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Bat Algorithm for Multi-objective Optimisation
Xin-She Yang
TL;DR
Engineering optimization often requires handling conflicting objectives and complex constraints, motivating efficient multiobjective methods. This paper extends the bat algorithm into MOBA, tests it on multiobjective functions, and applies it to engineering design benchmarks. Results suggest efficient optimization, including exponential convergence and a smooth welded-beam Pareto front, while the authors call for broader testing and comparisons.
Problem
Engineering optimization involves multiple conflicting objectives under complex nonlinear constraints, making multiobjective problems difficult to solve.
Method
The paper extends the bat algorithm into MOBA, validates it on multiobjective test functions, and applies it to engineering design benchmarks.
Results
MOBA achieves exponential convergence on the tested functions and generates a smooth approximate Pareto front for welded beam design after 1000 iterations.
Takeaways & Limitations
The results suggest that MOBA is an efficient multiobjective optimizer for tested functions and structural engineering design benchmarks.
Takeaways & Limitations
The authors state that additional tests and comparisons are needed, including wider test problems, discrete and mixed optimization, Pareto-front diversity, and comparisons with other methods.
Abstract
from arXiv · showhide
Engineering optimization is typically multiobjective and multidisciplinary with complex constraints, and the solution of such complex problems requires efficient optimization algorithms. Recently, Xin-She Yang proposed a bat-inspired algorithm for solving nonlinear, global optimisation problems. In this paper, we extend this algorithm to solve multiobjective optimisation problems. The proposed multiobjective bat algorithm (MOBA) is first validated against a subset of test functions, and then applied to solve multiobjective design problems such as welded beam design. Simulation results suggest that the proposed algorithm works efficiently.
1 Introduction
Engineering design optimization involves multiple often-conflicting objectives, nonlinear constraints, uncertainty, and limited resources. The paper extends the bat algorithm to multiobjective problems and evaluates it on test functions and engineering design.
- Engineering design optimization commonly involves multiple objectives under complex, highly nonlinear constraints, standards, material properties, resources, and costs.
- Multiobjective optimization requires approximating a Pareto front because no single solution is necessarily best, while maintaining uniform solution distribution and search diversity.
- Real-world optimization also involves uncertainty or noise, requiring designs that tolerate material variation and support decision-makers or designers.
- Nature-inspired metaheuristics have become prominent approaches for multiobjective global optimization and have been applied to nonlinear engineering design problems.
- The paper extends the bat algorithm into MOBA, validates it on multiobjective test functions, and applies it to engineering design optimization.
2 Bat Behaviour and Bat Algorithm
The bat algorithm idealizes echolocation as adaptive search: bats adjust frequency, velocity, loudness, and pulse emission while combining global movement with local random-walk search. The paper extends this single-objective framework toward multiobjective optimization.
- Bat Behaviour and Bat Algorithm: Echolocation lets microbats infer target distance, orientation, prey type, and prey speed from echo timing, interaural differences, and loudness variation.
- Bat Behaviour and Bat Algorithm: The bat algorithm is inspired by echolocation and uses idealized rules for bats searching through positions, velocities, frequencies, wavelengths, loudness, and pulse emission rates.
- Bat Behaviour and Bat Algorithm: The model simplifies bat behavior by omitting ray tracing and freely selecting frequency and wavelength ranges to suit different applications.
- Bat Behaviour and Bat Algorithm: Bat positions and velocities are updated using random frequencies and a global-best solution, with β drawn uniformly from [0, 1] to control movement.
- Bat Behaviour and Bat Algorithm: Local search generates solutions through random walks around a selected current best solution, using a random vector and average bat loudness.
- Bat Behaviour and Bat Algorithm: The algorithm combines particle-swarm-like movement with intensive local search, while loudness decreases and pulse-emission rate increases during iterations.
- Bat Behaviour and Bat Algorithm: The paper extends the promising bat algorithm for nonlinear global optimization to multiobjective optimization.
