Dynamic Resource Allocation in Full-Duplex Integrated Sensing and Communication: A Multi-Objective Memetic Grey Wolf Optimizer Approach

Abstract

To meet the dual demands of 6G cellular networks for high spectral efficiency and environmental sensing, this paper proposes a full-duplex (FD) integrated sensing and communication (ISAC) dynamic resource allocation framework. At the heart of the framework lies a dynamic frame structure that can self-adapt the time-domain resource ratio between sensing and communication, designed to flexibly handle complex traffic demands. In FD mode, however, the trade-off between communication and sensing performance, exacerbated by severe self-interference (SI), morphs into a non-convex, NP-hard multi-objective optimization problem (MOP). To tackle this, we propose an Adaptive Hybrid Memetic Multi-Objective Grey Wolf Optimizer (AM-MOGWO). Finally, simulations were conducted on a high-fidelity platform that integrates 3GPP-standardized channels, which was further extended to a challenging multi-cell interference scenario to validate the algorithm’s robustness. AM-MOGWO was systematically benchmarked against standard Grey Wolf Optimizer (GWO), random search (RS), and the genetic algorithm (GA). Simulation results demonstrate that in both the single-cell and the more complex multi-cell environments, the proposed algorithm excels in locating the Pareto-optimal solution set, where its solution set significantly outperforms the baseline methods. Its hypervolume (HV) metric surpasses the second-best approach by more than 93%. This result quantitatively demonstrates the algorithm’s superiority in finding a high-quality set of trade-off solutions, confirming the framework’s high efficiency in complex interference environments.

1. Introduction

As the 6G vision crystallizes, wireless networks are evolving from single-purpose communication services toward multifunctional convergence [1]. Across domains such as connected vehicles, smart factories, smart cities, and the metaverse, the demand for both reliable sensing and high-efficiency communication is surging [2,3]. Yet because communications and radar rely on separate hardware and platforms, they trigger severe contention for spectrum resources [4]. ISAC has emerged to reconcile soaring resource demands with holistic network capabilities; through their co-design, it promises to spark a new generation of intelligent applications [5].

For ISAC, much of the current research centers on partitioning communication and sensing functions by separating them in time or space. Reference [6] proposes a three-stage Time Division Duplexing (TDD) frame structure, while [7,8] subsequently design a dynamically adjustable frame to meet the joint communication-and-sensing demands in vehicular networks. Alternatively, by combining spatial and temporal division, configuring a single antenna array can also achieve switching between communication and sensing [9]. Although such designs avoid mutual interference [10], they also prevent ISAC systems from fully exploiting their spectral-efficiency potential. To meet 6G’s stringent demands on spectrum utilization, the FD mode [3] has now become the focal point of ISAC research. Unlike the interference-avoidance approaches mentioned earlier, FD mode allows a base station to transmit and receive simultaneously on the same time–frequency resources, maximizing resource utilization [11,12]. For instance, [13] repurposes downlink resources by having the BS serve as both radar transceiver and communication transmitter, whereas [14] reuses uplink resources for the same purpose. At the same time, FD operation introduces a thornier physical constraint—strong SI [11]—which turns what was once a straightforward resource-allocation task into a far more intricate problem. Furthermore, much of the existing research simplifies the network model to an idealized single-cell scenario, thereby overlooking the significant impact of inter-cell interference that is prevalent in practical deployments [15].

In ISAC-centric systems, resource scheduling is itself a pivotal lever for reconciling the fundamental tension between sensing accuracy and communication performance [16], and research into algorithms for this allocation problem continues to advance. Early efforts mostly leaned on classical mathematical-optimization tools: both [17,18] employed semidefinite relaxation (SDR) to optimize joint communication-and-radar beamforming under different application scenarios. Thereafter, a more mainstream approach has been the weighted-sum method, which linearly fuses multiple objectives into a single utility function [19], for instance, investigates joint beamforming design algorithms aimed at maximizing the weighted sum of sensing and communication performance. Reference [20] builds on non-orthogonal multiple access (NOMA) to formulate a beamforming problem that maximizes the weighted sum of communication throughput and effective sensing power. Yet this approach lacks flexibility, heavily relies on manually set weights, and cannot capture the complete Pareto frontier. To overcome this limitation, researchers have turned to multi-objective evolutionary algorithms (MOEAs) [21]; however, when applied to ISAC-specific problems, these methods often suffer from slow convergence and a tendency to fall into local optima. Recently, deep reinforcement learning (DRL) has also been brought into ISAC research: ref. [22] devises a hybrid DRL agent to maximize the system’s cooperative rate. Yet emerging DRL approaches still suffer from high training costs and unstable policy convergence.

Motivated by the issues above, this paper formulates an FD-mode ISAC system that explicitly accounts for self-interference and incorporates state-of-the-art self-interference cancelation (SIC) techniques [12]. To tackle the resulting NP-hard MOP [23] that is difficult to solve, we propose an AM-MOGWO. By integrating Lévy flights and an adaptive hybrid search, AM-MOGWO delivers strong global exploration while avoiding the high training costs of other approaches. The main contributions of this paper are summarized as follows:

• The proposed AM-MOGWO utilizes a problem-driven fusion of diverse strategies—including Lévy Flight, an adaptive hybrid search, and Memetic Computing—enabling it to exhibit superior global exploration capabilities and efficient convergence performance when solving complex NP-hard problems such as ISAC resource allocation. To ensure a rigorous and effective evaluation of the proposed optimization algorithms, this paper first establishes a high-fidelity system model for an ISAC network.
• This paper constructs a comprehensive ISAC system model that simulates a realistic operational environment by incorporating key physical factors—including 3GPP-standardized channels, the Doppler effect, environmental clutter, and bidirectional self-interference—thereby situating the resource allocation problem within a challenging and practical scenario for validation.
• The proposed algorithm’s performance is comprehensively evaluated in both a foundational single-cell environment and a more challenging, realistic multi-cell interference scenario. In both settings, the superiority of AM-MOGWO over baseline methods is systematically demonstrated through visual Pareto front dominance and quantitative metrics, yielding critical insights for the practical design and deployment of ISAC systems.

The remainder of this paper is organized as follows. Section 2 details the ISAC system model we have constructed, including the scenario description, channel model, and the multi-cell interference extension. Section 3 elaborates on the core mechanisms and pseudocode of the proposed AM-MOGWO algorithm. Section 4 presents and discusses the simulation results, including the performance comparison of the algorithms in both single-cell and multi-cell scenarios. Section 5 provides a summary of the main work in this study. Finally, Section 6 outlines future research directions. The overall framework of the proposed AM-MOGWO algorithm is illustrated in Figure 1.

2. Materials and Methods

This chapter establishes the system model for a multiuser FD ISAC network. The model is constructed to reflect a challenging and realistic operational environment by explicitly incorporating key physical phenomena: bidirectional SI resulting from FD operation, environmental clutter, and a composite channel structure that combines 3GPP-compliant, large-scale path loss with small-scale Rician fading [24]. Based on this comprehensive system model, we formulate the resource allocation challenge as a MOP. The primary goal is to jointly maximize two conflicting objectives—the aggregate downlink communication rate and the radar sensing mutual information—while adhering to the quality-of-service (QoS) requirements of all users [25].

