Table 2 Notations used in NF-MORL model.

(\(\:{s}_{t}\))

State at time t, representing environmental conditions, fog node status, and multi-objective context parameters.

(\(\:{a}_{t}\))

Action at time t, representing the task assignment decision among fog nodes.

(\(\:{R}_{t}\))

Reward at time t, integrating weighted objectives of makespan, energy efficiency, cost, and fault tolerance.

(\(\:\gamma\:\))

Discount factor adjusting the relative importance of future aggregated multi-objective rewards.

(\(\:{L}_{actor}\))

Actor loss computed by multi-objective policy gradient methods to optimize actions across Pareto fronts.

(\(\:{L}_{critic}\))

Critic loss used to estimate and stabilize multi-objective value functions during training.

(T)

Task length representing the computational complexity of each job.

(P)

Task priority determined dynamically through neuro-fuzzy inference.

(\(\:CP{U}_{i}\))

Current CPU utilization of fog node i.

(\(\:Me{m}_{i}\))

Available memory of fog node i.

(\(\:B{W}_{i}\))

Bandwidth capacity at fog node i.

(\(\:{E}_{i}\))

Current energy level at fog node i.

(S)

CPU processing speed at each fog node.

(\(\:{C}_{f}\))

Cost factor representing computation and transmission costs per task.

(\(\:{F}_{f}\))

Fault factor representing node reliability and failure probability.

(\(\:{L}_{t}\))

Task length in seconds, indicating computational workload.

(\(\:{D}_{t}\))

Task deadline, defining the maximum acceptable completion time.

(\(\:{R}_{q}\))

Resource quality factor combining CPU, memory, and bandwidth performance indices.

(\(\:{W}_{k}\))

Weight coefficient assigned to objective k in the multi-objective reward function.

(\(\:{\mu\:}_{j}\), \(\:{\sigma\:}_{j}\))

Mean and variance of fuzzy membership functions for linguistic variable j.

(\(\:{f}_{j}\left(x\right)\))

Fuzzy membership function used in the neuro-fuzzy inference layer.

(\(\:{\theta\:}_{\pi\:}\))

Parameters of the multi-objective policy network.

(\(\:{\:\theta\:}_{v}\))

Parameters of the multi-objective value network.

(\(\:{\theta\:}_{nf}\))

Parameters of the neuro-fuzzy inference subsystem.

(\(\:{h}_{\pi\:}({s}_{t},\:{a}_{t})\))

Action–value function of the multi-objective actor network under neuro-fuzzy control.