consider a scenario where 10 experts are assessing an attribute: 5 experts assign a value of 0.6, 2 experts give a value of 0.8, and 3 experts assign a value of 0.7. In this case, a simple fuzzy set representation might be {0.6, 0.8, 0.7}, and the final attribute score is calculated as 0.7. This method fails to capture the comprehensive information initially provided by the decision-making group and disregards the contributions of 7 experts during the information expression phase. Using the PHFs approach, the same example can be represented as {0.6 (0.5), 0.8 (0.2), 0.7 (0.3)}, allowing each expert’s opinion to be fully expressed within the same element. To calculate the final attribute score, we multiply the attribute scores by their corresponding probabilities, resulting in a more accurate final score of 0.67. Compared with the earlier approach, PHFs offer a representation that is much closer to the actual inputs of the group.

To effectively address the fuzziness in the representation of membership relationships in dealing with uncertain information in expert systems, a method more aligned with human cognitive processes is employed. This approach, known as the “seven linguistic variables based on fuzzy sets” method [57], captures the subtle differences and complexities of human cognition, as illustrated in

Table 4. Seven-scale linguistic variables.

Scale	Fuzzy Value
Ultra-low (UL)	(0, 0, 0.1)
Low (L)	(0, 0.1, 0.3)
Middle–low (ML)	(0.1, 0.3, 0.5)
Middle (M)	(0.3, 0.5, 0.7)
Middle–high (MH)	(0.5, 0.7, 0.9)
High (H)	(0.7, 0.9, 1.0)
Ultra-high (UH)	(0.9, 1.0, 1.0)

.

2.

Fusion Information Defuzzification

Defuzzification is a crucial step in using fuzzy sets, as it converts fuzzy outputs into precise output values, facilitating the identification of optimal alternatives for subsequent MADM problems. Given the characteristics of the “seven-degree linguistic variables”, we mapped their interval values using TFNs [58], as illustrated in Figure 6a.

Theorem 2.

The triangular fuzzy number $\tilde{A}$ is defined by ($A_l,A_m,A_r$), where ${\mu }_{\tilde{A}}\left(x\right)$ represents the membership function value. The membership function expression is illustrated in Equation (3).

$${\mu }_{\tilde{A}}(x)\left\{\begin{array}{c}0, x<A_l\\ \frac{x-A_l}{A_m-{ A}_l},{ A}_l\le x\le A_m\\ \frac{x-A_r}{A_m-A_r}, { A}_m\le x\le A_r\\ 0, x>A_r\end{array}\right.$$

(3)

where $A_l$ is the left distribution of the fuzzy number confidence interval,  $A_r$  is the right distribution of the fuzzy number confidence interval, and  $A_m$  is the peak point of the fuzzy number membership function, as illustrated in Figure 6b.

We employ the vertex computation method to defuzzificate [59]. The fuzzification values for each point are computed by Equation (4), representing their distances to the central point M.

$${\overline{A}}_{12}=d(\widetilde{A_1},\widetilde{A_2})=\sqrt{\frac{1}{3}\left[{\left(A_{1_l}-A_{2_l}\right)}^2+{\left(A_{1_m}-A_{2_m}\right)}^2+{\left(A_{1_r}-A_{2_r}\right)}^2\right]}$$

(4)

In Equation (4), ${\overline{A}}_{12}$ represents the distance between two points, while $\widetilde{A_1}(A_{1_l},A_{1_m},A_{1_r})$ and $\widetilde{A_2}(A_{2_l},A_{2_m},A_{2_r})$ denote the fuzziness values of point 1 and point 2, respectively.