2.1. Scenario Description

2.1.1. A Foundational Single-Cell Scenario

The operational environment for the ISAC system under investigation is a single-cell downlink scenario, as conceptually illustrated in Figure 2. In this framework, a central base station (BS) performs two concurrent tasks using a unified signal. For communication, the BS transmits downlink data to a set of mobile users. Simultaneously, for sensing, it leverages the same signal to detect and track a separate, non-communicating target, such as a vehicle. As the figure shows, the BS processes the echo signal reflected from the radar target to extract sensing information. For our performance evaluation, we define the specific parameters of this environment. The BS is situated at the center of a 50-m-radius cell at a height of 25 m. It serves ten mobile user equipments (UEs), which are randomly located within an annulus of 10 m to 50 m from the BS and move at speeds uniformly distributed between 30 and 60 km/h. To model a realistic sensing environment, the performance is primarily impeded by environmental clutter. We simulate this by distributing twenty static scatterers throughout the cell, each with a random radar cross-section (RCS) [26] in the range of [0.01, 0.1] m². These scatterers generate unwanted echoes, representing the main source of radar interference.

2.1.2. A More Challenging Multi-Cell Interference Scenario

To more accurately evaluate the performance and robustness of the proposed AM-MOGWO algorithm in practical network deployments, we construct a more challenging multi-cell interference scenario. This scenario adopts the classic hexagonal network topology, which is standard in cellular network research. The model comprises J = 7 cells: a central cell (indexed j = 0) and a full first tier of six interfering cells (indexed j ∈ {1, …, 6}). Based on our simulation parameters, the inter-site distance is set to 100 m.

Under this layout, the signal reception environment for a UE in the central cell becomes significantly more complex. In addition to the desired signal from the serving base station and the intra-cell self-interference introduced by the full-duplex mode, the UE will also continuously receive downlink signals from all six neighboring interfering base stations. This aggregate inter-cell interference is a key bottleneck affecting the performance of modern cellular networks. It makes the resource allocation problem in ISAC systems trickier and places higher demands on the optimization algorithm’s ability to find globally optimal solutions under strong interference.

2.2. Dynamic TDM Frame

Inheriting and evolving the flexible OFDM-based numerology of 5G New Radio (NR) [27], which supports multiple sub-carrier spacings (SCSs) to enable diverse service requirements, provides a robust foundation for integrating the advanced functionalities envisioned for 6G.

Building upon this principle, this work proposes a dynamic Time-Division Multiplexing (TDM) frame structure designed to facilitate adaptable resource allocation for ISAC [8]. As depicted in Figure 3, each transmission frame is temporally partitioned into two distinct, contiguous phases: an initial Sensing Phase comprising $N_s$ OFDM symbols, where the BS executes radar-centric tasks, followed by a Communication Phase of $N_{\mathrm{c}}$ symbols dedicated to multi-user downlink data transmission.

The cardinal advantage of this architecture lies in its intrinsic flexibility and adaptability. By dynamically tuning the ratio of $N_s$ to $N_{\mathrm{c}}$, the frame structure endows the system with the capability to manage the fundamental sensing-communication performance trade-off [1] in response to real-time service demands—shifting focus from high-precision sensing to high-throughput communication as needed. This adjustable time-domain partitioning thus constitutes a key degree of freedom (DoF) for the resource allocation optimization problem investigated herein.

2.3. Channel Model

The performance evaluation of resource allocation algorithms is critically dependent on the fidelity of the underlying channel model. For analytical tractability, many existing studies on ISAC resort to simplified channel assumptions, such as pure Line-of-Sight ($LOS$) propagation or otherwise idealized fading models. Such simplifications, however, risk overlooking the intricate propagation characteristics of real-world environments, potentially leading to overly optimistic conclusions regarding algorithmic performance. In stark contrast, to ensure that the robustness and effectiveness of our proposed algorithm are rigorously validated, this work establishes a high-fidelity, composite channel model. By integrating both large-scale fading effects, which govern the average path loss, and small-scale fading phenomena, which capture rapid multipath-induced fluctuations, we create a challenging yet realistic simulation environment that mirrors practical urban deployment scenarios.

2.3.1. Large-Scale Fading

This work adopts the 3GPP TR 38.901 path-loss model for the Urban Macro (UMa) scenario to ensure a realistic characterization of signal propagation. A defining feature of this model, which distinguishes it from simpler deterministic approaches, is its use of a probabilistic mechanism to determine whether a given user link is in a $LOS$ or Non-Line-of-Sight ($NLOS$) state.

The model elegantly assigns distinct roles to the two-dimensional distance and the three-dimensional distance. The two-dimensional distance serves as the criterion for determining the link state. In urban environments, the existence of a direct path depends more on the layout of ground-level obstacles such as buildings. Therefore, the horizontal distance between the user and the base station is a more effective indicator for assessing the likelihood of $LOS$, while the three-dimensional distance is used to compute the path loss. According to the fundamental physics of electromagnetic propagation, signal energy attenuates in direct proportion to the actual distance it travels through space—the three-dimensional slant range.

The $LOS$ probability for each user link, denoted as $P_{LOS}$, is a function of the two-dimensional ($2D$) horizontal distance $d_{2D}$, given by

$$\begin{array}{l}{P}_{LOS}(d_{2D})=\left\{\begin{array}{cc}1& if d_{2D}\le 18 m\\ {\displaystyle \frac{18}{d_{2D}}}+\left(1-{\displaystyle \frac{18}{d_{2D}}}\right)e^{-{\displaystyle \frac{d_{2D}}{63}}}& if d_{2D}>18 m\end{array}\right.\end{array}$$

(1)

The path loss ($PL$, in dB) for each link is then calculated by first probabilistically assigning a $LOS$ or $NLOS$ state and subsequently applying the corresponding formula.

The path loss for the $LOS$ case is formulated as ${PL}_{LOS}$:

$${PL}_{LOS}=28.0+22{log}_{10}\left(d_{3D}\right)+20{log}_{10}\left(f_c\right)$$

(2)

The path loss for the $LOS$ case is formulated as ${PL}_{NLOS}$:

$${PL}_{NLOS}=13.54+39.08{log}_{10}\left(d_{3D}\right)+20{log}_{10}\left(f_c\right)-0.6\left(h_{ue}-1.5\right)$$

(3)

The complete path loss model described in (1)–(3) is adopted from the 3GPP TR 38.901 standard for the UMa scenario [24]. Where $d_{3D}$ denotes the 3D slant range (m), f_c is the carrier frequency (GHz), and $h_{ue}$ represent the antenna heights of user equipment, respectively. To ensure the model remains physically consistent, the final $NLOS$ path loss value is taken as the maximum of the calculated $LOS$ and $NLOS$ values.

