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Fitness Dependent Optimizer: Inspired by the Bee Swarming Reproductive Process

Jaza M. Abdullah, Tarik A. Rashid

arXiv:1904.05226v1cs.NE

TL;DR

The paper addresses optimization problems where traditional methods struggle with complex, nonlinear, and multimodal search spaces. It proposes FDO, a bee-swarming-inspired optimizer using fitness values to generate movement weights, and reports better performance in most tested cases, comparative results elsewhere, and statistically significant results except TF2.

  • Problem

    Traditional optimization methods struggle with global search in complex, highly nonlinear, and multimodal problems, while no single algorithm is best for every problem.

  • Method

    FDO models scout-bee swarming and collective decision-making by using fitness values to generate weights that guide randomized artificial-scout movement during exploration and exploitation.

  • Results

    FDO generally outperformed comparison algorithms on the evaluated classical and CEC benchmark functions, with statistically significant results except TF2.

  • Takeaways & Limitations

    FDO effectively explored search spaces, improved solutions, avoided local optima, and converged fairly toward optimality under the reported measurements.

Abstract

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In this paper, a novel swarm intelligent algorithm is proposed, known as the fitness dependent optimizer (FDO). The bee swarming reproductive process and their collective decision-making have inspired this algorithm; it has no algorithmic connection with the honey bee algorithm or the artificial bee colony algorithm. It is worth mentioning that FDO is considered a particle swarm optimization (PSO)-based algorithm that updates the search agent position by adding velocity (pace). However, FDO calculates velocity differently; it uses the problem fitness function value to produce weights, and these weights guide the search agents during both the exploration and exploitation phases. Throughout the paper, the FDO algorithm is presented, and the motivation behind the idea is explained. Moreover, FDO is tested on a group of 19 classical benchmark test functions, and the results are compared with three well-known algorithms: PSO, the genetic algorithm (GA), and the dragonfly algorithm (DA), additionally, FDO is tested on IEEE Congress of Evolutionary Computation Benchmark Test Functions (CEC-C06, 2019 Competition) [1]. The results are compared with three modern algorithms: (DA), the whale optimization algorithm (WOA), and the salp swarm algorithm (SSA). The FDO results show better performance in most cases and comparative results in other cases. Furthermore, the results are statistically tested with the Wilcoxon rank-sum test to show the significance of the results. Likewise, FDO stability in both the exploration and exploitation phases is verified and performance-proofed using different standard measurements. Finally, FDO is applied to real-world applications as evidence of its feasibility.

I. INTRODUCTION

Optimization seeks suitable or optimal solutions, but traditional deterministic, local, problem-specific methods struggle with global optimality and highly nonlinear multimodal problems. Nature-inspired metaheuristics address these challenges by using stochastic, nature-derived search mechanisms, motivating FDO.

  • Motivation: Traditional algorithms perform local, often deterministic and problem-specific searches, providing no guarantee of global optimality.Their limited diversity also restricts the solutions obtained.
  • Motivation: Highly nonlinear multimodal problems are difficult for traditional algorithms because their search mechanisms do not handle multiple optima effectively.
  • Metaheuristics: Evolutionary algorithms use stochastic behavior, while heuristic and metaheuristic methods search through trial, error, randomization, and local-search mechanisms.The paper notes that heuristic and metaheuristic are often used interchangeably.
  • Nature-Inspired Algorithms: Nature-inspired algorithms model animal behaviors and natural phenomena because real-world complexity makes exhaustive search impractical in time, space, and cost.Examples include ants finding paths and groups of animals avoiding enemies or hunting prey.
  • FDO Motivation: FDO is proposed because no single optimization algorithm performs best on every problem, and it is inspired by bees’ reproductive swarming and collective decisions rather than ABC.The introduction frames this rationale through the No Free Lunch perspective.

II. BEE SWARMING

The paper abstracts honeybee reproductive swarming as a search process: scout bees explore candidate hive sites, communicate their evaluations, and collectively select a suitable new hive. FDO maps these biological elements to artificial solutions and fitness-based decision weights.