2.4.3. Normalization Method

Given the unique nature of evaluation criteria, it is crucial to determine the relative weight of each attribute indicator within the total indicator weight. Therefore, we utilize linear normalization as described in Equation (5) [60]. Furthermore, to address variations between different alternative solutions for the same attribute, dimensional considerations are taken into account, leading us to prefer vector normalization, as outlined in Equation (6) [61].

$$y_{nm}=\frac{x_{nm}}{{\sum }_{m=1}^m{x}_{nm}}$$

(5)

$$y_{nm}=\frac{x_{nm}}{\sqrt{{\sum }_{n=1}^n{x}_{nm}^2}}$$

(6)

2.4.4. MADM Algorithm Model

Based on the distance-based MADM evaluation model, this approach demonstrates robust objectivity and scalability. It constructs decision matrices by utilizing multi-source data from various solutions, aiming to reduce the subjectivity inherent in expert system evaluations and thereby enhancing the objectivity of the assessments. The computational steps are outlined as follows:

• Step 1: Construct the initial evaluation matrix.

Select alternative solutions that meet the requirements of the mission, as shown in

Table 5. Initial evaluation matrix.

	${\mathit{B}}_1$	${\mathit{B}}_2$	$\cdots$	${\mathit{B}}_{\mathit{m}}$
$A_1$	$x_{11}$	$x_{12}$	$\cdots$	$x_{1m}$
$A_2$	$x_{21}$	$x_{22}$	$\cdots$	$x_{2m}$
$⋮$	$⋮$	$⋮$	$\ddots$	$⋮$
$A_n$	$x_{n1}$	$x_{n2}$	$\cdots$	$x_{nm}$
$A_1$	$x_{11}$	$x_{12}$	$\cdots$	$x_{1m}$

.

According to the mission requirements, select a suitable set of attributes for alternative solutions. $A=\left\{A_1,A_2,...,A_n\right\}$ represents the set of alternative solutions, $\mathrm{B}=\left\{B_1,B_2,...,B_m\right\}$ denotes the set of attribute evaluation criteria, $X={\left(x_{ij}\right)}_{n\times m}$ signifies the defined evaluation matrix, and $x_{ij}$ stands for the initial elements of the evaluation matrix, indicating the jth evaluation value for the ith alternative solution.

2.

Step 2: Standardize the evaluation matrix.

Considering the diverse units of measurement in the initial matrix of attribute evaluations, it is necessary to standardize the data to mitigate the impact of unit discrepancies and other factors. We employ Equation (6) to compute the standardized attribute values $y_{ij}$.

3.

Step 3: Compute the value of each alternative solution.

Upon the foundation of the standardized decision matrix, the comprehensive scores for each alternative solution are computed utilizing MADM algorithms. (For further details, see Appendix B.)

3. Numerical Case Study

3.1. Case Study

This section presents a numerical case study to demonstrate the proposed decision-making algorithm for drone conceptual design. To illustrate the proposed method, under the mission requirements specified in Section 2.3.2, six types of small e-VTOLFW drones with distinctive features were selected [62], including dual-system (DS) [63], tilt-rotor (TR) [64,65], tilt-wing (TW) [66], mono thrust transitioning (MTT) [66,67,68,69], collective thrust transitioning (CTT) [70], and differential thrust transitioning (DTT) drones [71,72].

Table 6. Fundamental performance parameters of various e-VTOLFW drones [63,64,65,66,67,68,69,70,71,72,73].

Type	Maximum Range (km)	Payload Efficiency	Maximum Speed (km/h)	Endurance Capability (min)	Maximum Weight (kg)	Payload (kg)
DS	70	0.167	108	90	12	2
TR	50	0.152	84.6	35.5	22.992	3.5
TW	46.16	0.155	108	28.14	6.44	1
MTT	54	0.068	72	40	18	1.22
CTT	56.6	0.216	57.6	59	3.7	0.8
DTT	40	0.257	90	60	1.75	0.45

summarizes their fundamental performance metrics.