2.3.2. Small-Scale Fading

The small-scale fading effects, which arise from multipath interference, are modeled using a Rician distribution [28]. This choice is motivated by the operational context of ISAC systems, which frequently utilize higher-frequency bands such as millimeter-wave (mmWave). Propagation in such bands is characterized by a high probability of a stable $LOS$ path, a feature that the Rician model accurately represents.

The Rician fading model explicitly decomposes the channel coefficient into two constituent parts: a deterministic $LOS$ component and a random, scattered $NLOS$ component. To form the complete channel gain, these small-scale effects are scaled by the large-scale path loss discussed previously. The resulting composite channel gain, $h$, is formulated as follows [28]:

$$h = \sqrt{G_L}\left(\sqrt{{\displaystyle \frac{K}{K+1}}}{h}_{los}+ \sqrt{{\displaystyle \frac{1}{K+1}}}{h}_{nlos}\right)$$

(4)

where $G_L$ is the linear-scale, large-scale power gain, derived from the path loss in dB (${PL}_{dB}$) as $G_L={10}^{{\displaystyle \frac{-{PL}_{dB}}{10}}}$. The term K denotes the Rician K-factor, which is the power ratio of the deterministic $LOS$ component to the scattered components; for the special case where K = 0, the channel reduces to Rayleigh fading. The terms $h_{los}$ and $h_{nlos}$ represent the deterministic $LOS$ and the random scattered components, respectively, with $h_{los}$ being modeled as a circularly symmetric complex Gaussian (CSCG) random variable, i.e., $h_{nlos}=\mathcal{C}\mathcal{N}\left(\mathrm{0,1}\right)$ [29].

This synthesis of large-scale path loss and small-scale fading effects results in a high-fidelity channel model. This model serves as a realistic and robust foundation for evaluating our algorithm’s performance under practical propagation conditions.

2.4. System Performance Evaluation Indicators

In order to evaluate the performance of the dual functions of the proposed ISAC system, and to establish the objective function for the subsequent optimization problem, this section will, respectively, derive and define the core performance indicators for the two dimensions of sensing and communication.

2.4.1. Radar Metric

The sensing task is to reduce the prior uncertainty of the target state by measuring the target echo signals. Based on information theory [30], Mutual Information (MI) can directly quantify this reduction in uncertainty, and is therefore widely used as an effective metric for measuring the estimation performance of radar systems [31].

For a channel that can be approximated as Gaussian, the MI is a function of the available bandwidth and the Signal-to-Clutter-plus-Interference-plus-Noise Ratio (SCNR). Therefore, this work adopts the radar mutual information rate (in bit/s) as the sensing performance metric, which is calculated as follows [30]:

$$I^{rad} = B_{sens}·{\mathit{log}}_2(1 + \mathit{SCNR})$$

(5)

where $B_{\mathit{sens}}$ is the bandwidth utilized for sensing, and the SCNR is the ratio of the desired signal power to the aggregate power of all impairments, including environmental clutter, residual self-interference, and thermal noise.

2.4.2. Communication Metric

For the communication system, this paper adopts the Aggregate Downlink Rate of all downlink users as the key performance indicator. Its theoretical basis is the classic Shannon–Hartley theorem [32], which specifies the maximum rate at which information can be transmitted without error over a channel with a given bandwidth and signal-to-noise ratio. Therefore, the communication performance is defined as follows [7]:

$$C_{com}=\sum B{log}_2(1+SINR)$$

(6)

where $B$ represents the system bandwidth and $SINR$ represents the Signal-to-Interference-plus-Noise Ratio. In this ISAC system, the calculation of $SINR$ will comprehensively consider the effects of Inter-Carrier Interference (ICI) caused by user mobility, residual SI of the system, and thermal noise.

2.5. System Modeling

The core task of this chapter is to establish precise mathematical performance models for the two conflicting functions of communication and sensing based on the aforementioned system model, and ultimately to construct a multi-objective optimization problem. The basis for the modeling is the dynamic time-division duplexing frame structure operating in FD mode, which was defined in Section 2.2. Our goal is to express the system’s sensing performance (radar MI) and communication performance (downlink aggregate rate) as functions of resource allocation variables, which mainly include symbol allocation in the time domain ($N_s$, $N_c$) and the allocation scheme in the power domain.

2.5.1. Modeling in the Single-Cell Scenario

We first model the sensing performance. Within the sensing phase composed of $N_s$ symbols, the BS transmits sensing signals and processes radar echoes. The total signal ${\mathit{y}}_{rad}$ processed by its receiver can be decomposed into four parts: the desired target echo ${\mathit{s}}_{target}$, clutter from the environment ${\mathit{s}}_{clutter}$, residual self-interference ${\mathit{s}}_{SI}$ introduced by FD operation, and additive white Gaussian noise $\mathit{n}$:

$${\mathit{y}}_{rad}= {\mathit{s}}_{target}+ {\mathit{s}}_{clutter}+ {\mathit{s}}_{SI}+ \mathit{n}$$

(7)

To analyze this signal, we derive the power of its respective components. For a single target at a distance $d_t$ with a RCS of ${\sigma }_t$, according to the monostatic radar equation, its echo power $P_{\mathit{echo},n}$ can be expressed as [28]

$$P_{\mathit{echo},n}={\displaystyle \frac{P_{s,n}{G}_{\mathit{bs}}^2{\lambda }_c^2{\sigma }_t}{(4\pi {)}^3{d}_t^{4}L}}$$

(8)

where $P_{s,n}$ is the total sensing transmit power, $G_{bs}$ is the antenna gain, ${\lambda }_c$ is the carrier wavelength, and $L$ is the system loss.

Similarly, the total clutter power $P_{clutter,n}$ from $N_{cl}$ non-target scatterers in the environment is [28]

$$P_{clutter,n}=\sum _{j=1}^{N_{cl}}{\displaystyle \frac{P_{s,n}{G}_{bs}^2{\lambda }_c^2{\sigma }_{cl,j}}{(4\pi {)}^3{d}_{cl,j}^{4}L}}$$

(9)

To construct a high-fidelity simulation environment, our model explicitly considers the impact of environmental clutter, which is a primary source of interference in practical radar sensing. The parameters for the clutter scatterers are detailed in Section 4.

