Firefly algorithm is a nature-inspired metaheuristic algorithm inspired by the flashing behavior of fireflies. It is originally proposed for continuous problems. However, due to its effectiveness and success in solving continuous problems, different studies are conducted in modifying the algorithm to suit discrete problems. Many engineering as well as optimization problems from other disciplines involve discrete variables. Recent reviews on the application and modifications of firefly algorithm mainly focus on continuous problems. This paper is devoted to the detailed review of the modifications done on firefly algorithm in order to solve optimization problems with discrete variables. Hence, advances on the application of firefly algorithm for optimization problems with binary, integer as well as mixed variables will be discussed. Possible future works will also be highlighted.

Firefly algorithm is one of the new metaheuristic algorithms for optimization problems. The algorithm is inspired by the flashing behavior of fireflies. In the algorithm, randomly generated solutions will be considered as fireflies, and brightness is assigned depending on their performance on the objective function. One of the rules used to construct the algorithm is, a firefly will be attracted to a brighter firefly, and if there is no brighter firefly, it will move randomly. In this paper we modify this random movement of the brighter firefly by generating random directions in order to determine the best direction in which the brightness increases. If such a direction is not generated, it will remain in its current position. Furthermore the assignment of attractiveness is modified in such a way that the effect of the objective function is magnified. From the simulation result it is shown that the modified firefly algorithm performs better than the standard one in finding the best solution with smaller CPU time.

Firefly algorithm is one of the well-known swarm-based algorithms which gained popularity within a short time and has different applications. It is easy to understand and implement. The existing studies show that it is prone to premature convergence and suggest the relaxation of having constant parameters. To boost the performance of the algorithm, different modifications are done by several researchers. In this chapter, we will review these modifications done on the standard firefly algorithm based on parameter modification, modified search strategy and change the solution space to make the search easy using different probability distributions. The modifications are done for continuous as well as non-continuous problems. Different studies including hybridization of firefly algorithm with other algorithms, extended firefly algorithm for multiobjective as well as multilevel optimization problems, for dynamic problems, constraint handling and convergence study will also be briefly reviewed. A simulation-based comparison will also be provided to analyse the performance of the standard as well as the modified versions of the algorithm.