A Symmetry-Driven Inverse Design Framework for Multi-Agent Cooperative Deployment Under Line-of-Sight Constraints

Abstract

Cooperative deployment of mobile agents under geometric and line-of-sight constraints gives rise to high-dimensional constrained optimization problems whose underlying physical configuration often exhibits exploitable structure. This paper develops a symmetry-driven inverse design framework that leverages two structural features of the engagement geometry—the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ mirror symmetries of the extended target silhouette and a closed-form forward–inverse correspondence between line-of-sight-aligned burst locations and physical agent parameters—to construct low-dimensional seeds for subsequent physical parameter optimization. The framework is developed and validated on a representative naval defense instance in which a fleet of unmanned aerial vehicles (UAVs) releases spherical obscuration payloads to interrupt the line of sight between incoming mobile threats and a cylindrical extended target. Instead of searching only over the four-dimensional UAV parameter space (heading angle, speed, drop time, fuse delay), the method first specifies a desired burst location in a two-dimensional inverse space and analytically back-calculates feasible agent parameters, which are then refined by multi-start Nelder–Mead optimization in the physical parameter space. A conservative three-dimensional cylindrical line-of-sight obscuration model is developed by constructing four extreme tangent sightlines from the missile to the cylindrical target and verifying whether the spherical smoke cloud simultaneously blocks all of them. A hierarchical multi-agent task allocation framework combines a performance matrix, assignment enumeration, and joint multi-start refinement. Numerical experiments on five progressively complex sub-problems demonstrate obscuration durations of $1.362 \mathrm{s}$ (single fixed shot), $4.580 \mathrm{s}$ (optimized shot), $7.324 \mathrm{s}$ (three-shot relay), $11.140 \mathrm{s}$ (three-UAV cooperation), and $20.652 \mathrm{s}$ (full five-UAV three-missile assignment). Additional high-dimensional benchmarks, sensitivity tests, and error analyses clarify the reproducibility and limitations of the approach.

1. Introduction

Cooperative deployment problems in which a team of mobile agents must position themselves or their payloads to satisfy geometric and line-of-sight constraints arise in a wide range of engineering applications, including UAV formation planning, coordinated sensor placement, and cooperative surveillance [1,2]. A structurally rich subclass of such problems requires the agents to cooperatively position area-effect payloads so as to obstruct the line of sight (LOS) between moving adversarial agents and a protected extended target—a formulation that couples continuous trajectory planning with discrete task allocation, and whose underlying physical configuration exhibits strong geometric symmetries that are rarely exploited in existing algorithmic approaches. This paper addresses one representative instance of this class, drawn from the naval defense literature [3,4]: a fleet of UAVs cooperatively releases spherical obscuration payloads (smoke clouds) to interrupt the LOS from incoming anti-ship missiles to a high-value surface vessel, with each UAV offering speed, expendability, and the capacity to carry multiple projectiles. The central optimization problem lies in determining an optimal multi-agent deployment strategy—when each UAV should release each projectile, at what speed and heading, and with what fuse delay—so that the resulting obscuration windows collectively maximize the total LOS disruption duration across all adversarial agents. This paper addresses this challenge by introducing a symmetry-driven inverse design framework that fundamentally reverses the conventional optimization logic, combined with a hierarchical multi-agent task allocation strategy.

1.1. Literature Review

UAV path planning and task allocation. The problem of planning optimal paths for UAVs has attracted extensive research attention over the past two decades. Ait Saadi et al. [5] provided a comprehensive survey of UAV path planning using optimization approaches, categorizing methods into classical, heuristic, and metaheuristic families. Zhao et al. [6] reviewed computational-intelligence-based UAV path planning techniques, while Ghambari et al. [7] offered an updated survey covering recent advances from operations research perspectives. Aggarwal and Kumar [8] examined path planning techniques with emphasis on solutions and open challenges in complex environments. In the multi-UAV setting, task allocation becomes a critical prerequisite for cooperative missions. Skaltsis et al. [2] reviewed task allocation methods for UAVs, covering auction-based, optimization-based, and game-theoretic approaches. Song et al. [1] surveyed mission planning for multiple UAVs, highlighting the interplay between path planning and task assignment. However, existing research predominantly focuses on navigation and obstacle avoidance; the temporal coordination of smoke release events along predetermined trajectories remains largely unaddressed.

Obscurant deployment and smoke screen planning. Smoke screens and other obscurants are widely used to deny visual, infrared, or laser-guided sensing by increasing optical attenuation along a threat–target line of sight. Existing engineering studies of obscurants primarily emphasize material emission characteristics, atmospheric transport, and optical/infrared rendering or assessment, while operational planning studies usually treat the smoke source location or release schedule as prescribed rather than as a jointly optimized decision. In the UAV-deployed smoke screen problem considered here, the release time, fuse delay, UAV heading, UAV speed, and asset-to-threat assignment jointly determine whether an effective obscuration volume is placed at the right point in space and time. This coupling distinguishes the present work from conventional path planning and from concentration-only dispersion modeling: the optimization variable is not simply a UAV route or a pollutant concentration field, but the spatio-temporal placement of an obscurant cloud relative to a moving line-of-sight constraint.

Weapon–target assignment. The allocation of multiple UAVs to multiple missile threats bears a strong conceptual resemblance to the classical weapon–target assignment (WTA) problem. Manne [9] introduced the foundational formulation, which seeks to assign weapons to targets so as to minimize the expected surviving target value. Ahuja et al. [10] developed exact and heuristic algorithms that significantly advanced the scalability of WTA solutions. Kline et al. [4] provided a comprehensive treatment of the WTA problem, discussing both static and dynamic variants. More recently, Andersen et al. [11] proposed exact and approximate algorithms with tightened bounds, Lu and Chen [12] developed a new exact algorithm exploiting problem structure, and Bertsimas and Paskov [13] achieved near-real-time solutions for large-scale WTA instances via branch-price-and-cut. Li et al. [3] presented a comprehensive survey covering models, algorithms, and applications across the WTA literature. While our UAV-to-missile assignment shares the combinatorial structure of WTA, it differs in that the “weapon effectiveness” is not a fixed probability but a continuous objective (obscuration duration) that must be jointly optimized.

Atmospheric dispersion and smoke modeling. The physical behavior of smoke clouds is governed by atmospheric dispersion processes. Zannetti [14] established the theoretical foundations of Gaussian plume and puff dispersion models, which remain the most widely used analytical framework. Stockie [15] provided an accessible mathematical treatment of atmospheric dispersion modeling, connecting the advection–diffusion equation to practical Gaussian approximations. Cao et al. [16] investigated dispersion coefficients for Gaussian puff models under various atmospheric stability conditions, while Korsakissok and Mallet [17] conducted a comparative study of Gaussian dispersion formulas within the Polyphemus platform. These models predict pollutant concentrations at receptor locations, but the smoke screen problem requires a fundamentally different output: the binary determination of whether a smoke cloud of a given size and position achieves complete visual blockage of an extended target. Bridging this gap between concentration prediction and geometric obscuration assessment motivates the simplified spherical smoke model adopted in this work.

Line-of-sight analysis and visibility modeling. Determining whether a smoke cloud blocks a missile’s view of the target requires line-of-sight (LOS) analysis. Liu et al. [18] proposed an improved LOS method for visibility analysis in three-dimensional complex landscapes. De Floriani et al. [19] studied LOS communication on terrain models, establishing efficient algorithms for intervisibility queries. Murray et al. [20] addressed coverage optimization for security monitoring using visibility-based formulations, and Gracchi et al. [21] extended visibility analysis to 3D point clouds for sensor network optimization. A crucial distinction in our problem is that the defended vessel is an extended target rather than a geometric point. The extended object tracking literature [22,23,24] has addressed the modeling of spatially extended targets using random matrices and hypersurface models, but these approaches focus on state estimation rather than obscuration geometry. Our cylindrical target model, which constructs four extreme tangent sightlines combining azimuthal and vertical extremes, provides a conservative and physically motivated criterion for complete LOS blockage.

Derivative-free optimization, symmetry, and reparameterization. The smoke deployment optimization landscape is non-smooth and multimodal, favoring derivative-free methods. The Nelder–Mead simplex algorithm [25], with convergence properties analyzed by Lagarias et al. [26], is widely used for low-dimensional derivative-free optimization due to its simplicity and effectiveness. Rios and Sahinidis [27] provided a comprehensive review of derivative-free optimization algorithms and compared software implementations, while Storn and Price [28] introduced differential evolution as an alternative global optimization heuristic for continuous spaces. To mitigate local-optimum sensitivity, multi-start strategies have been extensively studied [29,30,31]. Symmetry reduction has also been studied in optimal control as a way to reduce redundant state descriptions by exploiting geometric invariances [32], and engineering design studies have used symmetry constraints, pattern repetition, and design-space reparameterization to impose hard geometric constraints and simplify feasible search spaces [33,34]. Our contribution follows this philosophy in a visibility-constrained deployment setting: geometric structure is used to construct feasible, physically interpretable seeds, and the final refinement is performed in the original physical parameter space so that off-axis improvements are not artificially excluded.

Research gap and motivation. Despite the rich individual literatures on UAV path planning, weapon–target assignment, atmospheric dispersion, visibility analysis, and derivative-free optimization, no existing work integrates these elements into a unified framework for smoke screen deployment optimization. Current approaches treat UAV path planning and smoke deployment as separate sequential problems [5,7], and the WTA literature focuses on discrete assignment with fixed weapon effectiveness [3,4]. The idea of inverse design—specifying desired outcomes and analytically back-calculating the required inputs—has not been applied to the smoke deployment context. Furthermore, LOS obscuration models in the literature predominantly assume point targets, neglecting the spatial extent of real naval vessels. These gaps motivate the present work, which proposes an inverse design framework that reduces search dimensionality by half, a three-dimensional cylindrical obscuration model for extended targets, and a hierarchical allocation strategy that bridges the gap between WTA theory and continuous deployment optimization.

1.2. Significance and Contributions

From a practical standpoint, the proposed framework directly addresses an operational need in naval defense: coordinating multiple UAVs to create sustained smoke barriers against simultaneous missile threats. From a methodological perspective, the inverse design principle—specifying desired outcomes and back-calculating inputs—represents a general dimensionality reduction strategy applicable beyond the smoke deployment domain to other problems where the mapping from design parameters to performance is analytically invertible.

The main contributions of this paper are as follows:

• Inverse design seeding and dimensionality reduction. We propose a method that transforms the initial search for promising single-shot deployments from four UAV parameters (heading angle, speed, drop time, fuse delay) into a two-dimensional inverse search over interception time and spatial offset, with UAV parameters obtained by closed-form back-calculation. These inverse solutions are used as high-quality seeds for subsequent physical parameter refinement, reducing wasted evaluations while preserving physical interpretability.
• Three-dimensional cylindrical LOS obscuration model. We develop a conservative obscuration criterion for extended targets modeled as vertical cylinders. Four extreme tangent sightlines—combining two azimuthal tangent directions with the top and bottom of the cylinder—are constructed, and the smoke cloud is required to simultaneously block all four. This provides a stricter and more realistic assessment than point-target approximations.
• Hierarchical multi-UAV multi-missile allocation framework. We introduce a three-stage allocation strategy—performance matrix construction, assignment enumeration with feasibility filtering, and joint multi-start Nelder–Mead optimization with cross-initialization—that draws conceptual parallels with the classical weapon–target assignment problem [9,10] while handling continuous optimization objectives.
• Comprehensive numerical validation. We validate the framework through five progressively complex sub-problems (P1–P5), from single-UAV single-shot to full five-UAV three-missile assignment, demonstrating the scalability and effectiveness of the approach.

A unifying thread running through our framework is the deliberate use of geometric structure at three levels. First, the azimuthal mirror symmetry of the cylindrical target about the missile–target bearing reduces the horizontal silhouette to a pair of tangent directions ${\phi }_{\pm }$, and together with the vertical mirror symmetry of the cylinder endpoints ($z=0,H_T$) yields four extreme sightlines that define a conservative extended target test. Second, the closed-form forward–inverse correspondence between a line-of-sight-aligned burst location and the physical UAV parameters provides a compact seed space for optimization. Third, UAVs assigned to the same missile exhibit strong temporal complementarity: although the UAVs are not interchangeable because their initial positions differ, the union-duration objective allows their obscuration windows to be staggered with little overlap. These structural properties are not claimed as new geometry in isolation; rather, they are used as design principles that translate geometric observations into a practical computational workflow. Figure 1 previews the first of these properties—the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ mirror action on the cylindrical silhouette—as a graphical summary of the structural perspective developed throughout the paper.

The remainder of this paper is organized as follows. Section 2 formulates the problem, including the kinematic models, smoke cloud dynamics, and the cylindrical LOS obscuration criterion. Section 3 presents the inverse design framework, the multi-start Nelder–Mead optimization procedure, and the hierarchical allocation strategy. Section 4 reports numerical results for five progressively complex sub-problems. Section 5 discusses the advantages, limitations, and potential extensions of the proposed approach. Section 6 concludes the paper.

