Mathematical Problems in Engineering

Volume 2018, Article ID 6906295, 16 pages

https://doi.org/10.1155/2018/6906295

## Biological Flower Pollination Algorithm with Orthogonal Learning Strategy and Catfish Effect Mechanism for Global Optimization Problems

School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Weijia Cui; nc.ude.aaub@aijiewiuc

Received 20 November 2017; Revised 13 February 2018; Accepted 25 February 2018; Published 8 April 2018

Academic Editor: Ricardo Soto

Copyright © 2018 Weijia Cui and Yuzhu He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The flower pollination algorithm (FPA) is a novel optimization technique derived from the pollination behavior of flowers. However, the shortcomings of the FPA, such as a tendency towards premature convergence and poor exploitation ability, confine its application in engineering problems. To further strengthen FPA optimization performance, an orthogonal learning (OL) strategy based on orthogonal experiment design (OED) is embedded into the local pollination operator. OED can predict the optimal factor level combination by constructing a smaller but representative test set based on an orthogonal array. Using this characteristic of OED, the OL strategy can extract a promising solution from various sources of experience information, which leads the population to a potentially reasonable search direction. Moreover, the catfish effect mechanism is introduced to focus on the worst individuals during the iteration process. This mechanism explores new valuable information and maintains superior population diversity. The experimental results on benchmark functions show that our proposed algorithm significantly enhances the performance of the basic FPA and offers stronger competitiveness than several state-of-the-art algorithms.

#### 1. Introduction

Conventional optimization methods face serious challenges in modern sciences because the characteristics of optimization problems are often noncontinuous, nonlinear, multivariate, or nonconvex [1]. Currently, swarm intelligence algorithms are efficiently used to solve complex optimization problems. Most swarm intelligence algorithms are developed by simulation of foraging behavior, migration patterns, or the evolutionary approach in natural species, and these algorithms include the genetic algorithm (GA) [2], particle swarm optimization (PSO) [3], differential evolution (DE) [4], shuffled frog leaping algorithm (SFLA) [5], biogeography-based optimization (BBO) [6], cuckoo search (CS) [7], krill herd algorithm (KH) [8], fruit fly optimization (FFO) [9], pigeon inspired optimization (PIO) [10], invasive weed optimization (IWO) [11], and bat algorithm (BA) [12].

As a novel heuristic algorithm, the flower pollination algorithm (FPA) is inspired by the pollination behavior of flowers. In nature, the pollination methods for flowers involve two main types: cross-pollination and self-pollination [13]. In cross-pollination, certain birds act as global pollinators that transfer pollen to the flowers of more distant plants. By contrast, in self-pollination, pollen is spread by the wind and only between adjacent flowers at the same plant. As such, the FPA is developed by mapping the two forms of cross-pollination and self-pollination into global pollination and local pollination operators, respectively. The FPA has attracted significant attention due to its merits of simple principles, few parameters, and ease of operation.

However, similar to other swarm intelligence algorithms, the FPA also suffers from a lack of perfect compromise between global exploration and local exploitation. Therefore, researchers have proposed various improved strategies to enhance the FPA search ability. Nabil [14] introduced the clone selection mechanism into the local pollination operator to obtain more accurate solutions. Hoang et al. [15] integrated the mutation, cross, and selection operators of DE to replace the original update method for local pollination. Zhou et al. [16] developed the elite opposition mechanism, the local self-adaptive greedy strategy, and a dynamic switch probability method to balance exploration and exploitation. Salgotra and Singh [17] analyzed the influence of the various mutation techniques on the performance of the basic FPA. These variants exhibited excellent performance on benchmark functions. Additionally, the FPA has achieved favorable results in selected practical engineering applications. Abdelaziz et al. [18] used the basic FPA to solve the economic load dispatch problem. Wang et al. [19] conducted cluster analysis using bee pollinator-based FPA. Zhang et al. [20] proposed a chaotic local search-based FPA to forecast wind speed. Xu et al. [21] hybridized the FPA with the good point set theory and Deb’s heuristic rules to identify multipass turning parameters. Salgotra and Singh [22] implemented the bat algorithm and the FPA in parallel to synthesize the linear antenna arrays.

