Component-wise analysis of metaheuristic algorithms for novel fuzzy-meta classifier

Metaheuristic research has proposed promising results in science, business, and engineering problems. But, mostly high-level analysis is performed on metaheuristic performances. This leaves several critical questions unanswered due to black-box issue that does not reveal why certain metaheuristic algorithms performed better on some problems and not on others. To address the significant gap between theory and practice in metaheuristic research, this study proposed in-depth analysis approach using component-view of metaheuristic algorithms and diversity measurement for determining exploration and exploitation abilities. This research selected three commonly used swarm-based metaheuristic algorithms – Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), and Cuckoo Search (CS) – to perform component-wise analysis. As a result, the study able to address premature convergence problem in PSO, poor exploitation in ABC, and imbalanced exploration and exploitation issue in CS. The proposed improved PSO (iPSO), improved ABC (iABC), and improved CS (iCS) outperformed standard algorithms and variants from existing literature, as well as, Grey Wolf Optimization (GWO) and Animal Migration Optimization (AMO) on ten numerical optimization problems with varying modalities. The proposed iPSO, iABC, and iCS were then employed on proposed novel Fuzzy-Meta Classifier (FMC) which offered highly reduced model complexity and high accuracy as compared to Adaptive Neuro-Fuzzy Inference System (ANFIS). The proposed three-layer FMC produced efficient rules that generated nearly 100% accuracies on ten different classification datasets, with significantly reduced number of trainable parameters and number of nodes in the network architecture, as compared to ANFIS

Advanced iterative procedures for solving the implicit Colebrook equation for fluid flow friction

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%