# Improving Monarch Butterfly Optimization Algorithm with Self-Adaptive Population
## 2 Related Work
[95] presented a novel binary MBO (BMBO) method used to address the 0-1 knapsack problem (0-1 KP). In BMBO, each butterfly individual is represented as a two-tuple string. Several individual allocation techniques are used to improve BMBO's performance. In order to keep the number of infeasible solutions to a minimum, a novel repair operator was applied. The comparativestudy of BMBO with other optimization techniques showed the superiority of the former in solving 0-1 KP. [PERSON] et al. [96; 97] put forward another variant of the MBO method, the GCMBO. In GCMBO, two modification strategies, including a self-adaptive crossover (SAC) operator and a greedy strategy, were utilized to improve its search ability. [PERSON] et al. [98] combined chaos theory with the basic MBO algorithm, and proposed a novel chaotic MBO (CMBO) algorithm. The proposed CMBO algorithm enhanced the search effectiveness significantly. In CMBO, in order to tune the two main operators, the best chaotic map is selected from 12 maps. Meanwhile, some of the worst individuals are improved by using a Gaussian mutation operator to avoid premature convergence. [PERSON] and [PERSON] [99] combined ABC with elements from MBO to proposed a new hybrid metaheuristic algorithm named hybrid ABC/MBO (HAM). The combined method uses an updated butterfly adjusting operator, considered to be a mutation operator, with the aim of sharing information with the employee bees in ABC. [PERSON] et al. [100] proposed a discrete version of MBO (DMBO) that was applied successfully to tackle the Chinese TSP (CTSP). They also studied and analyzed the parameter butterfly adjusting rate (BAR). The chosen BAR was used to find the best solution for the CTSP. [PERSON] et al. [101] proposed a type of multi-strategy MBO (MMBO) technique for the discounted 0-1 knapsack problem (DKP). In MMBO, two modifications, including neighborhood mutation and Gaussian perturbation, are utilized to retain the diversity of the population. An array of experimental results showed that the neighborhood mutation and Gaussian perturbation were quite capable of providing significant improvement in the exploration and exploitation of the MMBO approach, respectively. Accordingly, two kinds of NMBO were proposed: NCMBO and GMMBO. [PERSON] et al. [102] combined MBO with seven kinds of DE mutation strategies, using the intrinsic mechanism of the search process of MBO and the character of the differential mutation operator. They presented a novel DEMBO based on MBO and an improved DE mutation strategy. In this work, the migration operator was replaced by a differential mutation operator with the aim of improving its global optimization ability. The overall performance of DEMBO was fully assessed using thirty typical discounted 0-1 knapsack problem instances. The experimental results demonstrated that DEMBO could enhance the search ability while not increasing the time complexity. Meanwhile, the approximation ratio of all the 0-1 KP instances obtained by DEMBO were close to 1.0. [PERSON] et al. [103] proposed a new population initialization strategy in order to improve MBO's performance. First, the whole search space is equally divided into _NP_ (population size) parts at each dimension. Subsequently, two random distributions (the \(T\) and \(F\) distributions) are used to mutate the equally divided population. Accordingly, five variants of MBOs are proposed with a new initialization strategy. [PERSON] et al. [104] presented OMBO, a generalized opposition-based learning (OBL) [105; 106] MBO with Gaussian perturbation. The authors used the OBL strategy on the portion of the individuals in the late stage of evolution, and used Gaussian perturbation on the individuals with poor fitness in each evolution. OBL guaranteed a higher convergence speed of OMBO, and Gaussian perturbation avoided the possibility of falling into a local optimum. For the sake of testing and verifying the effectiveness of OMBO, three categories of 15 large-scale 0-1 KP cases from 800 to 2000 dimensions were used.