As is known to all, standard FPA adopts the differential learning method to achieve the global exploration and refine exploitation. However, this simple evolution pattern is inefficient and presents some drawbacks. First, the random search behavior around the current individual is lack of an effective mechanism to avoid the premature convergence; moreover, the update formula works on the complete solution vector rather than dimension by dimension; it may cause the consequence of “two steps forward, on step back” [23]. The main reason is that different solution vectors may contain different qualities of components; thus some high-quality components at some dimensions may improve the current individual, while some inferior components may result in a degeneration phenomenon at some certain dimensions. Through the above analysis, how to keep the population diversity and make better use of the various experience information to exploit the higher-quality solution becomes the critical factors of improving the optimizing capability.

Orthogonal experiment design (OED) as a typically analytical tool of experiment scheme can effectively predict the best combination of levels at different factors through choosing a smaller but most representative test set. In this paper, considering that potentially valuable information might exist in certain dimensions in different candidate solution vectors, an OL strategy based on orthogonal experiment design (OED) [24] is developed to obtain more promising candidate solution by combining the useful information among the best individual, current individual, and a random selected individual. In this way, the convergence accuracy and speed would be significantly improved. However, although OL strategy can improve the optimizing efficiency, the algorithm still suffers from the loss of diversity. To overcome this problem, the catfish effect mechanism [25, 26] is introduced to enhance the population diversity by replacing the worst individuals with their own opposition information. Specifically, as the iterations progress, the worst individuals are coming closer and closer to the best individual, and once the algorithm traps into the local convergence, catfish effect mechanism can drive the worst subpopulation to explore the new region. In this way, the population diversity is kept. In sum, the incorporation of OL strategy and catfish effect mechanism into the FPA guarantees the search stability of algorithm and can effectively improve the convergence performance. The experimental results demonstrate that our proposed algorithm is superior to the original FPA and other advanced evolutionary algorithms in testing on benchmark functions.

The remainder of this paper is described as follows. Section 2 describes the model of global optimization problems, Section 3 reviews the principle of FPA in detail, Section 4 elaborates on the OL strategy and catfish effect mechanism, Section 5 demonstrates the optimization performance of OCFPA on the benchmark functions, and Section 6 summarizes the conclusions.

#### 2. Global Optimization Problems

Many engineering problems can be considered as global optimization problems [27, 28], which can be described as follows:where is the vector form of the decision variable, is the number of decision variables, is the objective function, defines the decision space (i.e., the search space of the optimization algorithm), and and are the lower and upper limits of decision variable , respectively. Global optimization problem is to obtain the best decision variable* SX* by minimizing or maximizing the object function .

#### 3. Flower Pollination Algorithm

The FPA is a population-based global optimization algorithm. In [13], Yang et al. summarized the characteristics of the flower pollination process into four ideal rules as follows.

*Rule 1. *Cross-pollination conducted by the pollinators such as birds is viewed as a global pollination phase and the pollinators act with levy flight behavior.

*Rule 2. *Self-pollination occurring on the nearby flowers is viewed as a local pollination process.

*Rule 3. *The flower constancy is treated as the reproductive rate, which is direct ratio with the similar level of two involved flowers.

*Rule 4. *Global pollination and local pollination are implemented based on a switch probability.

According to the above rules, the FPA contains a global pollination operator and a local pollination operator. In the FPA, each pollen item is treated as a solution , and the solutions are initialized with random vectors in the feasible search space. The initial formula is given as follows:where , NP is the population size; is a -dimensional random vector in ; the lower limit of search space is Lower = , and the upper limit is Upper = .

In the global pollination operation, pollinators such as birds have a relatively large movement range and can carry pollen over a long distance. Thus, Rules 1 and 3 are formulated as follows:where is the solution at iteration , is the current global best solution, is a step factor, the flight feature of birds can be numerically imitated by a levy distribution denoted by in (3), and can also be considered a varying step factor to quantify the intensity of pollination. The levy distribution when can be described as follows:where is the standard gamma function with and is determined with two Gaussian distributions and as follows:where denotes the normal distribution with the mean value 0 and variance ; denotes the standard normal distribution.

If the pollination activities involve local pollination, then pollen are spread to a local neighbor, and the model can be formulated with Rules 2 and 3 as follows:where and are the pollen randomly selected from different flowers in the same plant, where and and is a -dimensional random vector in . In addition, from Rule 4, the two pollination activities occur randomly and are determined by a probability *. *In other words, if a random value* rand* in is smaller than , then global pollination is conducted, otherwise, vice versa. The flowchart of the FPA is shown in Figure 1.