The proliferation and technological advancement of unmanned aerial vehicles (UAVs) pose a significant and evolving threat to modern air defense architectures. Their characteristics—low cost, small radar cross-section, potential for swarming behavior, and increasing autonomy—challenge traditional air and missile defense paradigms. Consequently, countering UAVs, or anti-drone operations, has become an integral component of joint air and missile defense (JAMD) missions. Evaluating the effectiveness of anti-drone capabilities cannot be done in isolation; it must be assessed within the context of the broader JAMD system where capabilities are fused and synergized. Isolated assessments of individual anti-drone systems fail to capture the holistic combat power generated by network-centric integration. This study focuses on constructing a comprehensive evaluation framework and model for assessing anti-drone combat capability within a JAMD system. By employing a game theory-based combination weighting method that synthesizes subjective (Analytic Hierarchy Process) and objective (Entropy Weight) weights, we aim to derive a more scientific and rational set of indicator weights, providing a theoretical foundation for capability assessment and force development.
The cornerstone of any robust evaluation is a well-defined and hierarchical indicator system. For anti-drone combat capability within a JAMD framework, we propose a multi-tiered structure encompassing five primary capabilities, each decomposed into secondary and tertiary indicators. This structure captures the end-to-end process from system integration to sustained operations.
| Tier-1 Indicator | Tier-2 Indicator | Tier-3 Indicator |
|---|---|---|
| A. System Integration Capability | A1. Integration Stability | A11. Command/Info Transmission Continuity A12. Command/Info Transmission Accuracy |
| A2. Integration Depth | A21. Situational Awareness (SA) Sharing Capability A22. Execution Order Direct-Reach Capability |
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| A3. Integration Breadth | A31. Number of Integrated Anti-Drone System Types A32. Maximum Number of Connectable Anti-Drone Assets |
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| B. Detection & Tracking Capability | B1. Target Detection Capability | B11. Target Detection Range B12. Target Detection Altitude |
| B2. Target Tracking Capability | B21. Target Tracking Accuracy B22. Number of Simultaneously Tracked Targets |
|
| B3. Target Discrimination Capability | B31. Target Quantity Discrimination (within a swarm) B32. Target Type Recognition B33. Target Attribute/IFF Recognition |
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| C. Command & Control (C2) Capability | C1. Situational Understanding Capability | C11. Enemy Force Awareness C12. Friendly Force Awareness C13. Battlespace Environment Awareness |
| C2. Operational Planning Capability | C21. Pre-engagement Planning Effectiveness C22. In-engagement Planning (Re-tasking) Effectiveness |
|
| C3. Command Decision-Making Capability | C31. Force Employment Effectiveness C32. Order/Directive Issuance Accuracy C33. Order/Directive Issuance Timeliness |
|
| C4. Emergency Response Capability | C41. Response to Airborne Emergencies C42. Response to Ground-based Emergencies |
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| D. Multi-Effect Interception Capability | D1. Target Selection Capability | D11. Threat Level Assessment Accuracy D12. Target Value Assessment Accuracy D13. Intercept Feasibility Assessment Accuracy |
| D2. Force Application Capability | D21. Asset-to-Target Assignment Effectiveness D22. Target Kill/Neutralization Probability |
|
| D3. Comprehensive Cost-Effectiveness | D31. Cost-Benefit Ratio (Adversary Loss Cost vs. Friendly Expenditure Cost) | |
| E. Sustained Operations Capability | E1. Self-Protection Capability | E11. Counter-Reconnaissance Capability E12. Counter-Strike/Hardening Capability |
| E2. Integrated Support Capability | E21. Equipment Repair & Recovery Rate E22. Ammunition/Resource Resupply Rate |
Determining the relative importance (weight) of each indicator is critical. Sole reliance on subjective expert judgment (e.g., AHP) or purely objective data-driven methods (e.g., Entropy Weight) has limitations. We employ a three-step weighting methodology to achieve balance.
Step 1: Subjective Weighting via Analytic Hierarchy Process (AHP)
AHP uses pairwise comparisons based on expert judgment. For a set of criteria, a judgment matrix \( \mathbf{A} = (a_{ij})_{n \times n} \) is constructed, where \( a_{ij} \) represents the relative importance of criterion \( i \) over \( j \) using a 1-9 scale. The priority vector \( \mathbf{W}_{sub} = [w_1, w_2, …, w_n]^T \), representing the subjective weights, is derived as the principal eigenvector of \( \mathbf{A} \). Consistency is verified using the Consistency Ratio (\( CR \)):
$$ CR = \frac{CI}{RI}, \quad \text{where} \quad CI = \frac{\lambda_{max} – n}{n – 1} $$
Here, \( \lambda_{max} \) is the largest eigenvalue of \( \mathbf{A} \), and \( RI \) is the random index. A \( CR \leq 0.1 \) is acceptable.
Step 2: Objective Weighting via Entropy Weight Method
This method determines weights based on the information entropy inherent in the evaluation data, minimizing subjective bias. For \( m \) alternatives and \( n \) indicators, the standardized decision matrix is \( \mathbf{R} = (r_{ij})_{m \times n} \). The proportion of alternative \( i \) under indicator \( j \) is:
$$ p_{ij} = \frac{r_{ij}}{\sum_{i=1}^{m} r_{ij}} $$
The entropy \( e_j \) for indicator \( j \) is:
$$ e_j = -\frac{1}{\ln m} \sum_{i=1}^{m} p_{ij} \ln p_{ij} $$
The degree of divergence \( d_j = 1 – e_j \). The objective weight for indicator \( j \) is:
$$ w_j^{obj} = \frac{d_j}{\sum_{j=1}^{n} d_j} $$
The vector of objective weights is \( \mathbf{W}_{obj} \).