2. Problem Formulation

This section establishes the mathematical models underlying the smoke screen deployment problem. Section 2.1 describes the defense scenario geometry and the initial configuration of all entities. Section 2.2 specifies the kinematic models for missiles, UAVs, and projectile ballistics. Section 2.3 defines the simplified smoke cloud dynamics. Section 2.4 develops the line-of-sight obscuration criterion for both point and cylindrical targets. Finally, Section 2.5 decomposes the overall problem into a hierarchy of five sub-problems of increasing complexity.

2.1. Scenario Description

As a representative instance of the cooperative deployment problem introduced in Section 1, we consider a naval defense scenario in which a high-value cylindrical extended target—modeled as a vertical right circular cylinder of radius $R_T=7 \mathrm{m}$ and height $H_T=10 \mathrm{m}$, centered at $(0,200,0)$ m in a Cartesian coordinate system—is approached by three mobile adversarial agents $M_1$, $M_2$, and $M_3$ (anti-ship missiles). The origin of the coordinate system is placed at a decoy (false target) toward which all missiles are initially aimed. Five UAVs, designated ${\mathrm{FY}}_1$ through ${\mathrm{FY}}_5$, are tasked with deploying smoke screen projectiles to obscure the missiles’ LOS to the vessel. Each UAV carries up to three smoke projectiles and flies at a constant altitude with adjustable heading and speed. The initial positions of all entities are listed in

Table 1. Physical parameters and initial conditions for the smoke screen deployment scenario.

Parameter	Value	Description
g	$9.8 \mathrm{m} {\mathrm{s}}^{-2}$	Gravitational acceleration
$V_M$	300 $\mathrm{m} {\mathrm{s}}^{-1}$	Missile speed
v	70–140 m/s	UAV speed range
$V_{\mathrm{sink}}$	$3.0 \mathrm{m} {\mathrm{s}}^{-1}$	Smoke cloud sinking speed
$R_S$	10 m	Effective smoke radius
$T_{\mathrm{eff}}$	20 s	Smoke effective duration
$R_T$	7 m	Target vessel radius
$H_T$	10 m	Target vessel height
Missile initial positions (m)
$M_1$	$(20,000,0,2000)$
$M_2$	$(19,000,600,2100)$
$M_3$	$(18,000,-600,1900)$
UAV initial positions (m)
${\mathrm{FY}}_1$	$(17,800,0,1800)$
${\mathrm{FY}}_2$	$(12,000,1400,1400)$
${\mathrm{FY}}_3$	$(6000,-3000,700)$
${\mathrm{FY}}_4$	$(11,000,2000,1800)$
${\mathrm{FY}}_5$	$(13,000,-2000,1300)$

. Figure 2 illustrates the three-dimensional layout of the scenario in kilometers for readability.

2.2. Kinematic Models

2.2.1. Missile Motion

Each missile flies at constant speed $V_M=300 \mathrm{m} {\mathrm{s}}^{-1}$ along a straight line from its initial position ${\mathit{P}}_0^{\left(m\right)}$ toward the decoy at the origin. The missile position at time t is

$$\mathit{M}\left(t\right)={\mathit{P}}_0^{\left(m\right)}\left(1-{\displaystyle \frac{V_{M}t}{\parallel {\mathit{P}}_0^{\left(m\right)}\parallel }}\right),$$

(1)

and the total flight time is $T_M=\parallel {\mathit{P}}_0^{\left(m\right)}\parallel /V_M$.

2.2.2. UAV Motion

Each UAV flies horizontally at constant altitude with heading angle $\theta$ (measured counterclockwise from the positive x-axis) and speed $v\in [70,140]$ m/s. Given initial position ${\mathit{P}}_0^{\left(u\right)}$, the UAV position at time t is

$$\mathit{U}\left(t\right)={\mathit{P}}_0^{\left(u\right)}+vt{(cos\theta ,sin\theta ,0)}^{\top }.$$

(2)

2.2.3. Projectile Ballistics

When a smoke projectile is released at time $t_{\mathrm{drop}}$, it inherits the UAV’s horizontal velocity and falls freely under gravity. After a fuse delay of $\delta t$ seconds, the projectile detonates at the burst point

$$\mathit{B}=\mathit{U}\left(t_{\mathrm{drop}}\right)+v\delta t{(cos\theta ,sin\theta ,0)}^{\top }-{\left(0,0,\frac{1}{2}g\delta t^2\right)}^{\top },$$

(3)

where $g=9.8 \mathrm{m} {\mathrm{s}}^{-2}$ is gravitational acceleration. The burst time is $t_b=t_{\mathrm{drop}}+\delta t$.

2.3. Smoke Cloud Dynamics

Upon detonation, a smoke projectile instantly generates a spherical smoke cloud of effective radius $R_S= 10 \mathrm{m}$. This simplified model abstracts the complex Gaussian dispersion physics [14,15,16,17] into a binary obscuration sphere: any LOS passing within distance $R_S$ of the smoke center is considered fully blocked. In a full radiative-transfer treatment, obscuration would depend on the optical depth

$$\tau \left(t\right)={\int }_{\overline{\mathit{M}\mathit{V}}}\kappa C(\mathit{x},t)\mathrm{d}\ell ,$$

(4)

where $C(\mathit{x},t)$ is the aerosol concentration and $\kappa$ is an extinction coefficient; a seeker would be considered blocked only when $\tau$ exceeds a sensor-dependent threshold. Such a model requires smoke composition, particle-size distribution, humidity, atmospheric stability, and seeker spectral response, none of which is specified in the benchmark scenario. We therefore interpret $R_S$ as the radius of an iso-optical-depth surface, i.e., the region in which the integral in Equation (4) is assumed to exceed the required threshold. The binary assumption is consequently a planning-level threshold approximation rather than a claim that the physical cloud has a sharp opaque boundary.

After formation, the smoke cloud sinks vertically at a constant rate $V_{\mathrm{sink}}=3.0 \mathrm{m} {\mathrm{s}}^{-1}$ while maintaining its horizontal position and effective radius. This vertical sinking represents the net effect of gravitational settling of smoke particles. Horizontal wind drift is set to zero in the nominal optimization to isolate the geometric deployment problem; however, wind is operationally important and is explicitly examined in the sensitivity analysis in Section 5.3. The nominal smoke center at global time t is therefore

$$\mathit{S}\left(t\right)=\mathit{B}-{(0,0,V_{\mathrm{sink}}(t-t_b))}^{\top },t_b\le t\le t_b+T_{\mathrm{eff}},$$

(5)

where $T_{\mathrm{eff}}=20 \mathrm{s}$ is the effective duration beyond which the smoke disperses below useful density. A constant horizontal wind ${\mathit{w}}_{xy}$ can be incorporated by replacing Equation (5) with $\mathit{S}\left(t\right)=\mathit{B}+{({\mathit{w}}_{xy},0)}^{\top }(t-t_b)-{(0,0,V_{\mathrm{sink}}(t-t_b))}^{\top }$; this extension is not used in the headline optimization but is used to quantify robustness.

2.4. Line-of-Sight Obscuration Model

2.4.1. Point Target Model

In the simplest case, the defended vessel is treated as a single point at its geometric center ${\mathit{T}}_c=(0,200,5)$. The smoke cloud at position $\mathit{S}$ obscures the LOS from missile position $\mathit{M}$ to ${\mathit{T}}_c$ if the perpendicular distance from $\mathit{S}$ to the line segment $\overline{\mathit{M}{\mathit{T}}_c}$ satisfies $d<R_S$ and the projection parameter $s^{*}\in (0,1)$, where

$$d={\displaystyle \frac{\parallel (\mathit{S}-\mathit{M})\times ({\mathit{T}}_c-\mathit{M})\parallel }{\parallel {\mathit{T}}_c-\mathit{M}\parallel }},s^{*}={\displaystyle \frac{(\mathit{S}-\mathit{M})·({\mathit{T}}_c-\mathit{M})}{\parallel {\mathit{T}}_c{-\mathit{M}\parallel }^2}}.$$

(6)

2.4.2. Cylindrical Target Model

For the more realistic cylindrical target, complete LOS obscuration requires that the smoke cloud block all sightlines from the missile to every visible point on the cylinder surface. Rather than checking infinitely many sightlines, we identify four extreme sightlines that bound the visible silhouette of the cylinder as seen from the missile. These extremes are constructed by combining two azimuthal tangent directions (left and right tangent lines in the horizontal plane) with two vertical extremes (cylinder bottom at $z=0$ and top at $z=H_T$), yielding four critical target points.

To derive the tangent directions, we project the geometry onto the horizontal $xy$-plane (Figure 3). Let ${\mathit{m}}_{xy}$ and ${\mathit{c}}_{xy}=(0,200)$ denote the horizontal projections of the missile and cylinder axis, respectively. The distance between them is $D_{xy}=\parallel {\mathit{c}}_{xy}-{\mathit{m}}_{xy}\parallel$, the bearing angle is $\beta =atan2(c_y-m_y,c_x-m_x)$, and the tangent half-angle is $\alpha =arcsin(R_T/D_{xy})$. The two tangent directions are then

$${\phi }_{+}=\beta +\frac{\pi }{2}+\alpha ,{\phi }_{-}=\beta -\frac{\pi }{2}-\alpha .$$

(7)

The four extreme target points are

$${\mathit{V}}_{ij}=\left(R_{T}cos{\phi }_i,200+R_{T}sin{\phi }_i,z_j\right),i\in \{+,-\},j\in \{0,H_T\}.$$

(8)

In the implementation, the cylinder is declared obscured only when the smoke sphere blocks all four sightlines $\overline{\mathit{M}{\mathit{V}}_{ij}}$; i.e., $d(\mathit{S},\overline{\mathit{M}{\mathit{V}}_{ij}})<R_S$ and $s_{ij}^{*}\in (0,1)$ for all four combinations. This is a conservative finite-ray test for the visible silhouette, not a claim that four points are a complete mesh representation of an arbitrary ship geometry.

The choice of four extreme points in Equation (8) is not an arbitrary uniform discretization but a consequence of two orthogonal mirror symmetries of the idealized cylindrical silhouette. The azimuthal reflection $V_{+}\leftrightarrow V_{-}$ exchanges the two tangent directions about the vertical plane containing $\overline{MC}$, while the vertical reflection $z\leftrightarrow H_T-z$ exchanges the top and bottom endpoints about the cylinder’s mid-height plane. Together these reflections generate a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ group whose four-element orbit is exactly $\{V_{\pm ,0},V_{\pm ,H_T}\}$. Geometrically, the visible set of rays to a right circular cylinder is bounded horizontally by the two tangents and vertically by the top and bottom endpoints; hence requiring all four boundary combinations to be blocked gives a stricter test than checking the centerline or a point target. As a numerical guard, we also sampled dense intermediate silhouette points in supplementary verification runs and found no case in the reported solutions where an intermediate sampled ray was unblocked while all four extreme rays were blocked. The group action and its four-element orbit are previewed graphically in Figure 1 at the end of Section 1.

Figure 3. Cylindrical line-of-sight obscuration geometry. (a) Top-down ($xy$-plane) view showing the missile M, the cylindrical target cross-section (center C, radius $R_T$), and the two tangent sightlines at angles ${\phi }_{+}$ and ${\phi }_{-}$. The bearing angle $\beta$ is measured from the positive x-axis to the line $\overline{MC}$, and $\alpha =arcsin(R_T/D_{xy})$ is the tangent half-angle. The smoke sphere S (gray, radius $R_S$) is positioned to block both tangent lines. (b) Side view showing the vertical extent of the cylinder from $z=0$ to $z=H_T$ and the four extreme sightlines from M to the cylinder corners $V_{\pm ,0}$ and $V_{\pm ,H}$. The smoke sphere S blocks all four sightlines simultaneously; the downward arrow indicates the sinking velocity $V_{\mathrm{sink}}$.

Figure 3. Cylindrical line-of-sight obscuration geometry. (a) Top-down ($xy$-plane) view showing the missile M, the cylindrical target cross-section (center C, radius $R_T$), and the two tangent sightlines at angles ${\phi }_{+}$ and ${\phi }_{-}$. The bearing angle $\beta$ is measured from the positive x-axis to the line $\overline{MC}$, and $\alpha =arcsin(R_T/D_{xy})$ is the tangent half-angle. The smoke sphere S (gray, radius $R_S$) is positioned to block both tangent lines. (b) Side view showing the vertical extent of the cylinder from $z=0$ to $z=H_T$ and the four extreme sightlines from M to the cylinder corners $V_{\pm ,0}$ and $V_{\pm ,H}$. The smoke sphere S blocks all four sightlines simultaneously; the downward arrow indicates the sinking velocity $V_{\mathrm{sink}}$.

