Consider a set of N computational tasks. Task i is characterized by

where D i denotes the input data size, C i the required number of CPU cycles, τ i the maximum tolerated latency, and E i max the maximum device-side energy allowed for the task.

The execution decision for task i is represented by the ternary variable

where a i = 0 denotes local execution, a i = 1 MEC execution, and a i = 2 cloud execution. The complete offloading solution is denoted by

For local execution, let f i denote the CPU frequency of device i , and let κ be the effective switching-capacitance coefficient. The corresponding execution latency and energy consumption are

and

For offloaded tasks, the total uplink bandwidth B is shared equally among all tasks assigned to either MEC or cloud. Let N o f f ( A ) denote the number of offloaded tasks under assignment A . The bandwidth allocated to each offloaded task is therefore

The corresponding uplink transmission rate is

where P i is the transmit power of device i , h i is the channel power gain, and N 0 is the noise power spectral density. The uplink transmission time is then

For MEC execution, let F M E C denote the aggregate MEC computing capacity and N M E C ( A ) the number of tasks assigned to the MEC layer. Equal CPU sharing is assumed, giving

The MEC execution latency and device-side energy consumption are

and

For cloud execution, a fixed backhaul delay T b h and an effective per-task cloud computing rate f c l o u d are considered. The cloud CPU rate is not shared among cloud-assigned tasks; therefore, cloud-side CPU contention is not explicitly modeled. The resulting latency is

While the corresponding device-side energy consumption is

The result-download time is neglected because the returned output is assumed to be small relative to the uploaded task data. Energy consumption is evaluated from the IoT-device perspective; MEC and cloud infrastructure energy are, therefore, outside the scope of the present model. Future work will extend the model to include downlink transmission delay and infrastructure-side energy consumption at the MEC and cloud layers.

A key consequence of this formulation is that offloading decisions are mutually dependent. Changing the execution layer of one task may alter both the bandwidth available to all offloaded tasks and the computing capacity available to MEC-assigned tasks.

For a given assignment A , the latency T i ( A ) and energy E i ( A ) of task i correspond only to the execution layer selected by a i . A task is considered successfully served when it satisfies both its latency and device-side energy constraints:

The Quality of Experience (QoE) satisfaction ratio is then defined as

Because QoE is discrete, several different assignments may obtain the same satisfaction ratio. To distinguish such solutions during the search, a secondary normalized latency-energy term is introduced:

The optimization score is defined as

where η controls the contribution of the secondary normalized latency-energy term to the scalar optimization score. For η > 0 , candidate assignments are ranked according to the combined score S , and an assignment with slightly lower QoE may therefore be preferred if the latency-energy contribution is sufficiently favorable. The sensitivity of the results to η is examined in Section 5.2. In the final configuration selected on the disjoint validation seeds, η = 0 ; consequently, the final optimization runs rank candidate assignments according to QoE alone. Mean latency and mean device-side energy are reported separately as performance metrics.

MFO and PFOA operate in continuous search spaces, whereas the offloading problem considered here is ternary. To avoid introducing an implicit initialization bias toward any execution layer, each candidate solution is represented by a continuous vector X r = [ x r , 1 , ⋯ , x r , N ] , with x r , j ∈ [ 0 , 3 ) , and decoded only when its fitness is evaluated:

Accordingly, the intervals [ 0 , 1 ) , [ 1 , 2 ) , and [ 2 , 3 ) correspond respectively to local, MEC, and cloud execution. Since the three intervals have equal widths, uniform initialization over [ 0 , 3 ) gives the three execution layers equal initial selection probability.

The MFO backbone follows the search structure reported in [27]. Candidate positions are initialized uniformly over [ 0 , 3 ) . During exploration, each candidate is guided by a randomly selected population member with a better score; when no better member exists, the current global best is used instead. The resulting candidate is decoded and accepted only if its optimization score does not decrease.

The exploration update is

where u r , j ~ U ( 0 , 1 ) , I r , j ∈ { 1 , 2 } , and S S r , j denotes the j -th component of the selected guiding solution.

During exploitation, the candidate moves toward the current global best according to

where x j b e s t is the j -th component of the current best solution and t is the iteration index. As the search progresses, the 1 / t factor reduces the step magnitude and promotes progressively finer exploitation around the incumbent best solution.

The hybrid component introduces a separate refinement stage based on two mechanisms derived from PFOA [26]: nonlinear control and reverse-evasion direction. Rather than modifying the MFO population dynamics, this stage generates K peer-guided alternatives around the current best solution.

The nonlinear control factor is defined as

where T is the maximum number of iterations.

