Algorithm 1HS method.3. FA MethodFA [42] is a metaheuristic approach for optimization problems. The search strategy in FA comes from the fireflies swarm behavior [50]. There are two significant issues in FA that Perifosine clinical are the formulation of attractiveness and variation of light intensity [42].For simplicity, several characteristics of fireflies are idealized into three rules described in [51]. Based on these three rules, the FA can be described in Algorithm 2.Algorithm 2Firefly algorithm. FA method.For two fireflies xi and xj, they can be updated as follows:xit+1=xit+��0e?��rij2(xit?xjt)+����it,(3)where �� is the step size, ��0 is the attractiveness at r = 0, the second part is the attraction, while the third is randomization [50]. In our present work, we take ��0 = 1, �� [0, 1], and �� = 1 [50].

4. HS/FABased on the introduction of HS and FA in the previous section, the combination of the two approaches is described and HS/FA is proposed, which updates the poor solutions to accelerate its convergence speed.HS and FA are adept at exploring the search space and exploiting solution, respectively. Therefore, in the present work, a hybrid by inducing HS into FA method named HS/FA is utilized to deal with optimization problem, which can be considered as mutation operator. By this strategy, the mutation of the HS and FA can explore the new search space and exploit the population, respectively. Therefore, it can overcome the lack of the exploration of the FA. To combat the random walks used in FA, in the present work, the addition of mutation operator is introduced into the FA, including two detailed improvements.

The first one is the introduction of top fireflies scheme into FA to reduce running time that is analogous to the elitism scheme frequently used in other population-based optimization algorithms. In FA, due to dual loop, time complexity is O(NP2), whose performance significantly deteriorates with the increases in population size. This improvement is carried out by reduction of outer loop in FA. In HS/FA, we select the special firefly with optimal or near-optimal fitness (i.e., the brightest fireflies) to form top fireflies, and all the fireflies only move towards top fireflies. Through top fireflies scheme, the time complexity of HS/FA decreases from O(NP2) to O(KEEPNP), where KEEP is the number of top fireflies.

In general, KEEP is far smaller than NP, so the time used by HS/FA is much less than FA. Apparently, if KEEP = NP, the algorithm HS/FA is declined to the standard FA. If KEEP is too small, only few best fireflies Anacetrapib are selected to form top fireflies and it converges too fast, moreover, may be premature for lack of diversity. If KEEP is extremely big (near NP), almost all the fireflies are used to form top fireflies, so all fireflies are explored well, leading to potentially optimal solutions, while the algorithm performs badly and converges too slowly. Therefore, we use KEEP = 2 in our study.