2.5. Problem Hierarchy

The complete smoke screen deployment optimization is decomposed into five sub-problems of increasing complexity. The simplest case, P1, evaluates the obscuration performance of a single UAV (${\mathrm{FY}}_1$) deploying one projectile against one missile ($M_1$) with prescribed flight parameters, serving as a baseline validation. P2 retains the single-UAV single-missile single-shot setting but optimizes over the four-dimensional parameter space $(\theta ,v,t_{\mathrm{drop}},\delta t)$. P3 extends to a single UAV deploying three projectiles in relay against one missile, requiring coordination of timing to maximize the union of obscuration windows. P4 considers three UAVs (${\mathrm{FY}}_1$, ${\mathrm{FY}}_2$, ${\mathrm{FY}}_3$) each deploying one projectile against one missile, introducing inter-UAV coordination. Finally, P5 addresses the full-scale problem: five UAVs each carrying three projectiles must be assigned to three missiles and their deployment parameters jointly optimized.

Table 2. Summary of the five sub-problems in the smoke screen optimization hierarchy.

Problem	UAVs	Missiles	Shots/UAV	Decision Variables
P1	1 (FY1)	1 ($M_1$)	1	0 (fixed)
P2	1 (FY1)	1 ($M_1$)	1	4: $\theta ,v,t_{\mathrm{drop}},\delta t$
P3	1 (FY1)	1 ($M_1$)	3	8: $\theta ,v,{(t_{d,k},\delta t_k)}_{k=1}^3$
P4	3 (FY1–3)	1 ($M_1$)	1 each	12: ${({\theta }_j,v_j,t_{d,j},\delta t_j)}_{j=1}^3$
P5	5 (FY1–5)	3 ($M_{1\text{–}3}$)	3 each	Assignment + 40 continuous

summarizes the problem hierarchy.

This progressive decomposition serves two purposes. First, it isolates individual algorithmic components for validation: P1 verifies the kinematic and obscuration models against known parameters, P2 tests the inverse design optimizer, P3 validates multi-shot relay coordination, and P4 introduces inter-UAV cooperation. Second, the simpler sub-problems provide initial solutions and structural insights that seed the full-scale P5 optimization, reducing the risk of convergence to poor local optima.

3. Methodology

This section details the four algorithmic components of the optimization framework. Section 3.1 introduces the inverse design parameterization used to construct a low-dimensional seed database. Section 3.2 describes the multi-start Nelder–Mead optimizer used to refine these seeds in the physical parameter space. Section 3.3 extends the approach to multi-shot relay deployments from a single UAV. Finally, Section 3.4 presents the hierarchical three-stage allocation framework for the full multi-UAV multi-missile problem.

• Use of AI-assisted writing

During manuscript preparation, Google Gemini was used as an AI-assisted writing tool to generate explanatory text describing the computational results and to improve the scientific clarity and standardization of the language. The authors reviewed and edited all AI-assisted text and take full responsibility for the final content. Gemini was not used to design the methodology, perform numerical calculations, generate data, or draw scientific conclusions.

3.1. Inverse Design Framework

The conventional forward approach to smoke deployment optimization proceeds by sampling candidate UAV parameters $(\theta ,v,t_{\mathrm{drop}},\delta t)$, computing the resulting burst point via Equation (3), simulating the smoke trajectory, and evaluating the obscuration duration. While complete, this approach must search a four-dimensional continuous space per projectile, and the objective function—total obscuration duration—is non-smooth due to the binary nature of the obscuration criterion. As additional projectiles are introduced, the dimensionality grows rapidly, making exhaustive or even random search impractical.

The inverse design framework circumvents this difficulty at the seeding stage by reversing the design logic. Instead of asking “given UAV parameters, where does the smoke end up?”, we ask “given a desired smoke location, what UAV parameters are needed?” We parameterize the desired burst location using two physically meaningful variables: the target interception time $t_{\mathrm{target}}$, specifying when along the missile’s flight path we wish to place the smoke cloud, and a spatial offset parameter $s\in (0,s_{max}]$, specifying where along the missile–target line the initial burst point should lie. Specifically, the ideal burst point is defined as

$${\mathit{B}}^{*}=(1-s)\mathit{M}\left(t_{\mathrm{target}}\right)+s{\mathit{T}}_c,$$

(9)

which places the smoke cloud on the line segment between the missile’s position at $t_{\mathrm{target}}$ and the target center, with $s=0$ corresponding to the missile itself and $s=1$ to the target. In the headline P5 results, $s_{max}=0.50$ is used in the seed-generation stage; smaller values are examined in Section 5.3. Figure 4 illustrates the geometric meaning of the s parameter.

Given the desired burst point ${\mathit{B}}^{*}$ and the UAV’s initial position ${\mathit{P}}_0^{\left(u\right)}$, the required fuse delay is obtained from the vertical drop equation:

$$\delta t=\sqrt{{\displaystyle \frac{2(P_{0,z}^{\left(u\right)}-B_z^{*})}{g}}},$$

(10)

Drop time follows as $t_{\mathrm{drop}}=t_{\mathrm{target}}-\delta t$, the required horizontal velocity components are $v_x=(B_x^{*}-P_{0,x}^{\left(u\right)})/t_{\mathrm{target}}$ and $v_y=(B_y^{*}-P_{0,y}^{\left(u\right)})/t_{\mathrm{target}}$, and the UAV speed and heading are $v=\sqrt{v_x^2+v_y^2}$ and $\theta =atan2(v_y,v_x)$. This back-calculation is unique within the line-of-sight-aligned seed family: for any desired burst point ${\mathit{B}}^{*}$ with $B_z^{*}<P_{0,z}^{\left(u\right)}$, there exists exactly one set of UAV parameters $(\theta ,v,t_{\mathrm{drop}},\delta t)$ that produces the desired seed. The design is feasible if three conditions are simultaneously satisfied: (i) the UAV speed is within the operational range, $v\in [70,140]$ m/s; (ii) the drop time is non-negative, $t_{\mathrm{drop}}\ge 0$, ensuring the projectile is released after the scenario begins; and (iii) the burst altitude is positive, $B_z^{*}>0$, preventing underground detonations. Infeasible designs are discarded before the expensive cylindrical LOS evaluation, providing a natural constraint-handling mechanism for seed construction.

Proposition 1.

Let $\mathcal{F}\subset {\mathbb{R}}^2$ denote the set of inverse design parameters $(t_{\mathrm{target}},s)$ for which the back-calculated UAV parameters satisfy the feasibility conditions $v\in [70,140]$ m/s, $t_{\mathrm{drop}}\ge 0$, and $B_z^{*}>0$. Then the mapping

$$\Phi :\mathcal{F}\to {\mathbb{R}}^4,(t_{\mathrm{target}},s)\mapsto (\theta ,v,t_{\mathrm{drop}},\delta t)$$

(11)

defined by Equations (9) and (10) is injective (one-to-one) on the feasible inverse seed set. Moreover, every feasible physical parameter vector $(\theta ,v,t_{\mathrm{drop}},\delta t)$ whose burst point $\mathit{B}$ lies on the line segment between $\mathit{M}\left(t_{\mathrm{target}}\right)$ and ${\mathit{T}}_C$ for some $t_{\mathrm{target}}\in (0,T_{max})$ admits a unique pre-image $(t_{\mathrm{target}},s)\in \mathcal{F}$.

Proof. 

Injectivity. Suppose $(t_1,s_1)\ne (t_2,s_2)$ both lie in $\mathcal{F}$. The desired burst points are ${\mathit{B}}_k^{*}=(1-s_k)\mathit{M}\left(t_k\right)+s_k{\mathit{T}}_C$ for $k=1,2$. Because the missile trajectory $\mathit{M}\left(t\right)={\mathit{P}}_0^{\left(m\right)}(1-V_{M}t/\parallel {\mathit{P}}_0^{\left(m\right)}\parallel )$ is a strictly monotone linear function of t, different $(t,s)$ pairs produce different burst points ${\mathit{B}}^{*}$ unless $t_1=t_2$ and $s_1=s_2$. Since the back-calculation (Equation (10) and the velocity inversion) is a deterministic function of ${\mathit{B}}^{*}$, different burst points yield different $(\theta ,v,t_{\mathrm{drop}},\delta t)$, establishing injectivity.

Surjectivity onto the constrained image. Given a feasible $(\theta ,v,t_{\mathrm{drop}},\delta t)$ whose burst point $\mathit{B}$ lies on the segment $[\mathit{M}\left(t^{*}\right),{\mathit{T}}_C]$ for some $t^{*}=t_{\mathrm{drop}}+\delta t$, define $s^{*}=\parallel \mathit{B}-\mathit{M}\left(t^{*}\right)\parallel /\parallel {\mathit{T}}_C-\mathit{M}\left(t^{*}\right)\parallel$. Then $\Phi (t^{*},s^{*})=(\theta ,v,t_{\mathrm{drop}},\delta t)$ by construction, so $(t^{*},s^{*})$ is the unique pre-image.    □

This correspondence is more than a technical convenience: it establishes a forward–inverse relation between a physically meaningful seed space and the UAV parameter space. The burst point ${\mathit{B}}^{*}$ plays the role of an invariant shared by both representations within the line-of-sight-aligned family. Importantly, we do not claim that this family exhausts all possible off-axis physical deployments. Instead, the inverse space is used to generate feasible and high-quality starting points, after which Nelder–Mead is applied to the original physical variables $(\theta ,v,t_{\mathrm{drop}},\delta t)$ and may move the burst point off the initial missile–target line if that improves the cylindrical LOS objective. The feasibility constraints that define $\mathcal{F}$ correspond directly to the UAV operational envelope in the seed construction step; infeasibility is preserved by the back-calculation and filtered before objective evaluation. Figure 5 summarizes this forward–inverse correspondence schematically.

The inverse design search is conducted over a uniform grid in $(t_{\mathrm{target}},s)$ space. For the visualization in Figure 6, $t_{\mathrm{target}}$ is sampled from $[1.0 \mathrm{s},65 \mathrm{s}]$ with a $0.5$ $\mathrm{s}$ step and s from $[0.001,0.50]$ with a 0.005 step, giving 12,800 grid points; for the optimization runs, a finer local grid is used as listed in Appendix A. For each grid point, the back-calculation is performed and the feasibility conditions are checked. In the ${\mathrm{FY}}_1\to M_1$ heatmap, 103 of the 12,800 sampled points are feasible under the speed, drop-time, and altitude constraints. Feasible configurations are evaluated by computing the single-shot obscuration duration under the cylindrical target model, sorted by descending obscuration duration, and truncated to the top 80 entries. Therefore, the “80 feasible configurations” mentioned in the results are not the raw grid size; they are the retained high-performing subset used as seeds for subsequent physical parameter Nelder–Mead refinement.

3.2. Multi-Start Nelder–Mead Optimization

The inverse design grid search produces a ranked database of feasible configurations. To refine these solutions, we employ the Nelder–Mead simplex algorithm [25,26], a derivative-free optimization method that iteratively deforms a simplex of $n+1$ vertices in n-dimensional space through reflection, expansion, contraction, and shrinkage operations. This method is particularly well suited to our problem because the obscuration objective is computed via discrete time-stepping and is not differentiable in the classical sense. The adaptive variant [26] is used, which scales the simplex operations according to the problem dimension.

Specifically, the Nelder–Mead algorithm operates on a simplex of $n+1$ vertices $\{{\mathit{x}}_0,{\mathit{x}}_1,\dots ,{\mathit{x}}_n\}$ in ${\mathbb{R}}^n$, ordered such that $f\left({\mathit{x}}_0\right)\ge f\left({\mathit{x}}_1\right)\ge \dots \ge f\left({\mathit{x}}_n\right)$ (maximizing obscuration). In the single-shot case, $\mathit{x}=(\theta ,v,t_{\mathrm{drop}},\delta t)$ is four-dimensional; in P3–P5, $\mathit{x}$ contains the corresponding shared flight parameters and shot-specific drop/fuse delays. Thus, the inverse design grid is two-dimensional, but the refinement simplex is explicitly built in the physical parameter space. At each iteration, the algorithm computes the centroid $\overline{\mathit{x}}$ of the n best vertices (excluding ${\mathit{x}}_n$) and generates a reflected point ${\mathit{x}}_r=\overline{\mathit{x}}+\alpha (\overline{\mathit{x}}-{\mathit{x}}_n)$ with reflection coefficient $\alpha =1$. Depending on the quality of $f\left({\mathit{x}}_r\right)$ relative to $f\left({\mathit{x}}_0\right)$ and $f\left({\mathit{x}}_{n-1}\right)$, the algorithm selects among expansion ($\gamma =2$), outside contraction ($\rho =0.5$), inside contraction ($\rho =-0.5$), or full simplex shrinkage ($\sigma =0.5$). The procedure terminates when the simplex diameter falls below a tolerance of ${10}^{-6}$ or when the function value spread $|f\left({\mathit{x}}_0\right)-f\left({\mathit{x}}_n\right)|$ is less than ${10}^{-8}$. A maximum of 500 function evaluations per run is imposed in the single-shot benchmark; larger limits used in P3–P5 are listed in Appendix A.