For each refinement trial, a peer solution X r is selected from the population, and a component-wise random sign vector s ∈ { − 1 , + 1 } N is generated. The refined candidate is computed as

where ⊙ denotes element-wise multiplication and γ controls the refinement amplitude.

Each candidate generated by (20) is decoded using (16) and evaluated with the objective function defined in Section 3.3. The best refinement replaces the incumbent only if it strictly improves S ( A ) ; otherwise, the current best solution is retained.

The term X r − X b e s t provides a search direction informed by the current population rather than by an independent random perturbation. The sign vector s can reverse this direction independently across dimensions, thereby increasing the diversity of the refinement candidates. At the same time, α t decreases as the search progresses, so the refinement evolves from broader perturbations in early iterations toward a more localized search near convergence. The scale factor γ controls the overall strength of this mechanism and is selected using validation seeds that are separate from the final test set.

This design keeps the MFO search structure intact while using the PFOA-derived mechanism only as a selective refinement step. Its contribution can therefore be assessed directly through comparison with the MFO-only baseline. The complete procedure is summarized in Algorithm 1.

The main experiment considers 400 users, with one computational task generated

Algorithm 1.Proposed procedure.

for each user. For each random seed, a single workload instance is generated and reused across all methods to ensure paired comparisons under identical conditions. Each workload instance is treated as a static scheduling snapshot in which all considered tasks are assumed to be available within the same scheduling interval. Dynamic task arrivals and departures during the interval are not modeled.

Unless otherwise stated, task computation demand is sampled from a discrete uniform distribution between 100 and 1000 million CPU cycles, and input data size from a discrete uniform distribution between 50 and 500 KiB. The latency requirement is sampled from U ( 0.05 , 1.0 ) s, the device-side energy requirement from U ( 0.1 , 5.0 ) J, and the local CPU frequency from U ( 0.5 , 2.0 ) GHz. Channel gain is sampled in dB from U ( − 110 , − 60 ) and then converted to linear power gain. These distributions define a controlled synthetic workload and are not intended to reproduce measured application traces. The main simulation settings are summarized in Table 1.

Table 1. Main simulation settings.

The algorithmic parameters were examined using eight validation seeds (200-207), which are completely disjoint from the 30 seeds used in the final evaluation. The scale factor γ was evaluated at 0.2, 0.5, 0.8, 1.1, and 1.4; the number of refinement trials K at 4, 8, 12, and 16; and the weight η of the secondary latency-energy term at 0, 0.01, 0.02, and 0.05.

Table 2 reports the mean QoE and mean latency obtained for each tested parameter value on the validation seeds. Among the tested values, γ = 1.1 , K = 16 , and η = 0 yielded the highest mean QoE in their respective validation comparisons. These values were therefore fixed before the final 30-seed evaluation. The analysis is intended to assess parameter robustness and select a reasonable operating configuration rather than to perform an exhaustive hyperparameter search.

Table 2. Parameter sensitivity on validation seeds 200 - 207.

MFO is used as the direct ablation baseline because it corresponds to the population-search backbone without the selective refinement stage. HGSTO is implemented as a source-guided adaptation of the genetic-algorithm and simulated-annealing structure reported in [24] and mapped to the same ternary Device–MEC–Cloud decision space. The implementation uses tournament selection with tournament size 3, uniform crossover with a component-wise probability of 0.5, 10% elitism, an initial mutation probability of 0.30 that decreases linearly over generations, four hill-climbing trials when mutation is activated, and simulated-annealing refinement with initial temperature T 0 = 0.05 , cooling factor 0.90, and five neighborhood steps per generation. Discrete PSO uses integer-valued positions, inertia weight ω = 0.729 , cognitive and social coefficients c 1 = c 2 = 1.49445 , and velocities rounded and clipped to the interval [ − 2 , 2 ] before position updates. Greedy EDF (Earliest-Deadline-First) serves as a deterministic low-cost reference: tasks are processed in increasing deadline order, and each task is assigned to the execution layer that yields the best current global score among the three alternatives.

The main stochastic comparison uses a matched target budget of 11,250 fitness evaluations (NFE). The proposed method, MFO, HGSTO, and discrete PSO each use exactly 11,250 evaluations. Greedy EDF requires only 1200 evaluations and is therefore treated as a low-cost deterministic reference rather than as an equal-budget population-based competitor.