3 Multiobjective Bat Algorithm
MOBA adapts bat-inspired search to multiobjective optimization by handling Pareto optimality and generating diverse solutions. Its workflow is summarized as a population-based algorithm that can combine objectives with randomized weights.
- 3.1 Pareto Optimality: Multiobjective optimization requires approximating optimality fronts and modifying algorithms to accommodate multiple objectives.
- 3.1 Pareto Optimality: A solution dominates another when it is no worse in every objective and strictly better in at least one objective.
- 3.1 Pareto Optimality: A non-dominated solution has no other solution that dominates it, and the Pareto front consists of such solutions.
- 3.2 MOBA Algorithm: MOBA initializes bat positions and velocities while targeting N points on Pareto fronts or the corresponding Pareto-optimal set.
- 3.1 Pareto Optimality: Approximating a good Pareto front requires generating a diverse range of solutions with efficient techniques.
- 3.2 MOBA Algorithm: The proposed algorithm is summarized in pseudo-code as the multiobjective bat algorithm, MOBA.
- 3.2 MOBA Algorithm: MOBA combines objectives using a weighted sum, with randomly generated uniform weights intended to provide sufficient diversity for Pareto-front approximation.
- 3.2 MOBA Algorithm: In simulations, α = γ = 0.9 was used, and loudness and pulse-emission rates were updated only when new solutions improved.
4 Numerical Results
MOBA is evaluated on diverse multiobjective test functions and compared with true Pareto fronts using convergence measures. The reported results show almost exponential convergence across the tested functions.
- Experimental Setup: The experiments use a fixed stopping criterion of 5000 iterations, chosen as sufficient for most problems and adjustable when necessary.A fixed iteration count also facilitates comparison of Pareto-front closeness across functions.
- Multiobjective Test Functions: The benchmark subset covers convex, non-convex, and discontinuous Pareto fronts, plus functions with complex Pareto-optimal sets.This selection is intended to represent diverse front and Pareto-set properties.
- Evaluation Metric: The estimated Pareto front is compared with the corresponding true front using a least-square distance Ef over N points.The distance is defined between the estimated front PF_e and true front PF_t.
- Convergence Results: The ZDT1 results show an exponential-like decrease in Ef as iterations proceed, examined over the first 1000 iterations and on a logarithmic scale through 5000 iterations.The vertical axis represents f2 and the horizontal axis represents f1 in the reported figures.
- Overall Results: Exponential convergence is reported for all tested functions, with the simulations suggesting that MOBA efficiently handles nonlinear problems with complex constraints and diverse Pareto-optimal sets.Table 1 is identified as a summary of results.
5 Engineering Optimization
MOBA is applied to the welded-beam engineering benchmark, producing an approximate Pareto front while showing convergence on multiobjective test functions.
- The welded-beam benchmark minimizes fabrication cost and end deflection using four design variables: w, L, d, and h.
- The paper presents figures for the estimated and true ZDT1 fronts, MOBA convergence, exponential convergence, and the beam-design Pareto front.
- MOBA generates an approximate Pareto front from 50 non-dominated solutions after 1000 iterations.The results are consistent with earlier studies and are reported as smoother with fewer iterations.
- Simulations suggest that MOBA efficiently handles highly nonlinear problems with complex constraints and diverse Pareto-optimal sets.
6 Conclusions
The paper formulates MOBA from the bat algorithm, tests it on selected multiobjective functions, and applies it to structural-engineering benchmarks. The authors conclude that results suggest efficient optimization, while calling for broader testing and comparisons.
- The paper formulates the multiobjective bat algorithm by extending the recently developed bat algorithm.
- MOBA is tested on selected test functions and applied to structural-engineering design benchmarks.
- Results suggest that MOBA is an efficient multiobjective optimizer.
- Additional tests and comparisons are needed, including broader discrete and mixed-type optimization problems.
- Future work includes performance comparisons with popular multiobjective methods and possible hybridization with other algorithms.