Firefly-Analogy Ensemble (FAABE)

• The paper demonstrates that integrating the Firefly Algorithm with Analogy-Based Estimation reduces MMRE by up to 80% across varied software datasets.
• FAABE leverages a population-based metaheuristic to dynamically adjust feature weights, improving estimation accuracy through systematic error minimization.
• The framework adapts historical project similarities with optimized weight vectors, though it introduces additional computational overhead due to iterative convergence.

The Firefly-Analogy Ensemble (FAABE) is a hybrid computational framework for software effort estimation, integrating Analogy-Based Estimation (ABE) with the Firefly Algorithm (FA) to optimize feature weighting adaptively. This approach systematically addresses the inherent limitations of fixed-weight similarity functions in traditional ABE by introducing metaheuristic-driven learning and error minimization, resulting in substantial accuracy improvements across diverse software project datasets (Chintada et al., 29 Nov 2025).

1. Firefly Algorithm Component

The Firefly Algorithm (FA) incorporated in FAABE is a population-based, nature-inspired metaheuristic. Each firefly encodes a candidate weight vector $w = (w_1, ..., w_k)$ corresponding to the relative importance of features within the ABE similarity space. The algorithm models social behavior based on attractiveness proportional to brightness, which in turn is a function of the objective function to be optimized—here, predictive accuracy.

• Light Intensity and Attractiveness:

Light intensity at distance $r$ from the source is $$I(r)=I_0\,e^{-\gamma r^2}\quad \text{or} \quad I(r)=\frac{I_0}{1+\gamma r^2}$$ with absorption coefficient $\gamma>0$ and initial intensity $I_0$. Attractiveness between two fireflies declines with distance:

$\beta(r) = \beta_0\,e^{-\gamma r^2}$

where $\beta_0$ sets the baseline attractiveness.

• Distance and Movement:

The Euclidean distance between fireflies $i$ and $j$ is

$$r_{ij} = \|x_i - x_j\| = \sqrt{\sum_{d=1}^k (x_{i,d} - x_{j,d})^2}$$

Firefly $i$ updates its position if it is less bright:

$$x_i \leftarrow x_i + \beta_0\,e^{-\gamma r_{ij}^2}(x_j - x_i) + \alpha(\mathrm{rand} - \tfrac{1}{2})$$

with randomization $\alpha$. Each firefly’s brightness is governed by the inverse of an error metric (e.g., MMRE).

• Optimization Process:

Over $T$ iterations, fireflies migrate in the parameter space, continually updating weight vectors to reduce estimation error. The convergence criterion is satisfied when further iterations yield negligible improvement, with the brightest firefly yielding the optimal weight configuration for ABE.

2. Analogy-Based Estimation (ABE) Model

ABE estimates effort for a target project by reference to historical analogues, utilizing feature-based similarity and a solution adaptation strategy:

• Workflow:

1. Collect and preprocess historical project cases.
2. Extract relevant features.
3. Compute pairwise similarities between the target and historical cases using a weighted similarity function.
4. Adapt efforts from most similar cases for prediction.

• Similarity Functions:
  • Weighted Euclidean similarity:

  $$\mathrm{Sim}(p,p') = \frac{1}{\sqrt{\sum_{i=1}^k w_i\,\mathrm{Dis}(a_i,a_i')+\delta}}$$ - Weighted Manhattan similarity:

  $$\mathrm{Sim}(p,p') = \frac{1}{\sum_{i=1}^k w_i\,\mathrm{Dis}(a_i,a_i')+\delta}$$

Here, $\mathrm{Dis}(a_i,a_i')$ is $|a_i-a_i'|$ for numeric/ordinal features and $\{0,1\}$ for nominal; $\delta=10^{-4}$ regularizes the denominator.

• Solution Adaptation:

The inverse weighted mean (IWM) strategy is commonly employed:

$$\hat C(p) = \sum_{k=1}^S \frac{\mathrm{Sim}(p,p_k)}{\sum_{i=1}^S \mathrm{Sim}(p,p_i)} C_{p_k}$$

where $S$ denotes selected analogies and $C_{p_k}$ their known efforts.