In FD operation mode, the residual self-interference $P_{c\to s,n}$ refers to the interference leaked from the communication transmit link during the sensing slot. Its power is proportional to the total communication transmit power $P_c$ and is uniformly distributed on all $N_{sc}$ subcarriers:

$$P_{c\to s,n}={\displaystyle \frac{{\eta }_{c\to s}{P}_c}{N_{sc}}}$$

(10)

where ${\eta }_{c\to s}$ is the residual self-interference coefficient from communication to sensing. The thermal noise power of the receiver, $P_{noise,s}$, is

$$\begin{array}{c}{P}_{noise,s}=N_0\Delta f{F}_n\end{array}$$

(11)

where $N_0$ is the noise power spectral density, $\Delta f$ is the subcarrier bandwidth, and $F_n$ is the noise figure. Thus, the SCNR of the radar on subcarrier ${SCNR}_n$ can be precisely expressed as

$$\begin{array}{c}\begin{array}{c}{SCNR}_n={\displaystyle \frac{P_{echo,n}}{P_{clutter,n}+P_{c\to s,n}+P_{noise,s}}}\end{array}\end{array}$$

(12)

Based on this SCNR, we adopt the radar MI as the final sensing performance metric. For a Gaussian channel, and considering that the sensing task only occupies a time proportion of ${\displaystyle \frac{N_s}{N_s+N_c}}$, the total radar mutual information of the system, $I^{rad}$, can be obtained by accumulating the mutual information of all subcarriers:

$$I^{rad}={\displaystyle \frac{N_s}{N_s+N_c}}\sum _{n=1}^{N_{sc}}\Delta f{\mathit{log}}_2(1+{SCNR}_n)$$

(13)

where $\Delta f$ is the bandwidth of a single subcarrier. In the simulations of this paper, we make a common assumption that the sensing power is uniformly distributed on all $N_{sc}$ subcarriers. Under this condition, the SCNR values of each subcarrier are the same, denoted as SCNR. At this time, the above equation can be simplified to

$$\begin{array}{c}{I}_{rad}={\displaystyle \frac{N_s}{N_s+N_c}}·B·{\mathit{log}}_2\left(1+{SCNR}_n\right)\end{array}$$

(14)

where $B=N_{sc}\Delta f$ is the total system bandwidth.

In the considered cellular system, a BS simultaneously serves a set of K uplink users, indexed by $k\in \mathcal{U}=\left\{\mathrm{1,2},\dots ,\mathrm{K}\right\}$, and a set of L downlink users, indexed by $l\in \mathcal{L}=\left\{\mathrm{1,2},\dots ,L\right\}$.

During the communication phase, which consists of $N_c$ symbols, the signal $y_{\mathrm{c},l,\mathrm{n}}$ received by downlink user $l\in \mathcal{L}$ on its allocated subcarrier $n\in {\mathcal{L}}_l$ is modeled as a linear superposition of the desired signal, cross-phase SI, ICI, and AWGN:

$$\begin{array}{c}{y}_{c,l,n}=\sqrt{G_{bs}{G}_{ue}{h}_{l,n}{p}_{c,l,n}}{s}_{l,n}+i_{s\to c,n}+i_{ici,l,n}+z_{c,n}\end{array}$$

(15)

where $s_{l,n}$ is the data symbol with unit energy (i.e., $\begin{array}{c}\mathbb{E}\left[|s_{l,n}{|}^2\right]=1\end{array}$), $p_{\mathrm{c},l,\mathrm{n}}$ is the transmit power allocated to user $l$ on this subcarrier, $h_{l,n}$ is the channel gain, and $G_{bs}$, $G_{ue}$ are the antenna gains of the BS and the user, respectively. The average power of each signal component is then derived to construct the SINR expression, starting with the desired signal power, $P_{sig,l,n}$, which is given by the second-order moment of the first term in (16):

$$P_{sig,l,n}=\mathbb{E}\left[{\left|h_{l,n}\sqrt{G_b{G}_{ue}{p}_{c,l,n}}{s}_{l,n}\right|}^2\right]=G_{bs}{G}_{ue}{h_{l,n}|}^2{p}_{c,l,n}$$

(16)

Next, the interference term, $i_{s\to c,n}$, is identified as the cross-slot self-interference arising from the sensing hardware leakage during the communication reception period, with its per-subcarrier power given by

$$P_{s\to c,n}={\eta }_{s\to c}{\displaystyle \frac{P_s}{N_{sc}}}$$

(17)

where $P_s$ is the total sensing transmit power.

The ICI power, $P_{ici,l,n}$, stems from the Doppler effect induced by user mobility, which disrupts the orthogonality among OFDM subcarriers. Its value is approximated to be proportional to the desired signal power, $P_{sig,l,n}$, on the same subcarrier:

$$P_{ici,l,n}={\kappa }_{ici}\left(f_{d,l}\right)\cdot P_{sig,l,n}$$

(18)

where the dimensionless ICI coefficient, ${\kappa }_{ici}\left(f_{d,l}\right)$, is a function of the user’s Doppler shift ($f_{d,l}$) and the OFDM symbol duration ($T_{sym}$), and can be approximated as [29]

$${\kappa }_{ici}(f_{d,l})\approx {\displaystyle \frac{1}{12}}(\pi f_{d,l}{T}_{sym}{)}^2$$

(19)

The average thermal noise power, $P_{noise,c}$, over a single subcarrier bandwidth, $\Delta f$, is given by

$$P_{noise,c}=N_0\Delta f{F}_n$$

(20)

where $N_0$, $\Delta f$, and $F_n$ are the noise power spectral density, the subcarrier bandwidth, and the linear value of the receiver’s noise figure, respectively.

The SINR for user $l$ on subcarrier $n$, based on the preceding derivations, is defined as the ratio of the desired signal power to the total power of all interference and noise components:

$${SINR}_{l,n}={\displaystyle \frac{P_{sig,l,n}}{P_{s\to c,n}+P_{ici,l,n}+P_{noise,c}}}$$

(21)

Grounded in Shannon’s channel capacity theory, the aggregate downlink communication rate, $R_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, is the sum of the rates of all downlink users, scaled by the time proportion, ${\displaystyle \frac{N_c}{N_s+N_c}}$, allocated to the communication task:

$$R_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\displaystyle \frac{N_c}{N_s+N_c}}\sum _{l\in \mathcal{L}}\sum _{n\in {\mathcal{L}}_l}\Delta f{\log }_2(1+{\mathrm{S}\mathrm{I}\mathrm{N}\mathrm{R}}_{l,n})$$

(22)

The performance metrics, $R_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ and ${\mathrm{I}}_{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{r}}$, precisely derived in this section, form a solid basis for the subsequent formulation of the multi-objective optimization problem.

2.5.2. Modeling Extension for the Multi-Cell Scenario

While the model in Section 2.5.1 establishes a performance baseline in an ideal single-cell scenario, a more rigorous evaluation must account for interference from neighboring cells, a critical factor in dense network deployments. Therefore, this section extends the aforementioned model to a more challenging multi-cell interference scenario.

In this setting, a user in the central cell is subject not only to intra-cell self-interference but also to significant interference from the downlink transmissions of all neighboring base stations. Therefore, the SINR for user l on subcarrier n is formulated as

$${SINR}_{l,n}={\displaystyle \frac{P_{sig,l,n}}{P_{s\to c,n}+P_{noise,c}+P_{inter-cell,l,n}}}$$

(23)

where $P_{sig,l,n}$, $P_{s\to c,n}$, and $P_{noise,c}$ are the powers of the desired signal, intra-cell residual self-interference, and thermal noise, respectively. The inter-cell interference, $P_{inter-cell,l,n}$, is the aggregate power from all interfering base stations, given by

$$P_{inter-cell,l,n}=\sum _{j=1}^{J-1}{G}_{bs,j}{G}_{ue,l}|h_{j,l,n}{|}^2{p}_{c,j,n}$$

(24)

In this formula, $h_{j,l,n}$ is the channel gain from the j-th interferer to user l on subcarrier n, with $G_{bs,j}$ and $G_{ue,l}$ being the respective antenna gains. For a worst-case interference model, all interfering BSs transmit at maximum power ($P_{max}$), distributed evenly across the $N_{sc}$ total subcarriers. Thus, the transmit power per subcarrier is $p_{c,j,n}=P_{max}/N_{sc}$.