  • Bee Roles: Honeybee colonies contain queen, worker, and scout bees, with scouts exploring the environment and exploiting preferable targets.Swarming begins when colony and environmental conditions support reproduction.
  • Swarming Process: During swarming, the queen leaves the old colony with workers and scouts, which temporarily cluster before searching for a new hive.A swarm may include thousands to tens of thousands of bees.
  • Hive Selection: Scout bees evaluate potential hives using criteria including capacity, entrance size and location, and sunlight.The paper gives a minimum hive volume of 15 liters and a preferred volume of 40 liters.
  • Collective Decision-Making: Scouts communicate through bee dances and typically reach a decision when about 80% agree or when 20–30 scouts form a quorum.
  • Algorithmic Mapping: FDO represents each exploited hive as a possible solution, hive specifications as fitness functions, and collective scout decisions as fitness weights.

III. FITNESS DEPENDENT OPTIMIZER ALGORITHM

FDO is a swarm-inspired optimizer that moves artificial scouts by adding a pace to their positions. The pace is mainly determined by fitness-based weighting, while randomization controls movement direction and supports search behavior.

  • Biological Inspiration: FDO replicates scout bees’ search for a suitable hive among potential alternatives, treating hives as candidate solutions.
  • Search Rules: Artificial scouts are randomly initialized, then retain better newly found solutions while reusing prior directions or the best known solution when moves do not improve fitness.
  • Movement Mechanism: FDO updates each position as X_i,t+1 = X_i,t + pace, with pace magnitude mainly dependent on fitness weight and direction determined randomly.
  • Fitness Weight: The fitness weight uses the best global and current solution fitness values, with w_f controlling convergence and coverage behavior.w_f is either 0 or 1; setting w_f = 0 is described as providing a more stable search in some cases.
  • Special Cases: When fitness-weight conditions create edge cases, FDO uses pace = x_i,t* r, where r is a random number in [-1, 1], and Levy flight is selected for stability.
  • Complexity and Calculation: FDO has O(p*n + p*CF) time complexity per iteration and O(p*CF + p*pace) space complexity across iterations.Time complexity scales with iterations, while space complexity remains the same during iteration progress.
  • Complexity and Calculation: FDO calculates fitness weight and one random number per agent, fewer quantities than the global-best, agent-best, factors, and random values used by PSO.The paper also contrasts FDO’s mechanism with DA’s multiple parameter weights.

IV. FDO WITH SINGLE OBJECTIVE OPTIMIZATION PROBLEMS

FDOSOOP applies FDO to single-objective problems through iterative fitness-weighted scout movement, acceptance checks, and pace reuse. Maximization requires an inverse fitness-weight equation and reversed solution-selection comparisons.

  • FDOSOOP Procedure: FDOSOOP initializes scouts within problem bounds, selects the global best each iteration, and calculates each scout’s fitness weight.
  • FDOSOOP Procedure: For intermediate fitness weights, FDOSOOP generates pace using randomized alternatives before updating the scout position and testing solution improvement.
  • Acceptance and Pace Reuse: When a new solution is not better, FDO may reuse the previous pace only if it leads to improvement; otherwise, it maintains the current solution.Accepted pace values are saved for possible reuse in the next iteration.
  • Maximization: Maximization changes replace the minimization fitness-weight equation with its inverse and reverse the better-solution condition from < to >.
  • Pseudocode: The FDOSOOP pseudocode includes random initialization, divide-by-zero handling, fitness-weight branching, pace generation, position updates, and acceptance decisions.

V. RESULTS AND DISCUSSION

FDO is evaluated on classical and CEC benchmark functions against established and modern optimizers, with additional statistical and metric-based analyses. Across these evaluations, FDO generally outperforms or matches comparison algorithms and shows convergence and search-space improvement.