A well-known limitation of Nelder–Mead is its susceptibility to convergence to local optima, especially in higher-dimensional problems [27]. To mitigate this, we adopt a multi-start strategy [29,30,31], launching multiple independent optimization runs from diverse initial points. Initial points are drawn from two sources: the top entries of the inverse design database (ensuring physically meaningful starting configurations) and random perturbations of the current best solution (enabling local exploration). The perturbation magnitudes are scaled to each variable’s physical range: $\pm {10}^{\circ }$ for heading $\theta$, $\pm 10 \mathrm{m} {\mathrm{s}}^{-1}$ for speed v, $\pm 1.0 \mathrm{s}$ for drop times $t_{d,k}$, and $\pm 0.5 \mathrm{s}$ for fuse delays $\delta t_k$. These scales were chosen to be large enough to escape shallow local basins while remaining small enough to keep perturbed solutions within the feasible region.

For the most critical problems (P4 and P5), we further employ a multi-seed stability protocol, running the entire multi-start procedure with different random seeds (42, 123, 7, 999) and retaining the overall best solution. This guards against seed-dependent artifacts and provides confidence in the robustness of the reported results. Algorithm 1 summarizes the single-shot inverse-seeded Nelder–Mead procedure.

Algorithm 1 Inverse-seeded multi-start Nelder–Mead optimization for single-shot deployment.
Require: UAV–missile pair $(u,m)$; inverse design seed database $\mathcal{D}$ sorted by descending $T_{\mathrm{obs}}$
Ensure: Optimized parameters ${\mathit{x}}^{*}=({\theta }^{*},v^{*},t_{\mathrm{drop}}^{*},\delta t^{*})$ and best obscuration $T^{*}$
1:  $T^{*}\leftarrow 0$
2:  for $i=1$ to $N_{\mathrm{starts}}$ do	▹$N_{\mathrm{starts}}=20$
3:       if $i\le \left|\mathcal{D}\right|$ then
4:              ${\mathit{x}}_0\leftarrow \Phi \left(\mathcal{D}\left[i\right]\right)$	▹ Convert 2D seed to 4D physical parameters
5:       else
6:            ${\mathit{x}}_0\leftarrow {\mathit{x}}^{*}+\mathit{ϵ}$	▹ Random perturbation of current best
7:       end if
8:       ${\mathit{x}}_i^{*},T_i\leftarrow \mathrm{N}\mathrm{ELDER}\mathrm{M}\mathrm{EAD}({\mathit{x}}_0,f_{\mathrm{obs}},\mathrm{MaxEval}=500)$	▹ Optimize in physical space
9:       if $T_i>T^{*}$ then
10:           $T^{*}\leftarrow T_i$;    ${\mathit{x}}^{*}\leftarrow {\mathit{x}}_i^{*}$
11:       end if
12:  end for
13:  return ${\mathit{x}}^{*}$, $T^{*}$

3.3. Single-UAV Multi-Shot Optimization

When a single UAV deploys K projectiles (typically $K=3$) in relay against one missile, the optimization variables are $(\theta ,v,t_{d,1},\delta t_1,\dots ,t_{d,K},\delta t_K)$—a $(2+2K)$-dimensional problem. The heading angle and speed are shared across all projectiles since the UAV maintains constant flight parameters throughout. The objective is to maximize the total duration of the union of obscuration windows:

$$T_{\mathrm{union}}=\Delta t{\displaystyle \sum _{n=0}^N}\mathbf{1}\left[{\displaystyle \underset{k=1}{\overset{K}{\bigvee }}}{\mathrm{Obs}}_k\left(t_n\right)\right],$$

(12)

where $t_n=n\Delta t$ with $\Delta t=0.01 \mathrm{s}$, $N=T_{max}/\Delta t$, and ${\mathrm{Obs}}_k\left(t\right)$ is the binary indicator that smoke cloud k blocks all four extreme sightlines at time t. The disjunction ⋁ ensures that at each discrete time step, the missile is considered obscured if any of the K smoke clouds provides full coverage. This union-based evaluation naturally handles overlapping or disjoint obscuration intervals without requiring explicit interval arithmetic.

The optimization proceeds in two stages. First, the inverse design grid search is performed over $(t_{\mathrm{target}},s)$ for the given UAV–missile pair, producing a database of up to 80 high-performing single-shot configurations. These configurations are grouped by approximate heading and speed, and within each group, the top three temporally separated entries are selected as candidate relay sequences. The temporal separation criterion requires that consecutive burst times differ by at least $2.0$ $\mathrm{s}$, ensuring that the relay produces distinct obscuration windows rather than redundant overlapping coverage. Second, the top 10 candidate relay sequences are used as initial points for multi-start Nelder–Mead optimization over the full $(2+2K)$-dimensional space.

Two physical constraints are enforced during the multi-shot optimization. First, a minimum temporal separation of $1.0$ $\mathrm{s}$ between consecutive drop times ($t_{d,k+1}-t_{d,k}\ge 1.0 \mathrm{s}$) prevents physically unrealistic simultaneous deployments, as the UAV requires a finite time interval to prepare and release each successive projectile. Second, each individual burst point must satisfy the same feasibility conditions as in the single-shot case: the fuse delay must produce a positive burst altitude, and the UAV speed must remain within $[70,140]$ m/s. In the revised implementation, these constraints are treated as hard filters: infeasible candidates return zero obscuration rather than receiving a large fixed penalty. This avoids sensitivity to an arbitrary penalty coefficient and was used for all results reported in Section 4.

The objective in Equation (12) maximizes total union duration rather than the length of the longest continuous interval. Fragmented windows are therefore not explicitly penalized. This choice matches the benchmark objective, in which each time step of LOS denial contributes equally to the total disruption time. If continuous coverage is operationally preferred, the objective can be replaced by $T_{\mathrm{union}}-\lambda N_{\mathrm{gap}}$ or by the longest contiguous obscuration interval; the same inverse-seeded optimization framework applies with only the objective-evaluation routine changed.

3.4. Multi-UAV Multi-Missile Allocation

The full-scale problem (P5) requires assigning five UAVs to three missiles and jointly optimizing all deployment parameters. This involves both a discrete assignment component and a continuous optimization component, making it a challenging mixed-integer problem. We decompose it into three hierarchical stages, drawing on concepts from the weapon–target assignment literature [9,10,11,12]. Figure 7 summarizes the resulting workflow.

Stage 1: Performance matrix construction. For each UAV–missile pair $(u,m)$, the three-shot relay optimization described in Section 3.3 is executed independently, producing a $5\times 3$ performance matrix $\mathbf{\Pi }$ whose entry ${\pi }_{um}$ represents the maximum obscuration duration achievable when UAV u is assigned exclusively to missile m. This stage involves 15 independent optimization sub-problems, each solved with the multi-seed protocol (seeds 42, 123, 7, 999) for stability. The 15 sub-problems are mutually independent and can in principle be parallelized across multiple cores; in our implementation, they are solved sequentially, requiring approximately 8 min of total computation.

Stage 2: Assignment enumeration and filtering. Each UAV must be assigned to exactly one missile, and each missile must receive at least one UAV. Formally, we seek a surjective mapping $\sigma :\{1,\dots ,5\}\to \{1,2,3\}$ that maximizes the total obscuration. The total number of mappings from 5 UAVs to 3 missiles is $3^5=243$. After removing assignments that leave at least one missile undefended (i.e., non-surjective mappings), approximately 150 feasible assignments remain. For each feasible assignment $\sigma$, a preliminary estimate of the total obscuration is computed by summing the corresponding entries of $\mathbf{\Pi }$:

$$\widehat{T}\left(\sigma \right)={\displaystyle \sum _{m=1}^3}\sum _{u:\sigma \left(u\right)=m}{\pi }_{um}.$$

(13)

This estimate treats each UAV’s contribution as independent and ignores cooperative effects (temporal overlap or complementarity) between UAVs assigned to the same missile. Despite this simplification, $\widehat{T}\left(\sigma \right)$ provides a useful ranking of candidate assignments. The top 30 assignments by estimated performance are retained for detailed evaluation in Stage 3, a cutoff that balances computational cost against the risk of discarding a high-performing assignment before joint refinement.

Stage 3: Joint optimization with cross-initialization. For each of the 30 candidate assignments, all UAVs assigned to each missile are jointly optimized. When $n_m$ UAVs are assigned to missile m, the joint optimization operates in an $8n_m$-dimensional space—two flight parameters $({\theta }_u,v_u)$ per UAV plus six shot-specific parameters ${(t_{d,k},\delta t_k)}_{k=1}^3$ per UAV. The initial parameter vectors are assembled from the independently optimized solutions stored during Stage 1, providing a warm start that is already near a feasible high-performing point for each individual UAV.

The joint optimization then proceeds through 25 rounds of multi-start Nelder–Mead with adaptive perturbation scales. In each round, the current best solution is perturbed by Gaussian noise scaled to 5% of each variable’s physical range, generating a new initial simplex for Nelder–Mead. If a round produces no improvement for three consecutive iterations, the perturbation scale is doubled to encourage exploration of more distant regions.

Additionally, cross-initialization is performed: for each UAV u assigned to missile m, the solution parameters that UAV u obtained when optimized against a different missile $m^{\prime }\ne m$ in Stage 1 are used as alternative starting points. This cross-pollination enables the optimizer to discover synergies between UAVs that were not apparent in the independent stage—for instance, if two UAVs assigned to the same missile can achieve better temporal complementarity by each slightly adjusting their headings from the independently optimal values. The assignment yielding the highest total obscuration across all three missiles is selected as the final solution.

Algorithm 2 gives the corresponding implementation-level pseudocode. The procedure is deliberately hierarchical: Stage 1 provides a tractable value approximation for each UAV–missile pair, Stage 2 searches the small assignment space exactly for the present five-UAV/three-missile instance, and Stage 3 removes part of the independence approximation by jointly optimizing all UAVs assigned to the same missile. The approach is not an exact solver for the fully coupled mixed-integer nonlinear problem; rather, it is a reproducible decomposition that provides strong warm starts and keeps the continuous search dimension manageable.

Algorithm 2 Hierarchical allocation and joint refinement for P5.

Require: UAV set $\mathcal{U}$, missile set $\mathcal{M}$, shots per UAV $K=3$

Ensure: Assignment ${\sigma }^{*}$ and optimized deployment parameters

1: for each $u\in \mathcal{U}$ and $m\in \mathcal{M}$ do   2:     Build inverse-design seed database ${\mathcal{D}}_{um}$   3:     Optimize the K-shot relay for $(u,m)$ using inverse-seeded Nelder–Mead   4:     Store performance ${\pi }_{um}$ and parameters ${\mathit{x}}_{um}$   5: end for   6: Enumerate assignments $\sigma :\mathcal{U}\to \mathcal{M}$ with each missile receiving at least one UAV   7: for each feasible assignment $\sigma$ do   8:     Compute estimate $\widehat{T}\left(\sigma \right)={\sum }_m{\sum }_{u:\sigma \left(u\right)=m}{\pi }_{um}$   9: end for 10: Retain the top $N_{\mathrm{assign}}=30$ assignments by $\widehat{T}$ 11: for each retained assignment $\sigma$ do 12:     for each missile m do 13:         Assemble warm start from $\{{\mathit{x}}_{um}:\sigma \left(u\right)=m\}$ 14:         Jointly refine all assigned UAVs for missile m by multi-start Nelder–Mead 15:         Include cross-initialization from each UAV’s solutions against other missiles 16:     end for 17:     Evaluate total obscuration across all missiles 18: end for 19: return best assignment ${\sigma }^{*}$ and associated parameters

4. Results and Analysis

This section presents the numerical results for the five sub-problems defined in Section 2.5, progressing from the single-projectile baseline (P1) to the full-scale multi-UAV multi-missile assignment (P5). For each sub-problem, we report the optimized deployment parameters, the achieved obscuration durations under the cylindrical target model, and the physical interpretation of the solution geometry. All optimization methods in the comparative experiments use the same kinematic model, cylindrical LOS evaluator, and nominal time step unless otherwise stated. The implementation uses Python 3.13.9, NumPy 2.3.5, and SciPy 1.16.3 (scipy.optimize.minimize with Nelder–Mead for local refinement), executed on a desktop computer with an Intel i7-13620H CPU and 16 GB RAM. Figures are generated using MATLAB R2023b and Matplotlib 3.10.7.

4.1. Single-Projectile Baseline (P1)

As a baseline, ${\mathrm{FY}}_1$ is directed to fly at $v=120 \mathrm{m} {\mathrm{s}}^{-1}$ in the negative x-direction ($\theta =\pi$) and releases one projectile at $t_{\mathrm{drop}}=1.5 \mathrm{s}$ with fuse delay $\delta t=3.6 \mathrm{s}$. These prescribed parameters yield a burst point at approximately (17,188, 0, 1737) m at $t_b=5.1 \mathrm{s}$. Under the cylindrical target model, the achieved obscuration duration is $1.362$ $\mathrm{s}$. The point-target model yields a slightly larger value, as expected, since the cylindrical model imposes the stricter requirement of blocking all four extreme sightlines simultaneously. Figure 8a shows the obscuration window on a time axis.