2.6. Multi-Objective Problem Formulation

Building upon the detailed mathematical descriptions of the ISAC system model and performance metrics from the preceding chapter, this section formulates a MOP. The objective of this MOP is to find the Pareto-optimal tradeoff between the dual communication and sensing functions by strategically allocating time- and power-domain resources, subject to the system’s physical limitations and QoS requirements.

Following the performance metric derivations, the MOP is formulated to simultaneously maximize the communication performance, i.e., the aggregate rate $R_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, and the sensing performance, i.e., the radar mutual information $I_{rad}$. This is achieved by optimizing a vector of resource allocation variables, $x$, which encompasses the sensing duration $N_s$ and power allocation schemes, leading to the following formulation:

$$\begin{array}{l}\underset{x}{\mathit{max}}\hspace{1em}\{f_1(x)=R_{\mathit{total}}(x),\hspace{1em}{f}_2(x)=I_{rad}(x)\}\\ s.t. C1:P_C+P_S\le P_{max}\\ C2:R_{total}\ge R_{min}\\ C3:N_s\in \{N_{s,min},\dots ,N_{s,max}\},N_s\in \mathbb{Z}\\ C4:P_C\ge 0,P_S\ge 0\end{array}$$

(25)

The formulated optimization problem is governed by several key constraints. The maximum transmit power constraint (C1) reflects the physical limitation of the BS’s hardware, stipulating that the combined instantaneous power for communication ($P_C$) and sensing ($P_S$) cannot exceed the power amplifier’s maximum rating, $P_{max}$. In addition to this hardware limit, the QoS constraint (C2) guarantees a minimum user experience by requiring the aggregate data rate to remain above a threshold, $R_{min}$. Structurally, the integer symbol constraint (C3) dictates that the number of sensing symbols, $N_s$, must be an integer, as an OFDM symbol represents the fundamental discrete unit of time-domain resources. To ensure that neither function is deprived of resources, this value is also bounded within a feasible operational range, i.e., $N_{s,min}\le N_s\le N_{s,max}$. Finally, the non-negative power constraint (C4) enforces a fundamental physical law.

3. A Grey Wolf Optimizer for Dynamic Resource Allocation

The joint resource optimization for the ISAC system under investigation is a mathematically complex Mixed-Integer Non-Linear Programming (MINLP) problem. It is fundamentally NP-hard due to the non-convexity of the objective functions and the coupled nature of the constraints, rendering traditional gradient-based or convex optimization methods either inapplicable or unable to guarantee global optimality. Consequently, employing metaheuristic algorithms, such as the AM-MOGWO proposed herein, provides an effective pathway to finding a high-quality Pareto-optimal solution set within an acceptable computational complexity.

This chapter presents a detailed exposition of the AM-MOGWO framework, which is specifically designed for the unique characteristics of the ISAC problem. We begin with the fundamental mathematical principles of the standard GWO. Subsequently, we elaborate on the architecture of our proposed AM-MOGWO, detailing its multi-objective handling mechanism, the solution encoding scheme, and a series of strategies designed to enhance its performance.

3.1. Standard GWO

GWO is a metaheuristic algorithm inspired by the social hierarchy and hunting behavior of grey wolves [33], where in the search process is guided by the three most optimal wolves (the leaders) steering the entire population of search agents toward the most promising regions of the search space.

To emulate the social hierarchy of the wolf pack, each iteration of the algorithm involves sorting the population based on fitness values, where the top three solutions are designated as the alpha ($X_{\alpha }$), beta ($X_{\beta }$), and delta ($X_{\delta }$) wolves. These three leaders are considered to be the closest approximations to the prey (i.e., the optimal solution), while the remaining individuals, termed omega (ω) wolves, update their positions under the collective guidance of this leading trio.

The search process of the GWO emulates the pack’s hunting behaviors—such as encircling, chasing, and attacking the prey—wherein the position of each omega (ω) wolf is updated based on the collective guidance of the alpha (α), beta (β), and delta (δ) leaders. This process begins with the calculation of the distance vectors between the omega wolf and this leading trio [33]:

$$\left\{\begin{array}{c}\overrightarrow{D_{\alpha }}=|\overrightarrow{C_1}·\overrightarrow{X_{\alpha }}-\overrightarrow{X}\\ \overrightarrow{D_{\beta }}=|\overrightarrow{C_1}·\overrightarrow{X_{\beta }}-\overrightarrow{X}\\ \overrightarrow{D_{\delta }}=|\overrightarrow{C_1}·\overrightarrow{X_{\delta }}-\overrightarrow{X}\end{array}\right.$$

(26)

Using these distance vectors, the potential next positions towards the three leader wolves are then calculated as

$$\left\{\begin{array}{c}\overrightarrow{X_1}=\overrightarrow{X_{\alpha }}-\overrightarrow{A}·\overrightarrow{D} \\ \overrightarrow{X_2}=\overrightarrow{X_{\beta }}-\overrightarrow{A}·\overrightarrow{D}\\ \overrightarrow{X_3}=\overrightarrow{X_{\delta }}-\overrightarrow{A}·\overrightarrow{D}\end{array}\right.$$

(27)

The position of the omega (ω) wolf for the next iteration, $X(t+1)$, is then determined by averaging the three aforementioned potential positions:

$$X(t+1)={\displaystyle \frac{X_1+X_2+X_3}{3}}$$

(28)

where t indicates the current iteration, $\overrightarrow{X}(t)$ is the position vector of the omega wolf, and the coefficient vectors $\overrightarrow{A}$ and $\overrightarrow{C}$ are calculated via $\overrightarrow{A}=2\overrightarrow{a}\cdot {\overrightarrow{r}}_1-\overrightarrow{a}$ and $\overrightarrow{C}=2·\overrightarrow{r_2}$, with ${\overrightarrow{r}}_1$ and $\overrightarrow{r_2}$ being random vectors with elements drawn from [0, 1]. The parameter is linearly decreased from 2 to 0 over the course of iterations to balance the algorithm’s global exploration and local exploitation capabilities.

3.2. AM-MOGWO

The standard GWO is primarily designed for single-objective optimization and is prone to premature convergence to local optima when tackling complex, multi-modal problems such as the ISAC resource allocation task formulated herein. To address these limitations and efficiently solve the established multi-objective problem, we propose an AM-MOGWO. While this algorithm incorporates a Pareto-dominance-based mechanism to handle multiple objectives, its core innovation lies in a novel adaptive hybrid search framework constructed by fusing multiple advanced search strategies.

