Chaotic Gaussian-based Global ARO

• The paper introduces Chaotic Gaussian-based Global Artificial Rabbits Optimization (CGG-ARO) which enhances traditional ARO through chaotic initialization, Gaussian exploration, and global exploitation strategies.
• It combines structured mission assignment with a detailed ITS model, integrating route, dependency, timing, and budget constraints to optimize vehicle-task scheduling.
• Empirical results show that CGG-ARO improves completed missions and total benefits by around 7% compared to baseline methods, establishing it as a high-quality offline benchmark.

Searching arXiv for the specified papers and closely related work to ground the article. First, I’ll retrieve the cited CGG-ARO paper. Chaotic Gaussian-based Global ARO (CGG-ARO) is a metaheuristic optimization method introduced as the baseline one-slot optimizer within the Oranits framework for mission assignment and task offloading in an Open RAN-based intelligent transportation system. In that setting, it is defined as Chaotic Gaussian-based Global Artificial Rabbits Optimization, not as a variant of other algorithms that share the acronym ARO. Its purpose is to solve a constrained mission assignment and ordering problem at the cloud-side Non-RT layer by combining chaotic initialization, Gaussian exploration, and modified global exploitation and hiding strategies in order to improve the exploration–exploitation balance of Artificial Rabbits Optimization (Nguyen et al., 25 Jul 2025).

1. Definition and functional role

CGG-ARO is presented as an enhanced variant of Artificial Rabbits Optimization for the Oranits system. The enhancement is organized around three modifications: a chaotic initialization based on a Piecewise Chaotic Map (PCM), a Gaussian-based exploration step, and a global exploitation mechanism that combines opposition-based learning with attraction to the current best solution. Within Oranits, the algorithm operates as a metaheuristic baseline for one-slot optimization at the cloud, whereas the corresponding deep reinforcement learning method, MA-DDQN, is designed for faster adaptive decision-making (Nguyen et al., 25 Jul 2025).

Its operational context is a time-slotted intelligent transportation system in which autonomous vehicles cooperate with mobile edge computing infrastructure. Time is divided into periods of length $\tau$, missions arriving during a period are grouped into subsets of size $Z$ in a matrix $\mathbf{M}(\tau)$, and each subset is optimized separately. For a given subset, CGG-ARO determines which vehicle executes which mission and in what order missions are processed on each vehicle. The paper explicitly distinguishes this from server-selection logic: offloading itself follows a greedy policy, so CGG-ARO does not optimize which server a task is sent to. Instead, it searches over assignment and ordering decisions whose consequences propagate through the delay and budget model.

The objective of the resulting optimization is to maximize the number of missions completed before deadline under route, dependency, timing, and budget constraints. In the notation of the paper, the one-slot optimization problem is

$\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$

This objective makes CGG-ARO a feasibility-aware scheduler whose search space is combinatorial, dependency-constrained, and indirectly coupled to communication and computation costs.

2. Optimization model and constraint structure

The optimization model solved by CGG-ARO is defined over missions, vehicles, and mission-task attributes. For a given period $\tau$, a mission subset contains $Z$ missions denoted $M_i(\tau)$, each written as

$$M_i(\tau)\triangleq \left\langle r_i(\tau),\, T_i(\tau),\, B_i(\tau),\, \mathcal{M}_i^{-},\, \mathcal{M}_i^{+} \right\rangle,$$

where $r_i(\tau)$ is the route, $T_i(\tau)$ the deadline, $Z$0 the offloading budget, and $Z$1 and $Z$2 the predecessor and successor mission sets. Each mission contains tasks $Z$3, with each task represented by input size and CPU demand as $Z$4. The decision variables are the assigned vehicle $Z$5 and the execution order $Z$6 of mission $Z$7 (Nguyen et al., 25 Jul 2025).

