# Improving Monarch Butterfly Optimization Algorithm with Self-Adaptive Population
## 1 Introduction  
To optimize is to maximize or minimize given functions in a certain domain. In real life, human beings are driven to maximize profit or minimize cost. In mathematics and computer science, these real-world problems can be mathematically modeled, and then further tackled by various optimization techniques. In general, these optimization techniques can loosely be divided into two categories: traditional optimization methods and modern intelligent optimization algorithms. For each run, the traditional optimization methods will generate the same results under the same initial conditions; while modern intelligent optimization algorithms will generate fully different results even if the same conditions are provided. Since current problems are becoming more and more complicated, traditional optimization methods do not efficiently solve them. Therefore, more and more researchers have turned to modern intelligent optimization algorithms [1], which mainly include evolutionary computation [2], swarm intelligence, extreme learning machines [3], or artificial neural networks [4]. Among the different kinds of intelligent algorithms, swarm intelligence (SI) algorithms [5; 6; 7; 8] are one of the most representative paradigms. In 1995, particle swarm optimization (PSO) [9; 10; 11; 12; 13; 14; 15] was proposed based on the inspiration of bird flocking. In a sense, the development of PSO is one of the milestones in the history of swarm intelligence algorithms. Since then, many researchers have performed numerous in-depth studies on PSO, and it has been successfully used to solve various complicated engineering problems [16]. Such problems include gesture segmentation [17], scheduling [18], shape design [19], vehicle routing [20], test-sheet composition [21], malicious code detection [22], economic load dispatch [23; 24], IIR system identification [25], prediction of pupylation sites [26], target assessment [27; 28], unit commitment [29], path planning [30; 31; 32], directing orbits of chaotic systems [33], image processing [34], task assignment problem [35], floorplanning [36; 37], clustering [38], wind generator optimization [39], reliability problems [40], knapsack problem [41; 42; 43], and fault diagnosis [44]. Recently, immediately following PSO, many excellent SI algorithms have been put forward, including the ant colony optimization (ACO) [45; 46; 47], harmony search (HS) [48; 49], artificial bee colony (ABC) [50; 51; 52; 53; 54], cuckoo search (CS) [55; 56; 57; 58; 59; 60; 61], fireworks algorithm (FWA) [62], bat algorithm (BA) [63; 64; 65; 66; 67], fruit fly optimization algorithm (FOA) [68], earthworm optimization algorithm (EWA) [69], elephant herding optimization (EHO) [70; 71; 72], moth search (MS) algorithm [73; 74], biogeography-based optimization (BBO) [75; 76; 77], firefly algorithm (FA) [78; 79; 80], krill herd (KH) [81; 82; 83; 84; 85; 86; 87], and monarch butterfly optimization (MBO) [88]. These various algorithms are inspired by the swarm behavior of ants, honey bees, cuckoos, bats, grey wolves, krill, and butterflies. Recently, after a careful study of the migration behavior of monarch butterflies, [PERSON] et al. [88] designed a novel promising swarm intelligence-based optimization technique, called the monarch butterfly optimization (MBO). In MBO, all the monarch butterflies are located at land 1 and land 2, which are updated through implementation of the migration operator and butterfly adjusting operator at each generation. Based on thorough in-depth comparative studies of thirty-eight benchmark problems selected from previous literature, it was found that MBO significantly outperforms five other state-of-the-art metaheuristic algorithms. However, in the basic MBO algorithm, after implementing the migration operator, the generated monarch butterfly will be accepted as a new butterfly individual in the next generation regardless of whether it is better or worse.