Fuzzy FA: a modified Firefly Algorithm

The firefly algorithm (FA), which is usually used in optimization problems, is a stochastic, population-based algorithm inspired by the intelligent, collective behavior of fireflies in nature. In the standard FA, each firefly in each neighborhood is compared with other fireflies, and the less-bright firefly moves toward the brighter one (in the maximization optimization). In fact, in the standard FA, firefly movement is based on the local optima, and the global optima do not have any effect on the movement of fireflies. So, the exploration rate of the FA decreases. In this study, we propose a new, fuzzy-based, modified version of the standard FA—the fuzzy firefly algorithm (FFA)—to increase the exploration and improve the global search of the FA. In the FFA, to improve the speed of finding the global optima, in each iteration the global optima and some brighter fireflies have influence on the movement of fireflies. The effect of each firefly depends on its attractiveness, which is considered as a fuzzy variable in the FFA. In order to evaluate the proposed FFA, seven well-known benchmark functions, including Sphere, Rastrigin, Rosenbrock, Step, Schwefel’s P2.22, Ackly, and the Xin-She Yang functions are used in 10-, 20-, and 30-dimensional spaces. Also, to indicate the effectiveness of the proposed FFA, we utilize our approach in a multilevel image-thresholding application, which is one of the most important and challenging issues in image segmentation, and we propose the FFA-Otsu method. The experimental results show that our proposed method can be effective to find the global optima and can improve the global search and the exploration rate of the standard FA. Moreover, the obtained thresholding results show that the proposed FFA-Otsu algorithm is more efficient when compared with other benchmark approaches for image segmentation.