The constraints enforce a tightly structured schedule. A fixed number $Z$8 of vehicles is selected for each mission subset. Every mission must be assigned to exactly one vehicle. Each mission can occupy at most one order position, and missions assigned to the same vehicle must have unique orders. Predecessor and successor relations impose order consistency:

$Z$9

and

$\mathbf{M}(\tau)$0

The temporal model couples routing, communications, computation, and queueing. Communication rate from vehicle $\mathbf{M}(\tau)$1 to server $\mathbf{M}(\tau)$2 is modeled as

$\mathbf{M}(\tau)$3

with corresponding communication delay, optional fiber delay for cloud offloading, and total mission communication delay $\mathbf{M}(\tau)$4. Computation delay is

$\mathbf{M}(\tau)$5

and travel delay is

$\mathbf{M}(\tau)$6

The mission delay in isolation is then

$\mathbf{M}(\tau)$7

Completion time is lower-bounded by own delay plus same-vehicle predecessors plus dependency-induced waiting across vehicles. Budget feasibility is also explicit: with time-based cost rate $\mathbf{M}(\tau)$8, offloading cost is

$\mathbf{M}(\tau)$9

and the remaining budget must satisfy

$\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$0

This formulation makes the search problem more than a pure assignment problem. A candidate solution must reconcile queue positions, mission dependencies, mobility-induced travel time, communication and computation latency, and per-mission budget limits.

3. Solution encoding and evolutionary cycle

Each CGG-ARO individual, or rabbit, encodes one candidate solution for a single mission subset. The representation is

$\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$1

where $\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$2 is a permutation of the $\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$3 mission indices and $\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$4 is a vehicle-index vector of length $\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$5. The paper specifies that

$\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$6

and that each vehicle index in $\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$7 appears exactly $\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$8 times in $\mathscr{P}_{1}:\ \underset{\mathbf{D}(\tau)}{\max}\ \sum_{M_i(\tau)\in \mathbf{M}(\tau)} \mathds{1}_{\{\delta_i(\tau)\le T_i(\tau)\}}.$9, thereby enforcing the intended balanced use of the selected vehicles (Nguyen et al., 25 Jul 2025).

From this encoding, the assignment-order matrix row $\tau$0 is constructed by pairing the vehicle identity at each permutation position with the cumulative count of that vehicle’s prior appearances in the sequence. Operationally, the permutation determines which mission is considered at each position, the vehicle vector determines to which vehicle it is assigned, and the cumulative count induces the within-vehicle order. In this way, a single continuous-space search vector can be mapped into a discrete structured schedule.

Fitness evaluation uses the deadline-satisfaction objective directly:

$\tau$1

The paper also reports “completed missions” and “total benefits” as simulation metrics, although the exact scalar expression used for the reported fitness is not reproduced in the detailed summary. This suggests that the implementation evaluates both deadline-based feasibility and broader system benefit, even though the primary optimization target is the number of missions completed on time.

The iterative cycle follows the standard population-based structure of Artificial Rabbits Optimization. A population of $\tau$2 individuals is initialized, fitness values are computed, and the best-so-far solution $\tau$3 is tracked. At each generation, each rabbit is updated according to the current energy factor $\tau$4, which governs the switch between exploration and exploitation. The better of the old and new individuals is retained, and the global best is updated after processing the population.

4. Chaotic, Gaussian, and global mechanisms

The defining features of CGG-ARO are its modifications to initialization, exploration, and exploitation. The first is chaotic initialization through a Piecewise Chaotic Map. For a normalized scalar $\tau$5, the update is

$\tau$6

with $\tau$7. The stated effect is stronger mixing, non-repetition, and better diversity than pure random initialization, after which the resulting values are mapped back into valid mission and vehicle index ranges (Nguyen et al., 25 Jul 2025).

The second defining feature is Gaussian-based exploration. When the ARO energy factor indicates exploration and a Bernoulli choice selects the Gaussian branch, the update is

$\tau$8

where $\tau$9 is a binary random vector and

$Z$0

Because the perturbation scale is population-standard-deviation driven, exploration amplitude grows with diversity and shrinks as the population contracts. This creates an endogenous transition from wide search to finer search without introducing a separate annealing schedule.

The third feature is the global exploitation mechanism. One branch uses opposition-based learning plus attraction toward the incumbent best:

$Z$1

where

$Z$2

and the stochastic weight $Z$3 is

$Z$4

The opposition term promotes movement toward the opposite region of the search space relative to the current point, while the best-attraction term promotes convergence.

During the hiding or exploitation phase, CGG-ARO may also use an improved random hiding update:

$Z$5

for randomly chosen distinct indices $Z$6 and $Z$7. This turns hiding into a guided local search centered on the best solution but mediated by other individuals. The combined effect of PCM, Gaussian perturbation, opposition-based movement, and improved hiding is explicitly presented as a response to premature convergence, limited global search capability, and sensitivity to the initial population in the original ARO.