The proposed AM-MOGWO is a Pareto-dominance-based algorithm that employs a bounded external archive to store all non-dominated solutions discovered during the search process; this set of archived solutions constitutes the final output. For the purpose of guiding the population search and determining the alpha, beta, and delta leaders, the total utility function, W, as modeled in the preceding chapter, is adopted as a scalar fitness function. It must be emphasized, however, that this fitness function’s role is strictly confined to being an internal guidance mechanism for the algorithm.

Each solution to the optimization problem is encoded as a D-dimensional position vector, $\overrightarrow{X}=[N_s,P_c,P_s,\dots ]$, where each element corresponds to a resource allocation variable, with the sensing duration, $N_s$, being constrained as an integer.

To address the inherent challenge of balancing global exploration with local exploitation in standard GWO and to enhance its search efficiency within complex optimization spaces, the proposed AM-MOGWO integrates a series of performance-boosting strategies.

First, the Lévy Flight mechanism is incorporated to enhance the algorithm’s global exploration capability and effectively avert premature convergence [34]. This mechanism, a random walk model that emulates the foraging behavior of various organisms, is characterized by step lengths drawn from a heavy-tailed distribution, which enables a combination of fine-grained local searches and occasional long-distance jumps. The position update for a search agent executing a Lévy Flight is consequently formulated as [34]:

$$\overrightarrow{X}(t+1)=\overrightarrow{X}(t)+\alpha \oplus L(\lambda )$$

(29)

where α > 0 is a step-size control factor and the operator $\oplus$ denotes the entry-wise product (i.e., Hadamard product). The components of the Lévy step-length vector, $L\left(\lambda \right)$, are then generated as follows:

$$L(s)\sim {\displaystyle \frac{\lambda \cdot \Gamma \left(\lambda \right)\cdot \mathit{sin}(\pi \lambda /2)}{\pi }}\cdot {\displaystyle \frac{1}{s^{1+\lambda }}}$$

(30)

This entire mechanism, whose formulation involves the Gamma function, $\Gamma (\cdot )$, is designed to probabilistically guide a subset of the population towards large-scale exploration.

Second, the Opposition-Based Learning (OBL) strategy is incorporated to maintain population diversity within the search space [35]. This strategy is founded on the principle that a solution’s opposite may offer a better approximation to the optimum than the solution itself. Thus, for a given solution, $\overrightarrow{X}=[x_1,\dots ,x_D]$, in the D-dimensional space, its opposite counterpart, $\overrightarrow{X^{\prime }}$, is defined as [35]

$$X_j^{\prime }=l{b}_j+u{b}_j-x_j,j=1,\dots ,D$$

(31)

The OBL strategy is applied during both population initialization and random perturbation phases to prevent premature population aggregation, where $l{b}_j$ and $u{b}_j$ represent the lower and upper bounds of the search space in the j-th dimension, respectively.

Furthermore, to intensify the local exploitation capability around elite solutions, we integrate principles from Chaotic Search [36] and Memetic Computing [37]. This strategy leverages the ergodic and stochastic properties of chaotic maps, such as the Logistic map, to conduct a fine-grained local search in the vicinity of a wolf’s position. The iterative formula for the Logistic map is given by [38]

$$z_{k+1}=\mu \cdot z_k\cdot (1-z_k),z_k\in \left(\mathrm{0,1}\right)$$

(32)

where $\mu$ is the control parameter, and the resulting chaotic sequence is utilized to generate a set of high-quality neighboring solutions around a wolf’s position, thereby accelerating convergence to the optimum.

The fusion of the aforementioned enhancement strategies with the standard GWO hunting behavior results in a novel adaptive hybrid update framework. Within this framework, the final position update for an omega (ω) wolf is no longer determined by a simple average. Instead, an adaptive strategy selection mechanism is introduced, which intelligently switches among three distinct behaviors—standard hunting, global exploration (Lévy Flight), and local exploitation (Chaotic Search)—based on the current iteration phase and search performance. The core logic of this improved update can thus be summarized as

$$\overrightarrow{X}(t+1)=\left\{\begin{array}{cc}{\overrightarrow{X}}_{\alpha }-{\overrightarrow{A}}_1\cdot {\overrightarrow{D}}_{\alpha }+\alpha \oplus L(\lambda )& if p<p_m\\ {\displaystyle \frac{{\overrightarrow{X}}_1+{\overrightarrow{X}}_2+{\overrightarrow{X}}_3}{3}}+z_k({\overrightarrow{X}}_{\alpha })& if p\ge p_m\end{array}\right.$$

(33)

where $p$ is a random number drawn from $[0,1]$, while $p_m$ is an adaptive switching probability that can be adjusted based on the iteration count or population diversity. The first conditional line of the update rule represents an exploratory hunting behavior integrated with Lévy Flight. In contrast, the second line enacts a fine-grained exploitation, biasing the search towards the α wolf’s position while being perturbed by the chaotic sequence $z_k$. The intermediate position vectors, ${\overrightarrow{X}}_1,{\overrightarrow{X}}_2$, and ${\overrightarrow{X}}_3$, are computed as in the standard GWO. Through this hybrid update mechanism, AM-MOGWO maintains an efficient and dynamic balance between global exploration and local exploitation throughout the entire optimization process. The detailed implementation of this adaptive mechanism within the complete AM-MOGWO framework is formally outlined in Algorithm 1.

Algorithm 1 The Proposed AM-MOGWO Framework

$\mathbf{Input}: \mathrm{N}$$(\mathrm{Population} \mathrm{size}), {\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$$(\mathrm{Max} \mathrm{iterations}), \mathrm{l}\mathrm{b},\mathrm{u}\mathrm{b}$ (Search bounds)

$\mathbf{Output}: \mathcal{A}$ (The Pareto front)

$\mathbf{1}\mathbf{:} \mathrm{Initialize} \mathrm{population} \overrightarrow{\mathrm{X}}(0)$$\mathrm{with} \mathrm{OBL}, \mathrm{archive} \mathcal{A}\leftarrow \mathrm{\varnothing }$$; \mathrm{t}\leftarrow 0$

$\mathbf{2}\mathbf{:} \mathrm{Evaluate} \mathrm{objectives} \mathrm{and} \mathrm{fitness} \mathrm{for} \overrightarrow{\mathrm{X}}(0)$

$\mathbf{3}\mathbf{:} \mathcal{A}$$\leftarrow$$\mathbf{Update} \mathbf{Archive} (\mathcal{A},\overrightarrow{\mathrm{X}}(0)$)

4:  Repeat

$\mathbf{5}\mathbf{:}\hspace{1em}\hspace{1em}( \overrightarrow{{\mathrm{X}}_{\mathsf{\alpha }}}, \overrightarrow{{\mathrm{X}}_{\mathsf{\beta }}},\overrightarrow{{\mathrm{X}}_{\mathsf{\delta }}})$ $\leftarrow$$\mathbf{Select} \mathbf{Leaders} \left(\overrightarrow{\mathrm{X}}\right(\mathrm{t}),\mathrm{W}$)