3.2. Calculation of Alternative Index Values

We conducted a comprehensive literature search on VTOL review articles up to January 2024 using Google Scholar, Scopus, PubMed, and Web of Science. Drawing on the summary evaluations by Ducard et al. [62,74,75,76,77,78], we identified the distinct characteristics of various VTOL configurations, as shown in

Table 7. Fundamental performance parameters of various e-VTOLFW drones [62,74,75,76,77,78].

Type	Performance
DS	Merits: Maneuverability and stability, easy takeoff and landing, and simple transition mechanism Demerits: Extra unnecessary weight, low mode transition efficiency, and costlier maintenance
TR	Merits: Maneuverability and stability, easy takeoff and landing, and simple transition mechanism Demerits: Complex tilting mechanism, extra unnecessary weight, low mode transition efficiency, and costlier maintenance
TW	Merits: Good aerodynamic performance, and easy takeoff and landing. Demerits: Vulnerable to crosswinds, complex wing-tilting mechanism, extra unnecessary weight, low mode transition efficiency, and costlier maintenance
MTT	Merits: No extra actuators, easy takeoff and landing, and high mode transition efficiency Demerits: Vertical flight instability, lower payload and speed, vulnerable to crosswinds, complex power mechanism, high angle of attack transitions, and costlier maintenance
CTT	Merits: No extra actuators, easy takeoff and landing, and high mode transition efficiency Demerits: Vertical flight instability, vulnerable to crosswinds, and high angle of attack transitions
DTT	Merits: No extra actuators, agile maneuvering, high mode transition efficiency, and easy takeoff and landing Demerits: Horizontal flight efficiency reduced, vulnerable to crosswinds, and high angle of attack transitions

. The DS design increases dead weight, which negatively impacts economic efficiency but enhances stability and simplifies control during transitional flight phases. In contrast, TR and TW designs effectively reduce this additional dead weight. However, the TW configuration is particularly sensitive to crosswinds during takeoff and landing, complicating its operations. Tail-sitter designs (MTT, CTT, and DTT) eliminate the inherent extra dead weight in VTOL drone configurations, which is crucial for enhancing payload efficiency in small unmanned drones. Nevertheless, this advantage comes with challenges in nonlinear control at high angles of attack during transition phases, necessitating more sophisticated controller designs. Furthermore, all tail-sitter designs are highly susceptible to crosswinds during the VTOL phases.

According to the algorithm proposed in Section 2.4, we conducted a preliminary evaluation of the mission scenario and eight attributes presented in Section 2.3 to determine the weights of each decision attribute. To determine the required attributes (capabilities), the Delphi method was initially used. However, this method lacks consistency in the critical details needed for interpreting research results [79]. Therefore, a questionnaire survey was conducted to collect evaluations from decision-makers (DMs) regarding the attribute values of various scenarios and alternatives. These evaluations were then aggregated using the PHFs method. The initial weight assessment aims to quantify the relative importance of various criteria that may influence the decision-making process in different mission scenarios. Under each mission, the evaluation by the decision-makers (DMs) of the DSS is influenced by the mission scenario. The differing levels of importance assigned by the decision-makers reflect their nuanced understanding and diverse perspectives during the evaluation process, as shown in

Table 8. Attribute weighting assessment summary.

DMs	Scenario	Attribute Criteria
		B1	B2	B3	B4	B5	B6	B7	B8
DM1	1	H	L	H	H	M	H	MH	M
	2	UH	MH	L	H	ML	H	H	L
	3	M	H	H	M	UH	M	H	M
DM2	1	MH	M	M	M	M	H	ML	MH
	2	H	M	ML	M	L	H	MH	M
	3	MH	M	MH	M	H	M	H	MH
DM3	1	MH	H	L	M	ML	H	ML	M
	2	H	MH	L	UH	L	UH	MH	M
	3	M	H	UH	MH	UH	MH	H	M
DM4	1	MH	M	M	M	M	H	M	H
	2	H	MH	ML	H	L	H	MH	M
	3	MH	H	MH	H	MH	M	H	M
DM5	1	M	M	M	M	M	H	ML	MH
	2	MH	MH	ML	MH	ML	H	M	M
	3	M	MH	H	MH	H	M	H	ML
DM6	1	M	M	M	MH	M	H	ML	MH
	2	MH	MH	L	MH	L	MH	M	M
	3	MH	H	H	MH	H	M	H	ML

.