3. Integration of FA with Analogy-Based Estimation

FAABE’s hybridization occurs by embedding the FA as a meta-level optimizer for the similarity weights used by ABE:

• Objective Function:

Fireflies' brightness is linked to prediction error. Common objective metrics include:

$$\mathrm{MMRE}(w) = \frac{1}{N}\sum_{i=1}^N\left|\frac{A_i - \hat E_i(w)}{A_i}\right|,\qquad \mathrm{MAE}(w) = \frac{1}{N}\sum_{i=1}^N|A_i-\hat E_i(w)|$$

$$\mathrm{MSE}(w) = \frac{1}{N}\sum_{i=1}^N (A_i-\hat E_i(w))^2, \qquad \mathrm{RMSE}(w) = \sqrt{\mathrm{MSE}(w)}$$

with $A_i$ the actual and $\hat E_i(w)$ the predicted effort.

• Iterative Feature Selection and Optimization:

Initial features are filtered by Pearson correlation ($\rho \geq 0.5$) to eliminate irrelevant predictors. The weight vector for remaining features is encoded within each firefly. Iteratively, the algorithm seeks weights minimizing the selected error metric, with firefly updates as outlined in the FA component.

4. Empirical Evaluation and Comparative Results

Experiments assess FAABE and standard ABE across multiple publicly available datasets, processed and normalized as per protocol. The datasets are characterized as follows:

Dataset	Projects	Features
COCOMO81	64	16
Desharnais	81	12
China	499	14
Albrecht	24	8
Kemerer	15	7
Maxwell	62	27

After feature normalization and correlation-based selection, the models are evaluated on MMRE, MAE, MSE, and RMSE. A representative subset of results is as follows:

Dataset	Method	MMRE ↓	MAE ↓	MSE ↓	RMSE ↓
COCOMO81	ABE	3.2072	723.70	5.04×10⁶	2233.3
	FAABE	0.7188	62.24	3.63×10⁶	1905.8
Desharnais	ABE	0.7264	1938.1	2.70×10⁷	2629.5
	FAABE	0.4270	1283.6	2.79×10⁶	1670.2
Kemerer	ABE	0.3497	75.86	9.10×10³	94.31
	FAABE	0.1371	31.42	2.46×10³	49.60

Across all evaluated datasets, FAABE achieves consistent reductions in MMRE (typically by 40–80%), as well as improvements in MAE, MSE, and RMSE relative to baseline ABE.

5. Practical Implications and Limitations

FAABE provides notable advantage when baseline ABE incurs degraded performance stemming from suboptimal or nonuniform feature weighting, as is common in heterogeneous or noisy datasets. The metaheuristic search systematically tunes feature weights towards global error minimization.

However, the approach does introduce computational overhead: standard deployments utilize $N \approx 20$–50 fireflies over $T \approx 100$–200 iterations, translating to several minutes on medium-sized datasets. Parameter selection, particularly the randomization factor $\alpha$ and the absorption coefficient $\gamma$, is critical. A large $\alpha$ introduces excessive stochasticity and hinders convergence; an excessively small value risks stagnation in local minima. The absorption coefficient $\gamma$ governs the exploration–exploitation balance and is dataset-dependent.

6. Summary and Outlook

The Firefly-Analogy Ensemble (FAABE) realizes a systematic hybridization of analogy-based estimation with metaheuristic optimization, yielding substantial improvements in software effort estimation error metrics at moderate computational expense (Chintada et al., 29 Nov 2025). By optimizing feature similarity weights dynamically, FAABE offers robust performance across varied datasets, particularly where traditional ABE is sensitive to feature scale and noise. This suggests that metaheuristic-guided weight adaptation may be a generally useful paradigm in other nonparametric estimation regime subject to similar feature-weighting challenges. Further research may address more sophisticated adaptation, automated parameter selection, and extensions to multidimensional target variables.