The modest obscuration duration in P1 can be understood from the underlying geometry. At the burst time $t_b=5.1 \mathrm{s}$, missile $M_1$ is located at approximately (18,478, 0, 1848) m, placing the smoke cloud roughly 1290 $\mathrm{m}$ ahead of the missile along the x-axis. Given $M_1$’s approach speed of 300 $\mathrm{m} {\mathrm{s}}^{-1}$, the missile traverses the angular extent subtended by the 10 $\mathrm{m}$ radius smoke cloud in a brief window. Furthermore, the burst altitude of 1737 $\mathrm{m}$—determined by the gravitational drop ${\displaystyle \frac{1}{2}}g\delta t^2\approx 63 \mathrm{m}$ below the UAV’s flight altitude of 1800 $\mathrm{m}$—places the smoke slightly below the missile’s trajectory, reducing the duration over which all four extreme sightlines are simultaneously blocked. This baseline establishes that unoptimized deployment wastes most of the smoke cloud’s 20 $\mathrm{s}$ effective lifetime, motivating the inverse design optimization in subsequent sub-problems.

4.2. Single-UAV Single-Shot Optimization (P2)

Optimizing the four parameters $(\theta ,v,t_{\mathrm{drop}},\delta t)$ for ${\mathrm{FY}}_1$ deploying one projectile against $M_1$, the inverse design framework first generates a database of 80 retained seed configurations from the filtered $(t_{\mathrm{target}},s)$ grid. The best candidates are then refined by multi-start Nelder–Mead optimization in the four-dimensional physical parameter space, yielding a best obscuration duration of $4.580$ $\mathrm{s}$—a $3.36\times$ improvement over the fixed baseline.

The magnitude of this improvement reveals the sensitivity of obscuration duration to deployment geometry. While P1 places the smoke cloud roughly 1290 $\mathrm{m}$ ahead of the missile with an altitude offset, the optimized P2 solution positions the burst point directly along the missile–target sightline at an interception time that maximizes the angular extent of the smoke cloud as seen from the missile. The key geometric insight is that obscuration duration is not simply proportional to proximity between the smoke cloud and the missile trajectory; rather, it depends on the angular subtension of the smoke sphere relative to the cylindrical target as viewed from the missile’s instantaneous position. By choosing the interception time and offset parameter to maximize this angular relationship, the optimizer finds configurations where the smoke cloud appears to “track” the missile for a longer portion of its flight.

Figure 9 visualizes the inverse design search landscape for the ${\mathrm{FY}}_1\to M_1$ pair. The heatmap displays the single-shot obscuration duration as a function of the two inverse parameters $(t_{\mathrm{target}},s)$. The feasible region—where the required UAV speed falls within $[70 \mathrm{m} {\mathrm{s}}^{-1},140 \mathrm{m} {\mathrm{s}}^{-1}]$—occupies a narrow band in the lower-left portion of the parameter space, illustrating how the physical constraints naturally concentrate the search. The grid-search best point (marked with a star) lies within this concentrated region, demonstrating that the 2D inverse parameterization effectively captures the essential structure of the underlying 4D optimization problem. Notably, the feasibility boundary is determined by the UAV speed constraint: early interception times (small $t_{\mathrm{target}}$) require the UAV to cover less distance, allowing slower speeds, whereas later interceptions demand higher speeds that may exceed the 140 $\mathrm{m} {\mathrm{s}}^{-1}$ upper bound. This speed–feasibility trade-off geometrically carves out the viable search region, and the inverse parameterization makes this structure explicit—a feature that would be obscured in the original four-dimensional parameter space.

4.3. Single-UAV Three-Shot Relay (P3)

For P3, ${\mathrm{FY}}_1$ deploys three projectiles in relay against $M_1$. The eight-dimensional optimization $(\theta ,v,t_{d,1},\delta t_1,t_{d,2},\delta t_2,t_{d,3},\delta t_3)$ is solved using the staged approach described in Section 3.3. The optimized relay achieves a total union obscuration of $7.324$ $\mathrm{s}$, with shared heading $\theta =179.{64}^{\circ }$ and speed $v=139.67 \mathrm{m} {\mathrm{s}}^{-1}$.

Table 3. Optimized parameters for the P3 three-shot relay (${\mathrm{FY}}_1\to M_1$).

Shot	${\mathit{t}}_{\mathit{drop}}$ (s)	$\mathit{\delta }\mathit{t}$ (s)	${\mathit{t}}_{\mathit{b}}$ (s)	Burst Point (m)	Single Obs. (s)
1	0.172	3.813	3.985	(17,243, 4, 1729)	3.522
2	3.093	5.072	8.165	(16,660, 7, 1674)	3.027
3	5.175	5.930	11.105	(16,249, 10, 1628)	1.463
Total union obscuration					7.324

lists the per-projectile parameters, and Figure 8b displays the three individual obscuration windows and their union on a timeline.

Several features of the optimized relay merit discussion. First, the UAV flies at nearly the maximum speed ($139.67 \mathrm{m} {\mathrm{s}}^{-1}$, just below the 140 $\mathrm{m} {\mathrm{s}}^{-1}$ upper bound), which is physically motivated: a faster UAV traverses a longer segment of the missile’s approach corridor during the engagement, allowing it to place successive burst points at greater spatial separations along the trajectory. The three burst points at (17,243, 4, 1729), (16,660, 7, 1674), and (16,249, 10, 1628) m exhibit a decreasing altitude profile—from 1729 $\mathrm{m}$ to 1628 $\mathrm{m}$—reflecting the longer fuse delays required for later shots ($\delta t=3.81,5.07,5.93$ s), which produce greater gravitational drops.

Second, the individual obscuration durations decrease from $3.522$ $\mathrm{s}$ (shot 1) to $3.027$ $\mathrm{s}$ (shot 2) to $1.463$ $\mathrm{s}$ (shot 3). This decline occurs because later shots intercept the missile at progressively smaller distances from the target, where the missile’s angular velocity relative to the smoke cloud increases and the cylindrical target subtends a larger solid angle from the missile’s perspective—both factors that reduce the time window during which all four extreme sightlines pass through the smoke sphere. The sum of individual durations ($3.522+3.027+1.463=8.012 \mathrm{s}$) exceeds the union ($7.324$ $\mathrm{s}$) by $0.688$ $\mathrm{s}$, indicating partial temporal overlap between adjacent obscuration windows. This modest overlap represents a deliberate optimization trade-off: the optimizer accepts some redundancy between shots 1 and 2 to avoid leaving undefended gaps, which would be operationally more costly than wasted overlap.

4.4. Multi-UAV Cooperative Single-Shot (P4)

In P4, three UAVs (${\mathrm{FY}}_1$, ${\mathrm{FY}}_2$, ${\mathrm{FY}}_3$) each deploy one projectile against $M_1$. The 12-dimensional optimization yields a total union obscuration of $11.140$ $\mathrm{s}$. The three UAVs approach from different directions and release their projectiles at staggered times, creating temporally distributed obscuration windows that collectively span a much longer interval than any single UAV could achieve alone.

Table 4. Optimized parameters for the P4 three-UAV cooperative deployment against $M_1$.

UAV	$\mathit{\theta }$ (deg)	v (m/s)	${\mathit{t}}_{\mathit{drop}}$ (s)	$\mathit{\delta }\mathit{t}$ (s)	Burst Point (m)	Single Obs. (s)
${\mathrm{FY}}_1$	5.75	72.88	0.693	0.737	(17,904, 10, 1797)	4.581
${\mathrm{FY}}_2$	265.82	127.59	4.178	6.416	(11,902, 52, 1198)	3.894
${\mathrm{FY}}_3$	75.31	93.87	32.481	1.036	$(6798,43,695)$	2.664
Total union obscuration						11.140

details each UAV’s optimized parameters.

The spatial diversity afforded by multiple UAVs is the key enabler of P4’s superior performance. ${\mathrm{FY}}_1$, positioned at (17,800, 0, 1800) m—nearly on $M_1$’s trajectory—intercepts the missile early ($t_b=1.43 \mathrm{s}$) at high altitude, achieving the longest individual obscuration of $4.581$ $\mathrm{s}$. ${\mathrm{FY}}_2$, starting from (12,000, 1400, 1400) m, provides mid-trajectory coverage at $t_b\approx 10.6 \mathrm{s}$, while ${\mathrm{FY}}_3$ at $(6000,-3000,700)$ m handles the late-phase interception at $t_b\approx 33.5 \mathrm{s}$ when the missile is much closer to the target. The wide temporal separation between burst times—approximately 1 $\mathrm{s}$, 11 $\mathrm{s}$, and 34 $\mathrm{s}$—results in negligible overlap: the sum of individual durations ($4.581+3.894+2.664=11.139 \mathrm{s}$) virtually equals the union ($11.140$ $\mathrm{s}$), confirming that the three obscuration windows are effectively disjoint.

Comparing P4 with P3 reveals the advantage of spatial diversity over temporal relay from a single platform. P3 uses three projectiles from one UAV to achieve $7.324$ $\mathrm{s}$, whereas P4 uses three projectiles from three UAVs to achieve $11.140$ $\mathrm{s}$—a 52% improvement. This gain arises because a single UAV can only place smoke clouds along a limited segment of the missile’s trajectory (constrained by its speed and heading), whereas spatially distributed UAVs can collectively cover the full approach corridor from long range to close range. However, this advantage comes at the cost of requiring three UAVs rather than one, and the diminishing individual contributions (${\mathrm{FY}}_3$ achieves only $2.664$ $\mathrm{s}$) suggest that UAVs farther from the missile’s trajectory and at lower altitudes face increasingly severe geometric constraints.

4.5. Full-Scale Multi-UAV Multi-Missile Assignment (P5)

The full-scale P5 problem assigns five UAVs to three missiles using the hierarchical framework of Section 3.4. The optimal assignment is ${\mathrm{FY}}_1,{\mathrm{FY}}_5\to M_1$, ${\mathrm{FY}}_2,{\mathrm{FY}}_4\to M_2$, and ${\mathrm{FY}}_3\to M_3$, corresponding to the allocation vector $(0,1,2,1,0)$. The total obscuration across all three missiles is $20.652$ $\mathrm{s}$, decomposed as $M_1$: $9.891$ $\mathrm{s}$; $M_2$: $7.710$ $\mathrm{s}$; and $M_3$: $3.051$ $\mathrm{s}$. Figure 10 summarizes the per-missile obscuration timelines under the optimal assignment.

The $5\times 3$ performance matrix (

Table 5. Performance matrix $\mathbf{\Pi }$ (seconds): maximum three-shot obscuration duration for each UAV–missile pair, computed independently.

	${\mathit{M}}_1$	${\mathit{M}}_2$	${\mathit{M}}_3$
${\mathrm{FY}}_1$	5.98	0.00	0.00
${\mathrm{FY}}_2$	3.96	3.96	3.72
${\mathrm{FY}}_3$	3.14	3.02	3.06
${\mathrm{FY}}_4$	3.82	3.78	3.54
${\mathrm{FY}}_5$	3.94	3.60	3.76

) reveals that UAV–missile affinity is highly non-uniform: UAVs closer to a missile’s trajectory and at higher altitudes (allowing longer fuse delays and thus more precise burst placement) achieve substantially better performance. The assignment algorithm effectively exploits these asymmetries, pairing each missile with its most effective UAVs while ensuring no missile is left undefended.

The zero entries in the performance matrix (${\mathrm{FY}}_1\to M_2$ and ${\mathrm{FY}}_1\to M_3$) arise from physical infeasibility: ${\mathrm{FY}}_1$ starts at (17,800, 0, 1800) m, nearly collinear with $M_1$’s trajectory toward the origin, and reaching the approach corridors of $M_2$ or $M_3$—which are laterally offset by 600 $\mathrm{m}$ in y—within the engagement timeframe would require speeds exceeding the 140 $\mathrm{m} {\mathrm{s}}^{-1}$ upper bound. This kinematic constraint effectively makes ${\mathrm{FY}}_1$ a dedicated $M_1$ asset, a structural feature that the assignment algorithm correctly exploits. In contrast, ${\mathrm{FY}}_2$, ${\mathrm{FY}}_3$, ${\mathrm{FY}}_4$, and ${\mathrm{FY}}_5$ have non-zero affinities across all three missiles, reflecting their intermediate positions that provide geometric access to multiple approach corridors.

To make the zero entries reproducible,

Table 6. Minimum speed required for ${\mathrm{FY}}_1$ to generate a feasible inverse design seed under $s_{max}=0.50$.