  • Benchmark design: The evaluation covers 19 classical benchmark functions and 10 CEC-C06 2019 benchmark functions with comparisons against PSO, GA, DA, WOA, and SSA.Classical tests use unimodal, multimodal, and composite functions; CEC functions provide an additional single-objective evaluation.
  • Benchmark design: Each classical benchmark algorithm was tested 30 times with 30 search agents for 500 iterations, reporting average and standard deviation.FDO used w_f = 0 for all functions except TF2 and TF6, where w_f = 1.
  • Classical benchmark results: FDO generally outperformed DA, PSO, and GA on classical functions, including every case from TF14 to TF17 and TF19, while producing comparative or poorer results elsewhere.FDO was generally better on TF1–TF6 and outperformed the other algorithms on TF14–TF17 and TF19; TF8 was a poor case.
  • CEC-C06 2019 results: FDO outperformed the other algorithms on the CEC benchmarks except CEC06, with comparative results reported for CEC03, CEC05, and CEC09.WOA matched FDO on CEC03 and had standard deviation equal to 0 in that comparison.
  • Statistical analysis: Wilcoxon rank-sum tests found FDO statistically significant in all classical comparisons except TF2 and in most CEC comparisons, with specified exceptions involving CEC03, CEC04, and CEC08.For classical tests, TF2 was nonsignificant; for CEC tests, exceptions occurred for DA and WOA on CEC03 and WOA on CEC04 and CEC08.
  • Convergence and search behavior: Metric analyses showed steadily decreasing fitness values, reduced average fitness across agents, increasingly accurate global-best convergence, and effective exploration with avoidance of local optima.The reported metrics cover search history, agent trajectories, fitness values, and global-best convergence across unimodal, multimodal, and composite functions.

B- FDO ON FREQUENCY MODULATED SOUND WAVES

FDO is applied to a six-parameter frequency-modulated sound-wave synthesis problem, optimizing generated sound to resemble a target. With 30 agents over 200 iterations, the global best approached the near-global optimum from iteration 64.

  • FDO optimizes six FM synthesizer parameters to generate sound similar to a target sound.The parameter vector is X = {a1, w1, a2, w2, a3, w3}.
  • The parameter values are constrained to [−6.4, 6.35], with θ = 2π/100 and a fitness function based on squared-root differences over t = 100 turns.
  • Using 30 agents for 200 iterations, FDO generated a parameter set at iteration 200 and reached a near-global optimum from iteration 64.The reported parameter set was {a1 = 0.974, w1 = -0.241, a2 = -4.3160, w2 = -0.0193, a3 = -0.5701, w3 = 4.937}.

VI. CONCLUSION

The paper concludes that FDO, inspired by bee reproductive swarming and collective decision-making, performs better than competing algorithms in most tested cases and comparably in others. It also identifies a trade-off between search-agent count, accuracy, and computational cost.

  • FDO uses fitness-function values to generate weights that guide search agents during optimization.The algorithm is inspired by bee reproductive swarming and collective decision-making, without an algorithmic connection to ABC.
  • FDO was evaluated on 19 single-objective benchmark functions divided into unimodal, multimodal, and composite groups, plus 10 modern CEC-C06 benchmarks.
  • FDO outperformed competing algorithms in most cases and produced comparative results in the others, with statistical testing using the Wilcoxon rank-sum test.
  • FDO performance depended on the number of search agents: fewer than five notably reduced accuracy, while more agents improved accuracy but increased time and space costs.The conclusion links this trade-off partly to the algorithm’s reliance on fitness weight during search.
  • Future work will extend FDO to multiobjective and binary optimization and consider evolutionary-operator integration and hybridization with other algorithms.

VIII. APPENDIX

The appendix lists benchmark-function tables used to evaluate FDO, covering unimodal, multimodal, composite, and CEC-C06 benchmark groups. The CEC-C06 table is identified as “The 100-Digit Challenge.”

  • Table 6 lists the unimodal benchmark functions used in the classical benchmark evaluation.
  • Table 7 lists 10-dimensional multimodal benchmark functions for the classical evaluation.
  • Table 8 lists the composite benchmark functions used in the classical benchmark evaluation.
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