After establishing the weights for various attributes, a preliminary assessment of four attributes across six different e-VTOLFW drones was conducted by DMs, incorporating findings from Ducard et al.’s study, as shown in

Table 9. Initial values of subjective attributes of the alternatives.

DMs		B5	B6	B7	B8	DMs		B5	B6	B7	B8
DM1	DS	H	L	H	H	DM4	DS	L	UH	M	ML
	TR	UH	MH	L	H		TR	M	M	ML	M
	TW	M	H	H	M		TW	M	M	ML	L
	MTT	MH	M	M	M		MTT	L	ML	L	L
	CTT	H	M	ML	M		CTT	L	M	H	H
	DTT	MH	M	MH	M		DTT	H	H	MH	MH
DM3	DS	MH	H	L	M	DM5	DS	MH	UH	M	ML
	TR	H	MH	L	UH		TR	M	MH	M	M
	TW	M	H	UH	MH		TW	M	M	M	ML
	MTT	MH	M	M	M		MTT	ML	ML	ML	L
	CTT	H	MH	ML	H		CTT	L	M	H	H
	DTT	MH	H	MH	H		DTT	H	M	MH	MH
DM5	DS	M	M	M	M	DM6	DS	L	H	M	MH
	TR	MH	MH	ML	MH		TR	M	ML	M	ML
	TW	M	MH	H	MH		TW	ML	ML	ML	L
	MTT	M	M	M	MH		MTT	L	UL	L	M
	CTT	MH	MH	L	MH		CTT	ML	L	H	H
	DTT	MH	H	H	MH		DTT	H	M	M	MH

. These data reflect the subjective evaluations of different configurations by various experts, highlighting not only the differences in assessments by different experts but also the relative performance advantages and disadvantages of each configuration. These evaluations allow us to compare the acceptability of each configuration, thereby helping to determine which configurations may best meet specific standards or user needs. Additionally, this comparative approach may uncover variations in assessment outcomes within the same group of experts, likely due to individual differences in experience and biases.

Following the methods introduced earlier, we processed the data from Table 6, Table 8, and Table 9 through defuzzification and normalization to derive the initial evaluation matrices presented in

Table 10. E-VTOLFW drone initial evaluation matrix.

Type	B1		B2		B3		B4		B5		B6		B7		B8
DS	0.5196	1	0.3807	3	0.4977	1	0.6558	1	0.0881	6	0.6703	1	0.3707	3	0.3437	3
TR	0.3711	4	0.3465	5	0.3898	3	0.2586	5	0.4970	2	0.3546	3	0.2965	4	0.2492	4
TW	0.3426	5	0.3534	4	0.4977	1	0.2050	6	0.3691	3	0.2399	5	0.2471	5	0.1168	6
MTT	0.4008	3	0.1550	6	0.3318	4	0.2914	4	0.1965	5	0.1057	6	0.1175	6	0.1589	5
CTT	0.4201	2	0.4925	2	0.2654	5	0.4299	3	0.2343	4	0.3109	4	0.6588	1	0.6683	1
DTT	0.3711	4	0.5859	1	0.4147	2	0.4372	2	0.7178	1	0.5093	2	0.5153	2	0.5780	2

and

Table 11. Attribute standard weights.