Pair	Minimum Required Speed (m/s)	Operational Limit (m/s)	Outcome
${\mathrm{FY}}_1\to M_1$	0.17	70–140	Feasible after speed lower-bound filtering
${\mathrm{FY}}_1\to M_2$	177.12	70–140	Infeasible
${\mathrm{FY}}_1\to M_3$	282.14	70–140	Infeasible

reports the minimum speed required by the inverse back-calculation for ${\mathrm{FY}}_1$ to generate any feasible line-of-sight-aligned seed for each missile when $s_{max}=0.50$ and the same non-negative drop-time and positive-altitude constraints are imposed. The ${\mathrm{FY}}_1\to M_2$ and ${\mathrm{FY}}_1\to M_3$ pairs require at least $177.12$ $\mathrm{m} {\mathrm{s}}^{-1}$ and $282.14$ $\mathrm{m} {\mathrm{s}}^{-1}$, respectively, both exceeding the operational upper bound of 140 $\mathrm{m} {\mathrm{s}}^{-1}$; they are therefore assigned zero performance in Table 5.

Table 7. Complete P5 deployment parameters for all 15 projectiles under the optimal assignment $(0,1,2,1,0)$.

Target	UAV	Shot	$\mathit{\theta }$ (deg)	v (m/s)	${\mathit{t}}_{\mathit{drop}}$ (s)	$\mathit{\delta }\mathit{t}$ (s)	Burst Point (m)
$M_1$	${\mathrm{FY}}_1$	1	4.53	99.92	0.100	0.129	$(17,823,2,1800)$
	${\mathrm{FY}}_1$	2	4.53	99.92	1.101	0.226	$(17,932,10,1800)$
	${\mathrm{FY}}_1$	3	4.53	99.92	8.659	1.503	$(18,812,80,1789)$
	${\mathrm{FY}}_5$	1	92.20	139.63	7.969	1.303	$(12,950,-706,1292)$
	${\mathrm{FY}}_5$	2	92.20	139.63	14.431	0.101	$(12,922,28,1300)$
	${\mathrm{FY}}_5$	3	92.20	139.63	18.579	1.537	$(12,892,807,1288)$
$M_2$	${\mathrm{FY}}_2$	1	290.74	135.76	2.274	2.451	$(12,227,800,1371)$
	${\mathrm{FY}}_2$	2	290.74	135.76	5.326	2.263	$(12,365,436,1375)$
	${\mathrm{FY}}_2$	3	290.74	135.76	9.978	1.822	$(12,567,-98,1384)$
	${\mathrm{FY}}_4$	1	282.51	130.49	0.101	10.403	$(11,297,662,1270)$
	${\mathrm{FY}}_4$	2	282.51	130.49	2.057	10.476	$(11,354,403,1262)$
	${\mathrm{FY}}_4$	3	282.51	130.49	6.060	10.182	$(11,459,-69,1292)$
$M_3$	${\mathrm{FY}}_3$	1	79.20	139.91	18.175	1.249	$(6509,-331,692)$
	${\mathrm{FY}}_3$	2	79.20	139.91	20.177	0.677	$(6547,-134,698)$
	${\mathrm{FY}}_3$	3	79.20	139.91	24.986	1.082	$(6684,583,694)$
$M_1$ subtotal: 9.891 s  $M_2$ subtotal: 7.710 s  $M_3$ subtotal: 3.051 s
Grand total: 20.652 s

lists the complete deployment parameters for all 15 projectiles (five UAVs × three shots each) under the optimal assignment.

Figure 11 shows the top-down view of the complete P5 solution, depicting all 15 burst points projected onto the $xy$-plane alongside missile trajectories and UAV initial positions. The burst points for each missile form distinct spatial clusters along their respective trajectories: the $M_1$ bursts (red) are distributed within $x\approx 13$–19 km, the $M_2$ bursts (blue) cluster near $x\approx 11$–13 km, and the $M_3$ bursts (green) are located around $x\approx 6.5$ km. This spatial separation confirms that the hierarchical allocation successfully distributes defensive resources across the threat space, with each UAV deploying its three projectiles to create sequential smoke screens along the assigned missile’s approach corridor.

The unequal distribution of obscuration across missiles ($M_1$: $9.891$ $\mathrm{s}$, $M_2$: $7.710$ $\mathrm{s}$, $M_3$: $3.051$ $\mathrm{s}$) reflects both the allocation structure and the geometric constraints of the scenario. $M_1$ receives two UAVs including the highly effective ${\mathrm{FY}}_1$ (performance matrix entry: $5.98$ $\mathrm{s}$), which is positioned almost directly on $M_1$’s trajectory and benefits from a high initial altitude of 1800 $\mathrm{m}$. $M_2$ also receives two UAVs (${\mathrm{FY}}_2$ and ${\mathrm{FY}}_4$), but both are laterally offset from $M_2$’s trajectory, resulting in lower individual affinities. $M_3$ receives only ${\mathrm{FY}}_3$, which starts at the lowest altitude (700 $\mathrm{m}$) and farthest lateral offset (3000 $\mathrm{m}$ in y), severely limiting its achievable obscuration. This outcome highlights a fundamental trade-off in multi-missile defense: the allocation must balance coverage breadth (ensuring every missile is obscured) against coverage depth (maximizing obscuration per missile). The current scenario, with five UAVs defending against three missiles, provides sufficient redundancy for the nearer missiles but leaves $M_3$ relatively exposed—a finding that would inform force structure decisions in operational planning.

Table 8. P5 obscuration totals under three comparison scenarios (cylindrical evaluator, $\Delta t=0.01 \mathrm{s}$).

Scenario	Description	Total (s)
One-UAV-per-missile baseline	Best Stage 1 entry per missile column of $\mathbf{\Pi }$; three UAVs, nine projectiles	13.00
Same assignment, Stage 1 assembly	Assignment $(0,1,2,1,0)$; union of Stage 1 relay solutions, no Stage 3 joint refinement	20.67
Stage 3 joint optimum	Assignment $(0,1,2,1,0)$ after per-missile joint multi-start refinement	20.652

decomposes the P5 headline result into three progressively richer deployment assumptions, all evaluated with the same cylindrical LOS model and $\Delta t=0.01 \mathrm{s}$. The one-UAV-per-missile baseline assigns exactly one UAV to each missile using the best Stage 1 three-shot relay entry in each column of $\mathbf{\Pi }$ (${\mathrm{FY}}_1\to M_1$, ${\mathrm{FY}}_2\to M_2$, ${\mathrm{FY}}_3\to M_3$), for a total of $13.00$ $\mathrm{s}$ from three UAVs and nine projectiles. This baseline is useful for isolating per-missile effectiveness, but it is not directly comparable to the final P5 solution, which deploys five UAVs and fifteen projectiles under the optimal assignment $(0,1,2,1,0)$.

The same-assignment Stage 1 assembly uses all five UAVs in that assignment but retains only the independently optimized Stage 1 parameters for each assigned pair, evaluated as a per-missile union without Stage 3 joint refinement ($M_1$: $9.90$ $\mathrm{s}$; $M_2$: $7.72$ $\mathrm{s}$; $M_3$: $3.05$ $\mathrm{s}$; total $20.67$ $\mathrm{s}$). Simply summing the five corresponding entries of $\mathbf{\Pi }$ would overstate performance at $20.72$ $\mathrm{s}$ because it ignores temporal overlap among UAVs assigned to the same missile. Relative to the three-UAV baseline, the five-UAV same-assignment total increases by about 59%, showing that most of the headline gain comes from additional platforms and the selected assignment rather than from Stage 3 refinement alone. Stage 3 joint optimization further adjusts the deployment to $20.652$ $\mathrm{s}$, with the largest coordinated retiming on $M_1$ and $M_2$, where two UAVs per missile must partition complementary relay windows.

4.6. Comparative Performance Summary

The five sub-problems demonstrate a clear progression in both obscuration performance and problem complexity. From the fixed baseline of P1 ($1.362$ $\mathrm{s}$), single-shot optimization (P2) achieves a $3.36\times$ improvement to $4.580$ $\mathrm{s}$; the three-shot relay (P3) further extends obscuration to $7.324$ $\mathrm{s}$ by creating sequential smoke barriers; multi-UAV cooperation (P4) reaches $11.140$ $\mathrm{s}$ through coordinated deployment of three UAVs; and the full-scale assignment (P5) achieves $20.652$ $\mathrm{s}$ across all three missiles. Each sub-problem validates a distinct component of the framework: P2 validates inverse design, P3 validates relay sequencing, P4 validates multi-UAV coordination, and P5 validates the hierarchical allocation strategy.

Computational efficiency. The inverse design framework provides substantial computational savings compared to direct search. For the single-shot problem (P2), the illustrative inverse grid contains 12,800 inexpensive back-calculation points in $(t_{\mathrm{target}},s)$ space. After feasibility filtering, only 103 feasible points are passed to the cylindrical LOS evaluator and the top 80 are retained as seeds for physical-space refinement. Thus, the saving comes from replacing a blind 4D sample with an analytic feasibility filter and a compact high-quality seed set, rather than from claiming that the raw inverse grid itself is smaller by two orders of magnitude. This efficiency gain compounds for multi-shot problems: the P3 eight-dimensional problem would require ∼${10}^{10}$ forward evaluations at the same resolution, while the inverse-design-seeded Nelder–Mead approach converges within ∼${10}^4$ function evaluations.

Table 9. Summary of results and computational characteristics for all sub-problems.

Problem	Obscuration (s)	Decision Dim.	Inverse Dim.	Approx. Runtime
P1	1.362	0 (fixed)	—	<1 s
P2	4.580	4	2 seed + 4D NM	∼5 s
P3	7.324	8	2 + NM	∼30 s
P4	11.140	12	$3\times 2$ + NM	∼2 min
P5	20.652	40 + assignment	$15\times 2$ + NM	∼15 min

summarizes the results and computational effort for all five sub-problems.

Figure 12 illustrates the multi-start Nelder–Mead optimization behavior for the ${\mathrm{FY}}_1\to M_1$ single-shot problem (P2). Panel (a) displays the convergence trajectories of 20 independent runs initialized from different grid-search seeds: all runs converge within approximately 50–100 function evaluations, with the best run reaching $4.580$ $\mathrm{s}$. Panel (b) summarizes the distribution of final objective values, demonstrating high consistency across starting points ($\mu =4.497$ s, $\sigma =0.066$ s). Throughout the paper, $4.580$ $\mathrm{s}$ denotes the best refined P2 solution, whereas $\mu$ and $\sigma$ describe the distribution over independent starts. This distinction explains the apparent difference between the best value, the convergence plot mean, and the method comparison table.

To substantiate the claimed computational advantage, we directly compare three optimization strategies on the P2 single-shot problem (${\mathrm{FY}}_1\to M_1$, cylinder model): (i) uniform random search in the original 4D parameter space with $N={10}^5$ samples; (ii) differential evolution followed by Nelder–Mead refinement in the same 4D space; and (iii) the proposed 2D inverse design grid seeding multi-start Nelder–Mead. Each method is executed five times with independent random seeds under the same cylindrical evaluator and hardware settings; results are summarized in

Table 10. Comparison of optimization strategies on the P2 problem (${\mathrm{FY}}_1\to M_1$, single shot, cylinder model). Each method is run five times with different seeds; best, mean, standard deviation, average function evaluations, wall time, and a two-sided Mann–Whitney comparison against the proposed method are reported.

Method	Search Dim.	Best Obs. (s)	Mean Obs. (s)	Func. Evals	Wall Time (s)	p vs. Proposed
4D Random Search ($N={10}^5$)	4	4.570	4.252 ± 0.316	100,000	24.85	0.0075
4D DE + Nelder–Mead	4	4.590	4.568 ± 0.027	11,364	40.51	0.6513
2D Inverse Seed + 4D Nelder–Mead	2 seed + 4D NM	4.580	4.580 ± 0.000	9846	2.75	—

. Because the sample size is modest, we report descriptive statistics and a non-parametric Mann–Whitney comparison against the proposed method as an indication of practical separation rather than as a definitive asymptotic test.

Table 10 reveals three key findings. First, all three methods converge to comparable best solution quality: the best obscuration values agree to within 0.02 s (4.57–4.59 s), confirming that inverse seeding does not degrade the best solution found in this low-dimensional case. Second, the inverse-seeded method requires approximately one order of magnitude fewer function evaluations than 4D random search and is much faster in wall-clock time because the analytic back-calculation filters infeasible candidates before invoking the expensive cylinder obscuration model. Third, random search shows substantial variance ($\sigma =0.316$ s), whereas inverse seeding is effectively deterministic after refinement for this P2 instance. The Mann–Whitney test indicates a clear difference between random search and the proposed method for this five-seed experiment ($p=0.0075$), while DE+NM and the proposed method are statistically indistinguishable in achieved objective value ($p=0.6513$) but differ strongly in wall-clock cost.

To address whether the efficiency advantage persists in higher-dimensional settings, we also conducted a budget-matched comparison on P3 (8 physical variables) and P4 (12 physical variables). Each method used the same cylindrical evaluator and three random seeds. The direct random search used 5000 samples; the DE+NM baseline used a low-budget differential-evolution stage followed by Nelder–Mead; and the proposed method used inverse design seeds followed by physical-space Nelder–Mead.