All stochastic methods are evaluated on the same 30 workload seeds (0 - 29). For workload seed s , PiFBOA and MFO use solver seed 900000 + s ; PiFBOA additionally uses 1900000 + s for the refinement stage, while HGSTO and discrete PSO use 3000000 + s and 4000000 + s , respectively. Thus, the same workload instance is reused across methods, while solver randomizations remain independent. Paired t-tests and Wilcoxon signed-rank tests are used for pairwise comparisons. Holm correction is applied across the four proposed-method-versus-baseline comparisons for each reported metric. In addition, 95% confidence intervals for paired differences and Cohen’s d z are reported. Convergence is plotted against the number of fitness evaluations rather than iteration count so that search progress is compared on a common computational-effort axis.

Table 3 summarizes the main comparison over the 30 paired 400-user workloads. Under the matched search budget of 11,250 fitness evaluations, PiFBOA achieves a mean QoE of 0.5769, compared with 0.5719 for MFO, 0.5242 for HGSTO, and

Table 3. Main comparison over 30 paired 400-user workloads.

0.3643 for discrete PSO. Greedy EDF achieves a mean QoE of 0.5383 using only 1200 evaluations. PiFBOA and MFO exhibit nearly identical mean latency, at 0.4926 s and 0.4927 s respectively, while their mean device-side energy consumptions are also very close, at 0.0933 J and 0.0936 J.

Figure 1 compares convergence using the number of fitness evaluations as a common search-effort axis. PiFBOA maintains a small but persistent advantage over MFO after the early stage of the search. HGSTO improves more gradually, whereas discrete PSO converges to a lower QoE. Using NFE rather than iteration count provides a more comparable view of search progress across methods with different internal update structures.

Figure 1.Mean best QoE versus number of fitness evaluations.

Table 4 reports the paired QoE comparisons between PiFBOA and each comparator. The mean QoE difference between PiFBOA and MFO is 0.0050, with a 95% confidence interval of [0.0023, 0.0077]. The difference is statistically significant under both the paired t-test (p = 7.31 × 10⁻⁴) and the Wilcoxon signed-rank test (p = 7.29 × 10⁻⁴), and remains significant after Holm correction. In contrast, the differences in mean latency and device-side energy are not statistically significant, with Holm-adjusted p-values of 0.960 and 0.352, respectively.

Table 4. Paired QoE comparisons: PiFBOA minus comparator.

Table 5. Paired latency and energy comparisons: PiFBOA minus comparator.

As shown in Table 5, the latency and device-side energy differences between PiFBOA and MFO are not statistically significant after Holm correction. In contrast, PiFBOA shows statistically significant differences relative to HGSTO, discrete PSO, and Greedy EDF for both metrics.

To assess solution quality independently of the selected baselines, exhaustive search was performed on ten separate 9-task instances, for which exact enumeration of the ternary search space remains computationally feasible.

As summarized in Table 6, PiFBOA and MFO reached the exact optimum in all ten 9-task instances. Greedy EDF matched the exact optimum in seven of the ten instances. These results show that both stochastic optimizers consistently recover the exact QoE optimum on the tractable small-scale test cases, whereas the greedy heuristic does not always reach the optimum.

Table 6. Small-scale exhaustive validation (9 tasks, 10 seeds).

This experiment does not establish global optimality for the 400-user problem. Nevertheless, it shows that the implementation can recover the exact optimum on tractable instances and provides an additional sanity check on the search procedure.

The results indicate that the selective refinement mechanism provides a measurable improvement over the MFO backbone under the considered resource-coupled setting. Because offloaded tasks share uplink bandwidth and MEC processing capacity, individual decisions affect the resources available to other tasks. Within this coupled environment, the refinement stage improves QoE relative to MFO, while the differences in mean latency and device-side energy are not statistically significant.

The improvement over MFO remains modest, which is consistent with the design of the proposed method: MFO is intentionally retained as the main population-search mechanism, while the additional refinement stage acts only on the incumbent best solution. The empirical contribution should therefore be interpreted as evidence that this selective hybridization can improve search quality under the considered offloading model, rather than as evidence of a fundamentally new search paradigm.

The study also has several limitations. Workload parameters are synthetically generated and uniformly sampled rather than derived from measured application traces. Validation using measured IoT application traces and measured 5G channel conditions is left for future work. Uplink bandwidth and MEC computing capacity are shared equally instead of being jointly optimized. Future work will jointly optimize the discrete offloading decisions together with continuous communication and computing-resource allocation. Result-download time is neglected, cloud backhaul delay is fixed, and the energy model covers only the IoT device rather than MEC or cloud infrastructure. HGSTO is evaluated through a source-guided adaptation rather than the authors’ original implementation. Finally, exhaustive enumeration is practical only for small task counts, so optimality gaps at the 400-user scale remain unknown.