--- title: Randomized Memetic ABC (RMABC) Algorithm url: https://www.emergentmind.com/topics/randomized-memetic-artificial-bee-colony-rmabc type: topic --- # Randomized Memetic ABC (RMABC) Algorithm The Expert-Guided Memetic Walrus Optimizer (MWO) is an advanced evolutionary optimization algorithm designed for Adaptive Curriculum Sequencing (ACS), a multi-objective problem situated in personalized online learning environments. MWO integrates an expert-guided strategy with an agent aging mechanism, a nonlinear adaptive control signal framework, and a hierarchical educational priority mechanism to produce high-quality, stable, and contextually meaningful curriculum sequences. Empirical validation on the OULAD dataset and standard optimization benchmarks establishes its superior optimization stability, curriculum relevance metrics, and generalization to complex multi-objective scenarios [2506.13092]. ## 1. Multi-Objective Optimization Formulation MWO formulates ACS as a multi-objective binary selection problem, where the selection of educational materials for $T_s$ students and $T_m$ materials is encoded as $x\in\{0,1\}^{T_m}$. For each student, the overall loss function is \[ \min F(x) = \omega_1\,\mathcal{O}_1(x) + \omega_2\,\mathcal{O}_2(x) + \omega_3\,\mathcal{O}_3(x) \] where: - $\mathcal{O}_1$: concept coverage/redundancy, penalizing uncovered required concepts and minimally penalizing redundant coverage, \[ \mathcal{O}_1 = \varepsilon_1(|\mathcal{R}| - |\mathcal{R}\cap\mathcal{E}|) + \varepsilon_2(|\mathcal{E}| - |\mathcal{R}\cap\mathcal{E}|), \] with $\mathcal{R}=\bigcup_{j:x_j=1}Cm_j$, $\mathcal{E}=\bigcup_i C_i$. - $\mathcal{O}_2$: time constraint violation, \[ \mathcal{O}_2 = \begin{cases} \varepsilon_3, & \sum_{j=1}^{T_m}x_j\,Ts_j \notin [\underline{T},\overline{T}], \\ 0, & \text{otherwise}, \end{cases} \] - $\mathcal{O}_3$: learning-style compatibility, \[ \mathcal{O}_3 = \sum_{j=1}^{T_m} x_j \sum_{k=1}^4 |p_k - pm_{j,k}|, \] where $p_k$ is the $k$-th FSLSM coordinate of the student, $pm_{j,k}$ that of material $j$. Typical hyperparameter settings: $\omega_1=\omega_2=\omega_3=0.25$, $\varepsilon_1=1$, $\varepsilon_2=10^8$, $\varepsilon_3=10^3$. This combination strongly penalizes missing concepts, with lower cost for redundancy or time/style mismatches. ## 2. Expert-Guided Strategy and Aging Mechanism MWO maintains an expert pool comprising all agents, assigning each an exponentially decaying influence weight based on age-in-population. For agent $i$: \[ age_i \gets age_i + 1\ (\text{per iteration}),\quad age_i \gets 0\ \text{if }i\text{ is best or 2nd-best} \] \[ w_i = \begin{cases} 0, & age_i > 0.2T_{\max}, \\ \exp(-\lambda age_i), & \text{otherwise}, \end{cases}\quad \lambda=0.1 \] For updating, better-fitness experts $K=\{k| f_k < f_i\}$ are sampled with probability $P(k)=w_k/\sum_{j\in K}w_j$, and the update is \[ X_i^{new} = X_i + \mathrm{rand}(0,1) \times w_k (X_k - I X_i),\quad I\in\{1,2\} \] This mechanism improves exploitation and avoids premature convergence by favouring recent successful search directions. ## 3. Adaptive Control Signal Framework MWO employs two nonlinear control signals per iteration: - **Danger signal $D$**: Promotes exploration in early search phases. \[ E_1(t) = 2(1 - t/T_{\max})^{\pi t/T_{\max}},\quad D = E_1 \cdot E_0,\ E_0 \sim \mathcal{U}(0,1) \] - **Safety signal $S$**: Gradually increases exploitation. \[ \beta(t) = 1 - \frac{1}{1 + \exp(\frac{T_{\max}/2 - t}{10/T_{\max}})},\quad S = \beta \cdot r_2,\ r_2 \sim \mathcal{U}(0,1) \] Update strategies: 1. If $|D|\ge 0.5$: Migration update. \[ \Delta X=(\beta r_3^2)(X_{\mathrm{rand}_1}-X_{\mathrm{rand}_2}) \] 2. Else if $S\ge 0.5$: - Top-$P\%$ ("males"): Halton-based global moves. - Others: Local search around $X_{best}$. 3. Else: Combine best and second-best positions: \[ X_1 = X_{best} - a_1 \tan(\theta_1 \pi)|X_{best} - X_i| \] \[ X_2 = X_{second} - a_2 \tan(\theta_2 \pi)|X_{second} - X_i| \] \[ X_i = (X_1 + X_2)/2 \] This regime facilitates a dynamic and data-driven balance between global exploration and local refinement. ## 4. Three-Tier Curriculum Sequencing and Priority Assignment After selecting optimal material subsets, MWO applies a hierarchical sequencing: - **High priority ($P_i$):** Full concept coverage, within student ability, prioritized by prerequisite strength, \[ P_i = w_i \sum_{j\in Pre(i)} r_{ji} \] - **Medium priority ($M_i$):** Partial coverage, matched to ability, \[ M_i = \lambda d_i + (1-\lambda)\frac{t_i}{T_{\max}},\quad \lambda\in[0,1] \] - **Challenge set ($C_i$):** Slightly above ability, promotes progress, \[ C_i = \beta_d d_i + (1-\beta_d)\frac{|Pre(i)|}{|N|},\quad \beta_d = e^{-k/K} \] - **Final sequence score:** \[ S_i = \alpha_1 P_i + \alpha_2 M_i + \alpha_3 C_i,\quad \alpha_1 + \alpha_2 + \alpha_3 = 1 \] Materials are sorted by descending $S_i$, ensuring prerequisite constraints. ## 5. Algorithmic Structure and Computational Complexity MWO’s control flow follows the pseudocode: ``` Algorithm Memetic-Walrus-Optimizer (MWO) Input: N, Tmax, student set S, material set M Output: best solution X* and its sequence 1 Initialize population {X_i} (i=1…N), ages age_i=0 2 F_best←∞, F_second←∞ 3 for t=1 to Tmax do 4 Evaluate fitness F_i for each X_i 5 Update best/2nd‐best and reset age_i=0 6 age_i ← age_i+1 ; w_i ← exp(-λ·age_i) (or 0 if age_i>0.2Tmax) 7 Compute danger D and safety S signals 8 if |D|≥0.5 then 9 Migration update: X_i ← X_i + (β·r3^2)(X_rand1 - X_rand2) 10 else 11 for i=1…N do 12 Select expert k∈{k|F_k