5. Empirical results and computational profile

In the reported experiments, CGG-ARO is compared with original ARO, SHADE, L-SHADE, Equilibrium Optimizer, and Artificial Protozoa Optimizer. All methods use a population of $Z$8, run for $Z$9 iterations, and are repeated 15 times with different random seeds. On the reported benchmark, CGG-ARO achieves the best average values among all metaheuristics for fitness, completed missions, and total benefits: fitness $M_i(\tau)$0, completed missions $M_i(\tau)$1, and total benefits $M_i(\tau)$2. The corresponding ARO results are $M_i(\tau)$3, $M_i(\tau)$4, and $M_i(\tau)$5, respectively. APO, SHADE, L-SHADE, and EO all report lower means on the same metrics (Nguyen et al., 25 Jul 2025).

The paper’s aggregate summary states that CGG-ARO improves the number of completed missions and overall benefit by approximately 7.1% and 7.7%, respectively. The same study reports that MA-DDQN obtains larger aggregate improvements, namely 11.0% in completed missions and 12.5% in overall benefit. Across 15 mission sets, MA-DDQN performs better in most dynamic scenarios, but CGG-ARO surpasses it on some structured sets, specifically Sets 4, 5, 8, 11, and 15 in terms of missions or fitness. This pattern indicates that the evolutionary search remains competitive on some static instances even when learning-based control performs better on average.

The computational profile is a major part of the algorithm’s empirical characterization. The reported worst-case time complexity is

$M_i(\tau)$6

where $M_i(\tau)$7 is approximately $M_i(\tau)$8 and $M_i(\tau)$9 is the fitness-evaluation cost. The summary emphasizes that $$M_i(\tau)\triangleq \left\langle r_i(\tau),\, T_i(\tau),\, B_i(\tau),\, \mathcal{M}_i^{-},\, \mathcal{M}_i^{+} \right\rangle,$$0 dominates in practice because it includes delay calculations, dependency checks, and budget evaluation. For 1000 iterations, the reported runtime is approximately 100–180 seconds per iteration and around 2300 minutes total for a mission set. The paper therefore characterizes CGG-ARO as too slow for real-time ITS operation, which is precisely why it is positioned as an offline or infrequent cloud-side baseline rather than the online controller.

6. Interpretation, limitations, and acronym ambiguity

CGG-ARO’s strengths are tied directly to its design choices. Chaotic initialization is used to improve diversity and reduce dependence on the initial random seed. Gaussian exploration scales automatically with current population variance, which supports stronger early search and smoother late-stage refinement. Opposition-based learning and improved hiding increase the use of global information, especially the incumbent best solution, while retaining stochasticity. In the reported experiments, these mechanisms are associated with higher median performance, better convergence, and stronger average solution quality than the compared metaheuristics (Nguyen et al., 25 Jul 2025).

Its limitations are equally explicit. The algorithm has high computational cost, requiring continuous-to-discrete mapping, boundary handling, rounding, and repair to maintain valid permutations and balanced vehicle counts. Performance variability remains visible in the reported standard deviations, and learning-based control surpasses it in aggregate on dynamic mission sets. A plausible implication is that CGG-ARO is best understood as a high-quality reference optimizer for difficult static snapshots rather than as a deployable real-time policy.

A recurrent source of confusion is the acronym ARO itself. In the Oranits paper, CGG-ARO expands to Chaotic Gaussian-based Global Artificial Rabbits Optimization (Nguyen et al., 25 Jul 2025). By contrast, the portfolio-optimization paper “Markowitz-based cardinality constrained portfolio selection using Asexual Reproduction Optimization” uses ARO to mean Asexual Reproduction Optimization in an unrelated mean–variance portfolio-selection context (Mansouri et al., 2021). The shared acronym does not indicate a shared algorithmic lineage. Another common misconception is that CGG-ARO directly optimizes the task offloading destination; the paper states instead that offloading decisions are handled greedily, while CGG-ARO optimizes mission assignment and execution order under the induced communication, computation, and budget model.

Within the Oranits architecture, CGG-ARO therefore occupies a specific methodological niche: it is the cloud-side evolutionary engine for one-slot mission scheduling under dependencies, budgets, and mobility-induced delay, and it serves primarily as a strong offline baseline against which faster adaptive methods such as MA-DDQN are evaluated.