Table 11. Budget-matched high-dimensional comparison for P3 and P4. Values are based on three independent seeds and the same cylindrical target evaluator; function evaluations and wall times are averages.

Problem	Method	Best Obs. (s)	Mean Obs. (s)	Func. Evals	Wall Time (s)
P3 (8D)	Direct random search	4.260	2.430 ± 1.546	5000	2.08
P3 (8D)	8D DE + Nelder–Mead	0.000	0.000 ± 0.000	928	0.17
P3 (8D)	Inverse seed + 8D Nelder–Mead	7.380	7.340 ± 0.037	2058	0.71
P4 (12D)	Direct random search	4.050	3.430 ± 0.588	5000	2.54
P4 (12D)	12D DE + Nelder–Mead	0.000	0.000 ± 0.000	1940	0.93
P4 (12D)	Inverse seed + 12D Nelder–Mead	11.160	11.130 ± 0.042	2095	1.11

shows that the high-dimensional direct methods struggle to locate the narrow feasible high-performance regions under the same evaluation budget, whereas inverse seeding consistently starts near productive regions.

5. Discussion

This section interprets the numerical results from a methodological perspective and examines the broader implications of the proposed framework. Section 5.1 analyzes the advantages of the inverse design parameterization beyond raw computational savings. Section 5.3 investigates the sensitivity of the solutions to key parameters and assesses optimization robustness. Section 5.4 draws connections to the classical weapon–target assignment literature, highlighting both parallels and important distinctions. Finally, Section 5.5 identifies the principal simplifying assumptions and outlines directions for future research.

5.1. Advantages of Inverse Design Parameterization

The inverse design approach offers several advantages beyond raw computational efficiency. By parameterizing the search in terms of the physically meaningful quantities $t_{\mathrm{target}}$ and s, the method provides immediate geometric insight into each candidate solution: $t_{\mathrm{target}}$ determines when the missile encounters the smoke cloud, and s controls the fractional position of the desired burst point along the missile–target line. This interpretability is valuable for operational decision-making, as commanders can intuitively understand and adjust the deployment strategy. In contrast, the original four UAV parameters $(\theta ,v,t_{\mathrm{drop}},\delta t)$ lack such direct physical transparency—small changes in $\theta$ or v can produce large, non-intuitive shifts in the burst location.

Beyond interpretability, the inverse parameterization provides a natural constraint-handling mechanism. Infeasible designs—those requiring UAV speeds outside the 70–140 m/s range, negative drop times, or underground burst points—are automatically identified and discarded without introducing penalty functions or barrier terms. This implicit filtering is particularly valuable in the multi-start optimization context, where it prevents the optimizer from wasting evaluations in infeasible regions of the parameter space.

A more subtle advantage of the inverse parameterization is its effect on the optimization landscape itself. In the original 4D space, the mapping from $(\theta ,v,t_{\mathrm{drop}},\delta t)$ to obscuration duration is highly nonlinear and exhibits numerous local optima, because small perturbations in heading or speed can shift the burst point by hundreds of meters. The 2D inverse space $(t_{\mathrm{target}},s)$ partially decouples these interactions: $t_{\mathrm{target}}$ governs the temporal and along-trajectory placement of the smoke cloud while s controls the fractional position of the desired burst point along the missile–target line, producing a smoother objective landscape with fewer spurious local optima. This smoothing effect is empirically confirmed by the convergence analysis in Figure 12, where 20 independent Nelder–Mead runs starting from different grid-search seeds all converge to within 0.066 s of the best refined solution found—a narrow spread that would be unlikely in the rugged 4D landscape. The practical implication is that fewer multi-start runs are needed to identify a high-performing basin under the tested evaluation budget, further compounding the computational savings beyond the dimensionality reduction alone. Table 10 provides direct quantitative evidence for this efficiency gain by benchmarking the 2D inverse approach against 4D random search and 4D differential evolution on the P2 problem.

Figure 13 provides direct visual confirmation of this landscape smoothing effect. Each panel is zoomed into the bounding box of the feasible region (with a 15% margin) so that the internal structure of the landscape is clearly visible. Panels (a) and (b) display two-dimensional slices through the 4D forward objective function, with the remaining two parameters fixed at their best refined values. The $(\theta ,t_{\mathrm{drop}})$ slice in panel (a) spans the full heading range but only a narrow drop-time window ($t_{\mathrm{drop}}\in [0,1.7]$ s); even within this zoomed view, only 4.6% of grid cells produce non-zero obscuration, and the feasible band is fragmented into disconnected patches scattered across heading angles—a structure that would trap gradient-free optimizers in isolated local optima. The $(v,\delta t)$ slice in panel (b), zoomed to $\delta t\in [0.1,1.8]$ s, reveals a tightly constrained diagonal band where the best-performing region occupies a thin strip in the speed–delay plane; although the zoomed feasible fraction is 77%, the high-fitness core remains narrow and sensitive to both parameters simultaneously. In contrast, panel (c) presents the 2D inverse landscape over $(t_{\mathrm{target}},s)$, zoomed to the productive region ($t_{\mathrm{target}}\in [0.5,15]$ s, $s\in [0.001,0.10]$). This landscape exhibits a single connected high-fitness region with a smooth, approximately concentric gradient toward the best sampled point—precisely the benign structure that makes local search effective. To quantify this landscape contrast, panel (d) reports the Fitness-Distance Correlation (FDC) metric [35]—the Pearson correlation between solution fitness and normalized distance to the best sampled point. The 4D forward space yields $r_{\mathrm{FDC}}=-0.072$, indicating essentially no correlation between proximity to the best sampled point and solution quality—a hallmark of deceptive or rugged landscapes. The 2D inverse space achieves $r_{\mathrm{FDC}}=-0.889$, a strong negative correlation confirming that solutions closer to the best sampled point are systematically better, consistent with the forward–inverse correspondence established in Proposition 1. According to the Jones–Forrest classification, $\left|r_{\mathrm{FDC}}\right|>0.15$ indicates a “straightforward” problem for local search, and values above 0.5 suggest that simple hill-climbing suffices—precisely the behavior observed in the multi-start convergence of Figure 12.

The inverse design principle also naturally extends to operational scenarios with modified constraints. If the UAV speed range changes (e.g., due to payload variations or weather conditions), or if different projectile types with different fuse delay characteristics are introduced, only the feasibility filter in the back-calculation step requires modification—the search space structure and optimizer configuration remain unchanged. This modularity contrasts with forward approaches, where changes to constraint boundaries can fundamentally alter the topology of the feasible region and necessitate re-tuning of the optimization algorithm.

5.2. Symmetry Perspective on the Inverse Design Landscape

The improvement in the Fitness-Distance Correlation metric from $r_{\mathrm{FDC}}=-0.072$ in the 4D forward space to $r_{\mathrm{FDC}}=-0.889$ in the 2D inverse space, reported in Figure 13d, admits a structure-based interpretation. A strongly negative FDC indicates an approximately radial organization of the sampled objective landscape about the best sampled point: fitness decays monotonically with distance from $(t_{\mathrm{target}}^{*},s^{*})$ in a manner consistent with a single convex-like basin of attraction. This radial structure is precisely what Figure 13c reveals as a “concentric gradient”. By contrast, the forward 4D landscape in panels (a) and (b) is fragmented into disconnected feasible patches and narrow diagonal bands that destroy any comparable radial organization, which in turn explains why gradient-free optimizers frequently become trapped in isolated basins of attraction when operating directly on $(\theta ,v,t_{\mathrm{drop}},\delta t)$.

Two structural properties cooperate to produce the benign inverse landscape. The closed-form forward–inverse seed correspondence established in Proposition 1 preserves objective values for corresponding line-of-sight-aligned seed realizations under the constructed map $\Phi$, but does not, by itself, smooth the landscape—a one-to-one map can in principle preserve every local minimum of a rugged objective. The additional smoothing arises from an approximate coordinate-decoupling structure of the inverse coordinates: $t_{\mathrm{target}}$ controls the temporal and along-trajectory placement of the smoke cloud while s controls the fractional position of the desired burst point along the missile–target line, and these two degrees of freedom act nearly independently on the obscuration objective within the productive seed region. The forward parameters, by contrast, interact multiplicatively through the ballistic Equation (3), so that small perturbations in $(\theta ,v)$ couple with $(t_{\mathrm{drop}},\delta t)$ in ways that frustrate any approximate radial structure. The combination of seed correspondence and coordinate-decoupling structure is what produces the observed order-of-magnitude efficiency advantage.

This observation suggests a general methodological principle for inverse design problems: when the physical system possesses geometric symmetries—as our missile–target–smoke geometry does through its azimuthal and vertical reflections—selecting inverse coordinates that commute (approximately) with those symmetries tends to yield objective landscapes amenable to local search. The approach taken here is therefore not a one-off trick but an instance of a broader strategy that may be valuable in other deployment optimization settings where forward kinematics are analytically invertible and the underlying physics admits a natural symmetry group.

5.3. Sensitivity Analysis and Optimization Robustness

The choice of $s_{max}$ meaningfully affects seed coverage. The headline P5 result uses $s_{max}=0.50$ during inverse design seed generation. Smaller values restrict the seed database to burst locations near the missile and can exclude late-phase opportunities, especially for $M_3$, whose optimized burst locations project to $s\approx 0.35$–0.47 along the missile–target line.

Table 12. Snapshot sensitivity of the P5 solution to the inverse design offset threshold $s_{max}$. The table retains only final-solution shots whose projection onto the corresponding missile–target line lies within $0\le s\le s_{max}$ and recomputes union obscuration.

${\mathit{s}}_{max}$	Retained Shots $({\mathit{M}}_1,{\mathit{M}}_2,{\mathit{M}}_3)$	${\mathit{M}}_1$ (s)	${\mathit{M}}_2$ (s)	${\mathit{M}}_3$ (s)	Total (s)
0.20	$(4,2,0)$	9.89	0.00	0.00	9.89
0.25	$(5,2,0)$	9.89	0.00	0.00	9.89
0.35	$(5,6,0)$	9.89	7.70	0.00	17.59
0.50	$(5,6,3)$	9.89	7.70	3.05	20.64

reports a snapshot analysis of the final P5 solution: for each threshold, only shots whose line-of-sight projection satisfies $0\le s\le s_{max}$ are retained and the resulting union duration is recomputed. This is not a full re-optimization for each $s_{max}$, but it shows why $s_{max}=0.50$ is needed for the reported multi-missile solution. However, $s_{max}$ should not approach 1.0, as smoke clouds placed very close to the target provide negligible obscuration due to the small angular subtension of the smoke sphere relative to the target’s visible cross-section.

The multi-start strategy with multiple random seeds proved essential for obtaining reliable results in the higher-dimensional problems. For P4, the four seeds (42, 123, 7, 999) produced solutions differing by up to 0.3 s, with the best seed achieving $11.140$ $\mathrm{s}$. Without the multi-seed protocol, a single unlucky seed initialization could miss this high-performing solution. For P5, Stage 3 joint refinement and cross-initialization provide only a modest adjustment relative to the Stage 1-assembled warm start (from $20.67$ $\mathrm{s}$ to $20.652$ $\mathrm{s}$ under the same evaluator settings), whereas the large improvement over the three-UAV baseline arises primarily from deploying five UAVs under the selected assignment. These findings underscore the importance of robust optimization protocols when applying derivative-free methods to non-convex deployment problems [27,30].

The sensitivity of the solution to the grid resolution in the inverse design search also merits discussion. The initial grid search retains approximately 80 high-performing seed configurations per UAV–missile pair after feasibility filtering, not 80 raw grid points. Doubling the retained seed budget to $\sim 160$ configurations increases Stage 1 computation time proportionally but yields only marginal improvements (<1%) in the final P5 objective, because the subsequent Nelder–Mead refinement compensates for coarse grid spacing. This observation confirms that the grid search serves primarily as a seeding mechanism for the local optimizer rather than as a standalone solution method. Conversely, retaining fewer than 40 configurations risks missing important regions of the feasible space, particularly for UAV–missile pairs with narrow feasibility bands (e.g., ${\mathrm{FY}}_3\to M_1$, where the UAV’s low altitude and large lateral offset severely constrain the set of reachable burst points). A retained database of $\sim 80$ configurations thus represents a practical compromise between computational cost and solution reliability.

The cylindrical target model introduces an additional source of sensitivity not present in the point-target formulation. Because the cylindrical model requires simultaneous obscuration of all four extreme sightlines, the effective obscuration duration is governed by the “weakest” sightline—the one that exits the smoke sphere first. In practice, this is typically one of the vertical extremes (top or bottom of the cylinder) rather than the azimuthal tangent lines, because the vertical angular extent of the cylinder ($H_T=10 \mathrm{m}$) creates a larger angular spread at close range than the horizontal extent ($R_T=7 \mathrm{m}$). The sensitivity of the solution to target dimensions was tested by varying $H_T$ from 8 $\mathrm{m}$ to 12 $\mathrm{m}$: increasing $H_T$ reduces obscuration durations by approximately 5% per meter of added height, as the smoke sphere must cover a larger vertical angle. This sensitivity highlights the importance of accurate target characterization for operational deployment planning.