$\mathbf{6}\mathbf{:}\hspace{1em}\hspace{1em} \mathbf{For} \mathrm{i}=1$$\mathrm{to} \mathrm{N}$

7:   // Adaptive Hybrid Position Update Rule

$\mathbf{8}\mathbf{:} {\overrightarrow{\mathrm{X}}}_{\mathrm{i}}(\mathrm{t}+1)=\left\{\begin{array}{cc}{\overrightarrow{\mathrm{X}}}_{\mathsf{\alpha }}-{\overrightarrow{\mathrm{A}}}_1\cdot {\overrightarrow{\mathrm{D}}}_{\mathsf{\alpha }}+\mathsf{\alpha }\odot \mathrm{L}(\mathsf{\lambda })& \mathrm{i}\mathrm{f} \mathrm{p}<{\mathrm{p}}_{\mathrm{m}}\\ & \\ {\displaystyle \frac{{\overrightarrow{\mathrm{X}}}_1+{\overrightarrow{\mathrm{X}}}_2+{\overrightarrow{\mathrm{X}}}_3}{3}}+{\mathrm{z}}_{\mathrm{k}}({\overrightarrow{\mathrm{X}}}_{\mathsf{\alpha }})& \mathrm{i}\mathrm{f} \mathrm{p}\ge {\mathrm{p}}_{\mathrm{m}}\end{array}\right.$

9:   end for

$\mathbf{10}\mathbf{:}\hspace{1em}\hspace{1em} \mathrm{Enforce} \mathrm{boundary} \mathrm{constraints} \mathrm{on} \mathrm{the} \mathrm{new} \mathrm{population} \mathrm{X}(\mathrm{t}+1)$

$\mathbf{11}\mathbf{:}\hspace{1em}\hspace{1em} \mathrm{Evaluate} \mathrm{objectives} \mathrm{and} \mathrm{fitness} \mathrm{for} \mathrm{X}(\mathrm{t}+1)$.

$\mathbf{12}\mathbf{:}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathcal{A}$$\leftarrow$$\mathbf{Update} \mathbf{Archive} (\mathcal{A},\mathrm{X}(\mathrm{t}+1)$)

13:   // Memetic Step

$\mathbf{14}\mathbf{:}\hspace{1em}\hspace{1em} {\overrightarrow{\mathrm{X}}}_{\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{e}}\prime \leftarrow$$\mathbf{Chaotic} \mathbf{Local} \mathbf{Search} (\mathrm{Select} \mathrm{Elite} (\mathcal{A}$)).

$\mathbf{15}\mathbf{:}\hspace{1em}\hspace{1em} \mathcal{A}$$\leftarrow$$\mathbf{Update} \mathbf{Archive} (\mathcal{A},{\overrightarrow{\mathrm{X}}}_{\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{e}}\prime$)

$\mathbf{16}\mathbf{:}\hspace{1em}\hspace{1em} \mathrm{Update} \mathrm{control} \mathrm{parameters} a,p_m$$; \mathrm{and} t+1$.

$\mathbf{17}\mathbf{:} \mathbf{until} t\ge T_{\mathrm{m}\mathrm{a}\mathrm{x}}$

$\mathbf{18}\mathbf{:} \mathbf{return} \mathcal{A}$

4. Results

This section presents a comprehensive performance evaluation of the proposed AM-MOGWO algorithm, conducted through a two-stage simulation framework to assess its effectiveness and robustness.

In Section 4.1, we analyze the algorithm’s performance in a foundational single-cell scenario. In this setting, we investigate the impact of key system parameters, such as the self-interference coefficient, on the Pareto-optimal frontier and benchmark AM-MOGWO against several baseline methods using HV and Inverted Generational Distance (IGD) metrics. In Section 4.2, we validate the algorithm’s robustness and practical applicability in a more challenging multi-cell interference scenario. Here, we focus on demonstrating the superiority of AM-MOGWO over the baseline algorithms in a realistic, interference-limited environment. The key simulation parameters are listed in

Table 1. Key simulation parameters of ISAC system.

Parameter	Value
Network Layout	Hexagonal Grid
Number of Cells	7
Inter-Site Distance	100 m
$\mathrm{Carrier} \mathrm{Frequency} {\mathrm{f}}_{\mathrm{c}}$	28 GHz [4]
$\mathrm{System} \mathrm{Bandwidth} \mathrm{B}$	144 MHz [27]
$\mathrm{Number} \mathrm{of} \mathrm{Downlink} \mathrm{Users} \mathrm{L}$	10 [19]
Channel Model	3GPP UMa and Rician [24]
Rician K-factor K	0.1
$\mathrm{Max} \mathrm{BS} \mathrm{Transmit} \mathrm{Power} {\mathrm{p}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	40 W (46 dBm) [24]
$\mathrm{Antenna} \mathrm{Gains} (\mathrm{BS}/\mathrm{UE}) ({\mathrm{G}}_{\mathrm{b}\mathrm{s}}$$/{\mathrm{G}}_{\mathrm{u}\mathrm{e}}$)	25 dBi/5 dBi [24]
$\mathrm{Thermal} \mathrm{Noise} \mathrm{Density} {\mathrm{N}}_0$	−174 dBm/Hz [28]
$\mathrm{User} \mathrm{RCS} {\mathsf{\sigma }}_{\mathrm{t}}$	0.5 m2 [26]
Residual Self-Interference Coeff. $\eta$	0.01 [11]
$\mathrm{Symbols} \mathrm{per} \mathrm{Frame} \mathrm{N}$	140 [27]
$\mathrm{Min}. \mathrm{Communication} \mathrm{Rate} \mathrm{Constraint} {\mathrm{R}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$	0.4 Gbps
$\mathrm{Population} \mathrm{Size} {\mathrm{N}}_{\mathrm{p}}$	80
$\mathrm{Max} \mathrm{Iterations} T_{\mathrm{m}\mathrm{a}\mathrm{x}}$	200

.

4.1. Performance Analysis in the Single-Cell Scenario

Figure 4 illustrates the impact of SI on system performance: the system remains highly robust under low-to-moderate SI levels. The communication success rate stays at or above at 95% until the residual SI coefficient reaches a critical threshold of approximately $1\times 10^{-2}$. However, once this threshold is crossed, system performance deteriorates sharply, with the communication success rate plummeting toward zero in a cliff-like fashion. This demonstrates a pronounced threshold effect, rather than a gradual, linear decline.

To further elucidate this phenomenon, Figure 5 presents the CDFs of the communication rate under both high and low self-interference levels. The figure shows that high self-interference (orange line) causes the entire rate distribution to shift markedly toward the lower-rate region. Under low-interference conditions, the median communication rate (at a CDF of 0.5) is approximately 1.6 Gbps, whereas it drops to about 0.8 Gbps under high interference. Despite the performance degradation, the system still satisfies the 1.0 Gbps QoS target with a probability of approximately 25% under high-interference conditions.