Step 3: Optimized Combination Weighting via Game Theory
To reconcile \( \mathbf{W}_{sub} \) and \( \mathbf{W}_{obj} \), we seek a combined weight vector \( \mathbf{W}_{com} \) that minimizes the deviation from both basic vectors, modeled as a cooperative game:
$$ \mathbf{W}_{com} = \alpha \cdot \mathbf{W}_{sub}^T + \beta \cdot \mathbf{W}_{obj}^T $$
The coefficients \( \alpha \) and \( \beta \) are optimized by solving the minimization problem:
$$ \min \left\| \alpha \mathbf{W}_{sub}^T + \beta \mathbf{W}_{obj}^T – \mathbf{W}_{sub}^T \right\|_2 + \min \left\| \alpha \mathbf{W}_{sub}^T + \beta \mathbf{W}_{obj}^T – \mathbf{W}_{obj}^T \right\|_2 $$
This leads to the system of equations from the first-order derivative conditions:
$$
\begin{cases}
\alpha \mathbf{W}_{sub} \mathbf{W}_{sub}^T + \beta \mathbf{W}_{sub} \mathbf{W}_{obj}^T = \mathbf{W}_{sub} \mathbf{W}_{sub}^T \\
\alpha \mathbf{W}_{obj} \mathbf{W}_{sub}^T + \beta \mathbf{W}_{obj} \mathbf{W}_{obj}^T = \mathbf{W}_{obj} \mathbf{W}_{obj}^T
\end{cases}
$$
Solving for \( \alpha \) and \( \beta \), and normalizing, we get the final combination coefficients:
$$ \alpha^* = \frac{\alpha}{\alpha + \beta}, \quad \beta^* = \frac{\beta}{\alpha + \beta} $$
The final comprehensive weight vector is:
$$ \mathbf{W}_{com} = \alpha^* \mathbf{W}_{sub}^T + \beta^* \mathbf{W}_{obj}^T $$
Quantification models are essential for converting raw data into normalized scores for tertiary indicators. Most are ratio-based, measuring actual performance against an ideal or required standard. The aggregated score for an upper-level indicator \( Q_k \) is calculated as the weighted sum of its child indicator scores \( M_{ki} \):
$$ Q_k = \sum_{i=1}^{n} M_{ki} \cdot W_{ki} $$
where \( W_{ki} \) is the weight of the \( i \)-th child indicator under parent \( k \).
| Tier-3 Indicator | Exemplary Calculation Formula (Conceptual) |
|---|---|
| A11. Command Transmission Continuity | 1 – (Number of Interrupted Transmissions / Total Orders Sent) |
| B22. Number of Tracked Targets | Number of Tracks Maintained / Total Number of Airborne Threats |
| C32. Order Issuance Accuracy | Number of Correctly Executed Orders / Total Orders Issued |
| D22. Target Kill Probability | Number of UAVs Neutralized or Destroyed / Number of UAVs Engaged |
| E21. Equipment Repair Rate | Mean Time To Repair (MTTR) / Total Operational Time Window |
To demonstrate the model’s application, we consider an anti-drone scenario against a UAV swarm. Data from three simulation runs were used. First, expert judgment formed AHP matrices to compute subjective weights \( \mathbf{W}_{sub} \). Second, the simulation data was processed via the Entropy Weight method to obtain objective weights \( \mathbf{W}_{obj} \). Finally, the game theory combination model was applied. The derived combination coefficients were \( \alpha^* = 0.43 \) and \( \beta^* = 0.57 \), indicating a slight bias toward the objective data while retaining expert insight.
| Tier-1 Indicator | AHP Weight (\( \mathbf{W}_{sub} \)) | Entropy Weight (\( \mathbf{W}_{obj} \)) | Combined Weight (\( \mathbf{W}_{com} \)) |
|---|---|---|---|
| A. System Integration | 0.0764 | 0.2000 | 0.1500 |
| B. Detection & Tracking | 0.1611 | 0.1600 | 0.1600 |
| C. Command & Control | 0.2422 | 0.1600 | 0.2000 |
| D. Multi-Effect Interception | 0.4700 | 0.1600 | 0.2900 |
| E. Sustained Operations | 0.0503 | 0.3200 | 0.2000 |
The final aggregated anti-drone combat capability scores for the sample data, calculated using the three different weighting schemes, were:
- AHP-Only Score: 0.59
- Entropy-Only Score: 0.71
- Game Theory Combined Score: 0.66
The combined score of 0.66 represents a balanced assessment, mitigating the extreme emphasis on interception capability from pure AHP (weight of 0.47) and the heavy focus on sustainability from pure Entropy Weight (weight of 0.32). This result underscores the value of the combination approach, producing an evaluation that is both theoretically grounded and empirically informed, specifically tailored for the anti-drone mission in a JAMD context.
In conclusion, effectively countering modern drone threats requires their treatment as a core component of joint air and missile defense. This study established a hierarchical evaluation framework encompassing system integration, detection, command, interception, and sustainability. The proposed game theory-based combination of AHP and Entropy Weighting provides a robust and scientifically sound method for determining indicator importance, overcoming the biases inherent in using either method alone. The model offers a valuable tool for assessing the overall anti-drone combat capability of a JAMD system, identifying capability gaps, and supporting informed decision-making for force structure development and tactical training. Future work should involve broader expert consultation, dynamic adjustment of indicators and standards as technology evolves, and application to more complex, large-scale anti-drone scenarios, including heterogeneous swarms and adaptive adversaries.