	B1	B2	B3	B4	B5	B6	B7	B8
Scenario 1	0.1421	0.1074	0.1074	0.1278	0.1003	0.1868	0.0859	0.1421
Scenario 2	0.1760	0.1481	0.0455	0.1760	0.0395	0.1820	0.1400	0.0928
Scenario 3	0.1067	0.1387	0.1480	0.1177	0.1564	0.0949	0.1547	0.0830
Overall score	0.1438	0.1265	0.0988	0.1395	0.0955	0.1642	0.1175	0.1141

. Figure 7 provides a more intuitive representation, illustrating the discrepancies between each solution and the positive ideal solution (C⁺), the negative ideal solution (C⁻), and the median values. From Table 10 and Figure 7, it can be concluded that the data evaluated by the DMs of the DSS generally align with the perspectives described textually by Ducard et al. [62,74,75,76,77,78].

As shown in Table 8 and Table 9, based on the methods described in Equations (4)–(6), linguistic terms were defuzzified to obtain crisp values and underwent normalization to derive initial scores for the six drones and weights for each mission scenario. Figure 7 illustrates the values of the attributes of each alternative compared with the positive ideal solution (C⁺), negative ideal solution (C⁻), and mean value of the attributes. To implement a PHFs-MADM-based combination method, we developed a parameterized DSS model. In this model, users input numerical values related to the comparison criteria of the studied drones and define the relevant important factors for each criterion. Finally, quantitative rankings for each configured drone are obtained.

For different mission requirements, there are specific priorities and constraints associated with the various performance aspects of unmanned aerial drones. Decision-making algorithms, therefore, aim to determine the most suitable outcome within these constraints. Typically, during urban patrol missions, it is crucial to ensure flight stability (B₆). For environmental mapping tasks, both stable flight capabilities (B₆), maximum range (B₁), and endurance capability (B₄) are prioritized. In military applications, such as wide-area reconnaissance, flight maneuverability (B₅) and portability (B₇) become critical factors. Table 11 effectively illustrates these requirements in a more intuitive manner.

3.3. Method Verification

To validate the effectiveness and sensitivity of the PHFs-MADM method, we employed a multi-dimensional validation approach, grounded in the three-pronged evidence set theory [80]: content validity, construct validity, and criterion validity. This framework, also used by Kwok et al. [81], substantiates our methodology effectively. Content validity ensures the relevance and appropriateness of the input data, with the fundamental performance parameters of drones derived from scholarly articles and actual testing data provided by manufacturers confirming the robustness of our approach. Furthermore, these parameters are consistent with those outlined by Ducard et al., aligning with the established literature. Construct validity is supported by both empirical and theoretical evidence, as detailed in Section 2.4. of our paper, which elucidates the logical and causal underpinnings of the PHFs-MADM method. Finally, criterion validity assesses whether the method produces results that are consistent with established benchmarks or expected outcomes.

3.3.1. Procedures

This experiment comprised four sets of experiments. As outlined in Section 3.1., the selection of drone configurations was based on six distinct pendant configurations and eight criteria. The importance weights for each criterion under different mission scenarios, as specified in Table 10, were applied to rank the alternative solutions using the PHFs-MTOPSIS, PHFs-TOPSIS, PHFs-VIKOR, and PHFs-EDAS methods. Additionally, in the VIKOR method, the weight distribution parameter r = 0.5 [82].

3.3.2. Results and Analysis

The comparative analysis and final rankings of alternative solutions across various mission scenarios are presented in

Table 12. Comparison of PHFs-MTOPSIS, PHFs-TOPSIS, PHFs-VIKOR, and PHFs-EDAS.