A deeper insight is that, even under ideal self-interference suppression (the leftmost region of the x-axis in Figure 4), the inherent stochastic fading of the channel itself constitutes another fundamental bottleneck. Therefore, to achieve the ultra-high-reliability communications envisioned for 6G, it is imperative to integrate advanced interference-cancelation techniques with more robust channel-enhancement technologies.

To evaluate the proposed algorithm’s performance, we compare it against the standard GWO, the GA, and RS. Qualitative analysis in Figure 6 shows that the proposed AM-MOGWO algorithm successfully produces a broad and continuous Pareto front, whereas all benchmark algorithms converge to a single point that is dominated by the AM-MOGWO front. This visually demonstrates AM-MOGWO’s ability to thoroughly explore the entire multi-objective solution space rather than prematurely converging to a single local optimum.

To quantitatively assess the overall quality of the solution sets, we adopt the HV metric, whose higher values indicate better comprehensive performance. Figure 7 presents the HV distributions of the four algorithms over 30 independent runs. In terms of performance, the median HV of AM-MOGWO is consistently around 1.18, whereas all benchmark algorithms fall within the 0.2–0.25 range. These gaps confirm that the Pareto front discovered by AM-MOGWO is of substantially higher quality. In terms of stability, AM-MOGWO’s boxplot is noticeably more compact, indicating that the algorithm exhibits highly consistent performance and strong robustness across multiple runs. Although the other algorithms also converge reliably, their HV values remain very low, reflecting the inherent limitation of single-point solutions under a multi-objective evaluation framework. Thus, the HV-metric comparison further corroborates the exceptional performance and high stability of the AM-MOGWO algorithm.

We also computed the IGD metric to gauge how closely the obtained solution set approximates the true Pareto-optimal front; lower IGD values indicate higher-quality solutions that lie closer to the optimal frontier. A lower value of this metric signifies a higher-quality solution set that lies closer to the true Pareto front. Figure 8 compares the IGD-value distributions of the algorithms. The median IGD value of AM-MOGWO is approximately 0.3, whereas the IGD values for the other three algorithms all exceed 0.8. This result indicates that, on average, the solution set identified by AM-MOGWO is the closest to the true Pareto front, demonstrating the best convergence among all methods. The consistently high and stable IGD values of standard GWO, GA, and RS precisely illustrate that their single-point solutions remain persistently and reliably distant from the full Pareto-optimal front. Although AM-MOGWO achieves the lowest mean IGD, its distribution spans a wider range, reflecting minor, normal fluctuations in the shape of the obtained front across runs. Nevertheless, given its orders-of-magnitude advantage in IGD values, AM-MOGWO still delivers the best overall performance in approximating the true Pareto-optimal front.

4.2. Robustness Validation in the Multi-Cell Interference Scenario

Figure 9 plots the generated Pareto Front against the average performance points of the three baseline algorithms. It is clear from the figure that the Pareto Front formed by AM-MOGWO completely dominates the average solutions of all baseline algorithms in the objective space. For a precise quantitative comparison, we analyzed the specific performance data of the baseline algorithms: the best-performing baseline was the Genetic Algorithm, with its average performance point located at a communication rate of 0.231 Gbps and a radar mutual information of 55.5 Mbps. However, at the same communication rate of 0.231 Gbps, the Pareto solutions offered by the AM-MOGWO algorithm can achieve a radar performance of over 150 Mbps, representing a performance gain of more than 170%. In contrast, the performance points of the other two baseline algorithms, Standard GWO and Random Search (approximately 49.7 Mbps and 49.5 Mbps, respectively), are dominated by AM-MOGWO with an even larger margin. More critically, the AM-MOGWO algorithm successfully delineates a clear performance trade-off boundary, providing decision-makers with a series of Pareto-optimal solutions to choose from to adapt to different task priorities. This stands in stark contrast to the baseline algorithms, which only converge to a single, suboptimal point, highlighting its fundamental advantage in solving multi-objective problems.

For a more precise quantitative analysis, Figure 10 presents the average performance of each algorithm on the HV metric. The proposed AM-MOGWO algorithm achieved the best average value of 7.931 × 10¹⁷, an improvement of approximately 0.35% compared to the second-best Random Search algorithm (7.903 × 10¹⁷), demonstrating its advantage in the coverage and breadth of the solution set.

Figure 11 presents the average performance of each algorithm on the IGD metric. The advantage of AM-MOGWO is even more significant on the IGD metric, which evaluates algorithm convergence. Its average IGD value was 0.511790, an improvement of approximately 3.7% compared to the second-best Genetic Algorithm (0.531435). This result demonstrates AM-MOGWO’s superior ability to guide the search process towards the true Pareto front.

5. Conclusions

This paper investigates the dynamic resource allocation problem for FD ISAC systems in 6G networks. We analyze the trade-off between communication throughput and sensing accuracy, which, under severe SI, becomes a complex non-convex MOP. To solve this, we propose an efficient and robust intelligent optimization algorithm, AM-MOGWO.

The proposed AM-MOGWO enhances the standard GWO by integrating OBL and Lévy flight strategies. This approach improves the initial population’s quality and diversity and strengthens the algorithm’s global search capability. The algorithm also incorporates chaotic search and memetic computing to intensify local refinement around elite solutions. An adaptive switching mechanism integrates these strategies, ensuring a dynamic balance between global exploration and local exploitation.

We validated the algorithm on a 3GPP-compliant, high-fidelity simulation platform against benchmarks, including standard GWO, GA, and RS. The results highlight AM-MOGWO’s superior performance. For the HV metric, AM-MOGWO achieved a median value of approximately 1.18, over 93% higher than the runner-up’s 0.25. The IGD metric was 0.3, far surpassing the benchmarks, all of which scored above 0.8. These findings confirm the algorithm’s excellent convergence and diversity in approximating the Pareto front.

6. Future Works and Outlook

This study provides a robust theoretical framework and an effective optimization algorithm for the FD-ISAC resource allocation problem. Future works can further enhance the framework’s practical applicability in several key directions.

A primary research direction is the evaluation of the proposed AM-MOGWO algorithm’s hardware feasibility and computational complexity. Real-time ISAC applications, such as vehicular networks, require low computational cost for practical deployment. Therefore, future studies should port the algorithm to specific hardware platforms like FPGAs or embedded GPUs. This would enable a quantitative analysis of its runtime, power consumption, and resource utilization in real physical systems.

Another important direction is extending the framework to more complex dynamic scenarios to comprehensively verify its universality. This includes high-speed mobility environments, such as high-speed rail or drone communications, to study the impact of fast time-varying channels and large Doppler shifts. It also involves exploring the algorithm’s scheduling strategies and scalability in dense user environments to meet the demands of massive device access.

The multi-cell interference model established in this study serves as a foundation for future network-level ISAC research. Based on this model, advanced technologies like Coordinated Multi-Point (CoMP) or Inter-Cell Interference Coordination (ICIC) can be further investigated. The aim is to proactively manage inter-cell interference, breakthrough performance bottlenecks, and improve the overall efficiency of the entire cellular network.