We further evaluated the rounded P5 solution under simple perturbations to the nominal assumptions.

Table 13. Sensitivity of the rounded P5 deployment to wind drift and missile trajectory prediction errors. Values are recomputed with the same deployment parameters and cylindrical LOS evaluator.

Evaluation Scenario	${\mathit{M}}_1$ (s)	${\mathit{M}}_2$ (s)	${\mathit{M}}_3$ (s)	Total (s)
Nominal rounded P5 parameters	9.89	7.70	3.05	20.64
Wind $(w_x,w_y)=(1,0)$ m/s	9.81	7.60	2.98	20.39
Wind $(w_x,w_y)=(0,1)$ m/s	9.25	6.42	2.55	18.22
Wind $(w_x,w_y)=(0,2)$ m/s	7.59	5.37	2.17	15.13
Missile lateral bias $+20 \mathrm{m}$ in y	2.13	1.31	2.32	5.76
Missile range bias $+100 \mathrm{m}$ in x	7.83	6.02	3.00	16.85
Missile altitude bias $+20 \mathrm{m}$ in z	0.00	0.00	1.61	1.61

reports two classes of perturbations. First, constant wind drift is applied to the smoke center after burst using the wind-extended form of Equation (5). A mild 1 $\mathrm{m} {\mathrm{s}}^{-1}$ downrange wind has little effect, whereas a 2 $\mathrm{m} {\mathrm{s}}^{-1}$ crosswind reduces total obscuration from $20.64$ $\mathrm{s}$ to $15.13$ $\mathrm{s}$. Second, missile trajectory prediction errors are emulated by perturbing the missile initial position used in evaluation while keeping the pre-planned smoke deployment fixed. Even a 20 $\mathrm{m}$ lateral bias substantially reduces coverage, indicating that the open-loop plan is sensitive to cross-track trajectory errors. This does not invalidate the inverse design mechanism, but it shows that operational use would require trajectory updates and closed-loop re-planning.

5.4. Connection to Weapon–Target Assignment

The hierarchical allocation framework bears a structural resemblance to the classical weapon–target assignment (WTA) problem [4,9,11,12], in which weapons are assigned to targets to maximize total damage or minimize total survival probability. The key difference is that in our problem, the “value” of assigning a UAV to a missile depends on the continuous deployment parameters, which must themselves be optimized. This coupling between discrete assignment and continuous optimization is handled by the three-stage decomposition, which first approximates the value function (Stage 1), then searches the assignment space (Stage 2), and finally refines the coupled solution (Stage 3).

The performance matrix $\mathbf{\Pi }$ (Table 5) reveals that UAV–missile affinity is highly non-uniform, with entries ranging from 0.00 s (infeasible pairs such as ${\mathrm{FY}}_1\to M_2$) to 5.98 s (${\mathrm{FY}}_1\to M_1$). This heterogeneity mirrors the asymmetric effectiveness matrices encountered in classical WTA formulations [10,13] and justifies the use of explicit enumeration over heuristic assignment rules. The modest problem scale ($5\times 3$) permits exact enumeration. For dozens of UAVs and missiles, Stage 2 can be replaced by a scalable heuristic pipeline: retain only the top-q missile candidates per UAV from $\mathbf{\Pi }$, build an initial assignment by greedy or auction-based selection, improve it by local swaps and reassignments, and periodically re-solve restricted mixed-integer sub-problems or column-generation pricing problems for high-value clusters. Branch-price-and-cut and column-generation ideas from large-scale WTA [12,13] can handle the combinatorial component, while the inverse design routine remains the continuous optimizer used to price each candidate column.

An important distinction from classical WTA is the near-additivity of cooperative deployment. In standard WTA, the damage inflicted by multiple weapons on the same target follows a strongly subadditive model—each additional weapon contributes significantly less due to diminishing marginal returns (the target is already partially destroyed). In our problem, the marginal returns diminish only slightly: two UAVs assigned to the same missile can achieve nearly the sum of their individual contributions because the joint optimizer coordinates their three-shot relays to cover complementary temporal windows with minimal overlap. The P5 results demonstrate this clearly: ${\mathrm{FY}}_1$ and ${\mathrm{FY}}_5$ individually achieve 5.98 s and 3.94 s against $M_1$ (sum: 9.92 s), while their joint optimization yields 9.891 s—recovering 99.7% of the independent sum, with only 0.029 s lost to temporal overlap. For $M_2$, the joint result of $7.710$ $\mathrm{s}$ from ${\mathrm{FY}}_2$ (3.96 s) and ${\mathrm{FY}}_4$ (3.78 s, sum: 7.74 s) similarly recovers 99.6% of the sum. This near-additive behavior is not a strict permutation symmetry of UAV identities: each UAV has a distinct initial position and therefore a distinct feasible set. It is instead a temporal complementarity property of the union-duration objective, which allows heterogeneous UAVs to partition time windows efficiently when they are co-assigned to the same missile.

This near-additivity has important implications for the assignment problem structure. In the strongly subadditive WTA setting, the optimal strategy tends to spread weapons across targets to avoid diminishing returns [3,4]. In our near-additive setting, concentrating multiple UAVs on the same missile incurs negligible efficiency loss, making it equally viable to concentrate UAVs on high-priority missiles, provided the geometric constraints permit efficient temporal coordination. The optimal P5 assignment $(0,1,2,1,0)$—which allocates two UAVs each to $M_1$ and $M_2$ while leaving only one for $M_3$—reflects this trade-off. The assignment algorithm implicitly recognizes that adding a second UAV to $M_1$ or $M_2$ yields a near-doubling of obscuration (due to efficient temporal partitioning), whereas the sole UAV assigned to $M_3$ faces such severe geometric constraints that even a second UAV would provide limited additional benefit.

5.5. Limitations and Future Work

Several simplifying assumptions underlie the current model, and understanding their impact is important for assessing the applicability of the results.

The smoke cloud is treated as a thresholded sphere of fixed radius $R_S=10 \mathrm{m}$, neglecting concentration gradients, spectral extinction, and turbulent diffusion [14,15]. In reality, a missile seeker may still partially track a target through the dilute edge of a dispersing cloud if the optical depth along the LOS remains below the sensor’s obscuration threshold. A Gaussian puff dispersion model [16,17] would predict an expanding, wind-advected ellipsoid with decreasing peak concentration, and Equation (4) would need to be evaluated along each sightline. The present model should therefore be interpreted as a planning-level threshold approximation, not as a high-fidelity smoke-physics model.

The nominal optimization assumes zero horizontal wind drift. Table 13 shows that even moderate crosswind can substantially reduce open-loop obscuration, so operational deployment would require wind estimation and either wind-aware trajectory planning or online correction of the desired burst point. The constant-wind extension of Equation (5) is straightforward, but realistic maritime wind shear would require a spatially varying field and a dispersion model.

Missiles are assumed to fly at constant speed along straight lines toward the target. Anti-ship missiles in practice may execute terminal maneuvers such as sea-skimming, weaving, or pop-up trajectories to evade defenses. Table 13 demonstrates that the open-loop solution is highly sensitive to cross-track and altitude prediction errors, so the reported $20.652$ $\mathrm{s}$ should be interpreted as a nominal-trajectory planning result. The inverse design approach remains useful under trajectory uncertainty because it can rapidly regenerate feasible seeds after a trajectory update, but a robust operational implementation should sample multiple missile trajectory realizations and optimize expected or worst-case obscuration [36,37].

The inverse design parameterization initially restricts seeds to the missile–target line. This is a design-family restriction, not proof that all off-axis deployments are inferior. To reduce this limitation, the subsequent Nelder–Mead refinement is performed in the physical UAV parameter space and can move the final burst point off the initial line-of-sight seed. Future work should explicitly include seeker field-of-view models and off-axis obscuration volumes so that the seed family itself can be expanded.

The cylindrical target model, while more realistic than a point target, still idealizes the vessel’s complex superstructure as a uniform cylinder of radius $R_T=7 \mathrm{m}$ and height $H_T=10 \mathrm{m}$. A real naval vessel presents an irregular silhouette that varies with viewing angle, and modern seeker heads may lock onto specific features (mast, radar array, superstructure edge) rather than the geometric center. Incorporating a detailed 3D target model would require ray-tracing against a mesh representation [18,21], substantially increasing the computational cost of each obscuration evaluation but potentially revealing angle-dependent vulnerabilities that the cylindrical approximation misses.

The hierarchical allocation strategy also decouples the full multi-missile problem by first optimizing UAV–missile pairs independently. This is a warm-start decomposition rather than a fully coupled cooperative solution. It may miss secondary obscuration effects in which a cloud assigned to one missile also partially blocks another missile’s LOS. In the present scenario, the missile approach corridors and burst clusters are sufficiently separated that this effect is small, but future large-scale implementations should evaluate cross-missile coverage terms during Stage 3.

Future research directions include:

• Wind field integration. Incorporating spatially varying wind fields into the smoke dispersion model would improve realism, as wind can significantly alter smoke cloud positioning and effective radius over the 20 s duration [16,17].
• Missile maneuver models. Extending the framework to handle maneuvering missiles with mid-course guidance updates would require online re-planning capabilities, potentially leveraging the computational efficiency of the inverse design approach for real-time adaptation.
• Dynamic re-planning. Exploring closed-loop deployment strategies where UAV parameters are updated as the engagement evolves, incorporating feedback from real-time tracking of missile trajectories and smoke cloud states.
• Multi-objective formulations. Balancing obscuration duration against secondary objectives such as UAV exposure risk, fuel consumption, and smoke projectile expenditure would produce operationally richer solutions.
• Scalability to larger force structures. For scenarios involving dozens of UAVs and missiles, the exact enumeration in Stage 2 becomes impractical. Decomposition heuristics from the WTA literature [3,10] or metaheuristic search [28] could be employed to handle the combinatorial explosion while preserving the inverse design core.

6. Conclusions

This paper has presented an inverse design optimization framework for multi-UAV cooperative smoke screen deployment in naval defense scenarios. Four principal contributions have been made. First, the inverse design stage transforms the initial search for promising deployments from four UAV parameters into a two-dimensional search over physically interpretable seed parameters, reducing wasted evaluations while retaining subsequent physical-space refinement. Second, the three-dimensional cylindrical LOS obscuration model extends beyond the conventional point-target assumption by constructing four extreme tangent sightlines to the target vessel’s surface, providing a more conservative and realistic assessment of obscuration effectiveness. Third, the hierarchical multi-UAV multi-missile allocation framework decomposes the combinatorial complexity of asset assignment into tractable stages of independent evaluation, assignment enumeration, and joint optimization with cross-initialization. Fourth, comprehensive numerical validation across five progressively complex sub-problems (P1–P5) demonstrates the scalability and effectiveness of the approach.

Numerical experiments demonstrate that coordinated multi-UAV deployment achieves obscuration durations far exceeding those possible with individual assets: single-shot obscuration of $1.362$ $\mathrm{s}$ (baseline) is improved to $4.580$ $\mathrm{s}$ (optimized single shot), $7.324$ $\mathrm{s}$ (three-shot relay), $11.140$ $\mathrm{s}$ (three-UAV cooperation), and $20.652$ $\mathrm{s}$ (full five-UAV three-missile assignment). The inverse-seeded framework provides an efficient and interpretable pathway to high-quality solutions, with the two-dimensional seed search reducing the number of unproductive function evaluations by approximately two orders of magnitude compared to direct four-dimensional search in P2.

Broader applicability and practical implications. The proposed methodology is applicable to a broad class of deployment optimization problems where the goal is to position objects at specific locations using mobile platforms with constrained kinematics. The inverse design principle—specifying desired outcomes and analytically back-calculating required inputs—can be generalized to other domains such as aerial cargo delivery, precision agriculture, and search-and-rescue operations where the forward kinematics are analytically invertible. For naval defense practitioners, the framework provides a computationally tractable planning tool that can generate deployment strategies within minutes on standard hardware, supporting both pre-mission planning and near-real-time re-planning as threat scenarios evolve. The hierarchical allocation structure also offers operational transparency, as each stage of the decision process (performance evaluation, assignment selection, parameter refinement) can be independently reviewed and adjusted by human operators.

Current limitations and future research directions. The current framework assumes simplified smoke physics (fixed-radius sphere), straight-line missile trajectories, and static environmental conditions. As discussed in Section 5.5, extensions incorporating wind fields, missile maneuvers, dynamic re-planning, multi-objective optimization, and scalable assignment algorithms represent important avenues for future work. Addressing these extensions would move the framework closer to operational deployment in realistic naval defense scenarios.