Type	PHFs-MTOPSIS		PHFs-TOPSIS		PHFs-VIKOR		PHFs-EDAS
	Index Value	Ranking	Index Value	Ranking	Index Value	Ranking	Index Value	Ranking
Scenario 1
DS	0.5672	2	0.6110	2	0.3538	3	0.0914	2
TR	0.3984	4	0.3916	4	0.2661	4	−0.0335	4
TW	0.2867	5	0.2676	5	0.1196	5	−0.0894	5
MTT	0.1169	6	0.1094	6	0.0568	6	−0.1567	6
CTT	0.5564	3	0.5439	3	0.3832	2	0.0539	3
DTT	0.7397	1	0.7298	1	0.5653	1	0.1343	1
Scenario 2
$\mathrm{DS}$	0.6277	2	0.6929	1	0.4742	3	0.1174	1
$\mathrm{TR}$	0.3622	4	0.3713	4	0.1737	4	−0.0490	4
$\mathrm{TW}$	0.2588	5	0.3470	5	0.0164	6	−0.1027	5
$\mathrm{MTT}$	0.1076	6	0.1048	6	0.0720	5	−0.1597	6
$\mathrm{CTT}$	0.5981	3	0.5890	3	0.4357	2	0.0770	3
$\mathrm{DTT}$	0.7097	1	0.6860	2	0.6541	1	0.1170	2
Scenario 3
$\mathrm{DS}$	0.4962	3	0.4663	3	0.2241	3	0.0463	3
$\mathrm{TR}$	0.4280	4	0.4459	4	0.0910	4	−0.0223	4
$\mathrm{TW}$	0.3311	5	0.3522	5	0.0265	6	−0.0643	5
$\mathrm{MTT}$	0.1260	6	0.1291	6	0.0316	5	−0.1562	6
$\mathrm{CTT}$	0.5539	2	0.5448	2	0.2675	2	0.0505	2
$\mathrm{DTT}$	0.7609	1	0.7723	1	0.3901	1	0.1460	1
Scenario 4 (Overall Score)
DS	0.5658	2	0.6059	2	0.3855	3	0.0886	2
TR	0.3956	4	0.3864	4	0.2101	4	−0.0354	4
TW	0.2902	5	0.2759	5	0.0621	5	−0.0874	5
MTT	0.1165	6	0.1116	6	0.0559	6	−0.1574	6
CTT	0.5671	3	0.5615	3	0.3986	2	0.0598	3
DTT	0.7359	1	0.7224	1	0.560873	1	0.1319	1

. This table illustrates the evaluation scores and rankings derived from four distinct methodologies: PHFs-MTOPSIS, PHFs-TOPSIS, PHFs-VIKOR, and PHFs-EDAS. The results from these methods offer a comprehensive framework for assessing the relative effectiveness of each option. By incorporating multiple evaluation techniques, the analysis provides a robust and diversified approach to decision making. This highlights the strengths and weaknesses of each option under consideration. Such methodological diversity is essential for deriving reliable and insightful conclusions in complex decision environments.

In Scenario 1, as shown in Figure 8a, the results indicate that the DTT configuration is the most suitable solution for the drone, closely followed by the dual-system DS configuration and the tail-sitter CTT configuration. This outcome is understandable given that attributes B₁, B₆, and B₈ are ranked as the top three among eight considered attributes. Although the DS configuration scores the highest in two attributes and third in another, the DTT configuration demonstrates a more balanced performance, particularly since one attribute of the DS-configured drone ranks last among all alternatives.

In Scenario 2, as shown in Figure 8b, when attributes B₁, B₄, and B₆ are prioritized, the top three configurations are DTT, CTT, and DS. However, there is a divergence in determining the optimal configuration. Under the methodologies of PHFs-TOPSIS and PHFs-EDAS, the scores for the first and second positions are closely matched, with a difference of less than 1%, while the gap between the fifth and sixth positions exceeds 52%. This indicates that both algorithms need to be optimized based on specific performance metrics.

In Scenario 3, as shown in Figure 8c, where portability and flight maneuverability are prioritized, the DTT configuration emerges as a leader in comprehensive performance evaluations across various algorithms. This consistent high performance underscores its superior ability to meet the demands of both portability and mobility, making it exceptionally well-suited for tasks requiring these attributes. This analysis suggests that the DTT configuration is optimally tailored for environments where rapid deployment and ease of transport are critical. Conversely, the CTT and DS configurations represent viable alternatives, offering robust potential solutions where slightly different balances of these key attributes might be needed. These insights are crucial for strategic decision making in dynamic operational contexts where agility and adaptability are paramount.

Scenario 4, as shown in Figure 8d, was designed to select an optimal solution that performs comprehensively well across the previously mentioned three scenarios. Given the frequency of Scenario 1 at 50%, Scenario 2 at 30%, and Scenario 3 at 20%, the DTT configuration is identified as the top performer. This configuration demonstrates consistent superiority across multiple decision-making methods, underscoring its capability to balance the integrated demands effectively. Such uniform excellence renders the DTT configuration exceptionally suitable for environments that require robust performance levels. Meanwhile, the DS and CTT configurations are identified as strong contenders, offering viable alternatives where slightly different performance equilibria are desired.

Based on the comparison of results obtained using these four algorithms across four mission scenarios, it is evident that all four algorithms yield similar outcomes. According to the standard validity theory, this indicates the effectiveness of the algorithms.

4. Discussion

In this study, we built on the recent MADM literature addressing decision-making problems in the aviation field to propose a method for predicting the completion rate of different drones under various missions using mission efficiency. By defining mission efficiency, we interpreted it as a trade-off between conflicting attributes and mapped it onto flight performance, environmental adaptability, and cost. Finally, we quantified these relationships, and the results were used to support the selection of drones with similar functionalities.

The decision model designed in this study combines PHFs and seven-degree linguistic variables with the MADM method, preserving the diverse fuzzy opinions of different experts when evaluating subjective indicators and integrating these opinions into a single index. To validate the effectiveness of the proposed method, we considered and compared several small e-VTOLFW drones with different configurations. To assess the reliability of the proposed method and its sensitivity to weight changes, we considered the capability requirements under various mission scenarios. The results indicate that the PHFs-MADM model can accurately evaluate and rank the mission efficiency of small e-VTOLFW drones. The model provides robust results in complex mission scenarios and under multiple criteria, demonstrating strong applicability.

This study provides actionable insights into the design and optimization of drones in the context of complex mission requirements. However, a notable challenge is the inherent uncertainty in the rankings derived from the solutions due to assumptions made during evaluation. This uncertainty stems from the complex nature of drone design, which encompasses multiple disciplines and is limited by the available quantifiable technical parameters. The uncertainties primarily involve aspects such as controller design, energy technology, and materials. From the perspective of overall drone design, the flight performance of drones heavily depends on their aerodynamic design; however, the performance of control systems also significantly impacts flight capabilities. Therefore, it is assumed that drones with different configurations have consistent control system performance. Additionally, since different energy sources significantly affect aspects such as range, we chose to adopt electric propulsion designs for evaluation. Materials and manufacturing processes were also kept consistent. This also suggests that if future research papers on drone design can ensure the comprehensiveness and accuracy of data, the precision of decision-making systems will correspondingly improve.

5. Conclusions

This paper addresses the issue of the lack of clear mapping between system designs and model libraries in MBSE-based UAV designs. By deriving capability requirements from mission requirements, we established a mapping relationship between different tasks and capability attributes. We successfully developed a PHFs-MADM model to evaluate UAV mission efficiency in complex tasks. The current popular e-VTOLFW drones were used as the research subject to validate the rationality of the evaluation model in complex mission scenarios.

This model also provides a new approach for the rapid development of other similar products. By constructing the “demand–function” mapping relationship, the corresponding “product–market demand” early evaluation model was established. This approach reduces development costs, avoids mismatches between products and market demands, and facilitates rapid product development.

In the era of artificial general intelligence, by constructing knowledge graphs for related products and embedding comprehensive evaluation models within these graphs, we can leverage big data and powerful computing capabilities to derive effective conclusions in complex, multi-source decision-making environments. This approach can even determine the design parameter ranges for related products, thereby achieving true intelligent manufacturing.