Shock and Vibration

Volume 2015, Article ID 518692, 10 pages

http://dx.doi.org/10.1155/2015/518692

## A Comparative Study of Genetic and Firefly Algorithms for Sensor Placement in Structural Health Monitoring

^{1}College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China^{2}School of Civil Engineering, Dalian University of Technology, Dalian 116023, China

Received 11 August 2014; Accepted 1 October 2014

Academic Editor: Bo Chen

Copyright © 2015 Guang-Dong Zhou et al. 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

Optimal sensor placement (OSP) is an important task during the implementation of sophisticated structural health monitoring (SHM) systems for large-scale structures. In this paper, a comparative study between the genetic algorithm (GA) and the firefly algorithm (FA) in solving the OSP problem is conducted. To overcome the drawback related to the inapplicability of the FA in optimization problems with discrete variables, some improvements are proposed, including the one-dimensional binary coding system, the Hamming distance between any two fireflies, and the semioriented movement scheme; also, a simple discrete firefly algorithm (SDFA) is developed. The capabilities of the SDFA and the GA in finding the optimal sensor locations are evaluated using two disparate objective functions in a numerical example with a long-span benchmark cable-stayed bridge. The results show that the developed SDFA can find the optimal sensor configuration with high reliability. The comparative study indicates that the SDFA outperforms the GA in terms of algorithm complexity, computational efficiency, and result quality. The optimization mechanism of the FA has the potential to be extended to a wide range of optimization problems.

#### 1. Introduction

The performance deterioration and the total collapse of large-scale civil infrastructures induced by the environment and service loads highlight the importance of structural health monitoring (SHM) as a significant approach for the safe operation and the reasonable maintenance of structures. SHM, which involves an array of sensors to continuously monitor structural behavior, along with the extraction of damage-sensitive features from these measurements and the evaluation of current system health by analysis methods, can be used for rapid condition screening and aims to provide reliable information regarding the integrity of the structure in near real time [1–3]. At present, successful deployment and operation of long-term SHM systems on newly constructed structures and existing structures have been reported throughout the world [4–7]. In an SHM system, the sensor network provides original information indicating structural behavior for further parameter identification; therefore, the efficiency of an SHM system relies heavily on the reliability of the acquired data measured by the sensor networks on the structure. For the complexity of large-scale structures, such as long-span bridges and high-rise buildings, the degrees of freedom (DOFs) used to characterize structural performance count are on the order of thousands to tens of thousands. It is impossible to distribute sensors on all of the DOFs because of the high costs of data acquisition systems (sensors and their supporting instruments) and technology limitations [8, 9]. Therefore, selecting optimal sensor placement (OSP) is a critical task before a sophisticated SHM system is designed and implemented on a real structure [10].

The problem of determining OSP has been investigated using a large number of interesting approaches and criteria in the past few decades, which can be seen from the abundance of literature. Among them, conventional gradient-based local optimization methods were unable to efficiently handle multiple local optima and may present difficulties in estimating the global minimum. They lack reliability in dealing with the OSP problem, because convergence to the global minimum is not guaranteed [11, 12]. Thus, the shift of OSP research away from classical deterministic optimization methods toward the use of combinatorial optimization methods based on biological and physical analogues has been motivated by the high computational efficiency and success rate of intelligent optimization methods. Many contributions regarding the adoption of intelligent optimization methods to the OSP problem have been recently made. The genetic algorithm (GA) based on the Darwinian principle of natural selection is a representative example and has proved to be a powerful tool for OSP. Yao et al. [13] demonstrated that the GA can replace the effective independence (EfI) method when using the determinant of the Fisher information matrix (FIM) as the objective function. Subsequently, a number of improvements have been employed to overcome drawbacks of the original GA. To accelerate convergence, the simulated annealing (SA) algorithm was integrated into the GA by Worden and Burrows [14] and Hwang and He [15] to extract the OSP in structural dynamic tests. To keep the sensor number constant during the genetic operation, the coding system was replaced by decimal two-dimensional array coding [16] or dual-structure coding [17]. With the purpose of improving the quality of solutions and convergence speed, two-quarter selection was adopted by Yi et al. [18, 19]. The GA was also extended to the optimal wireless sensor placement, which has many constraints [20–22]. Particle swarm optimization (PSO), which is inspired by the movement of organisms in bird flocking or fish schooling, is another stochastic search technique and was successfully applied to the OSP problem [23, 24]. Furthermore, the monkey algorithm (MA), which imitates the mountain-climbing process of monkeys, is considered to be an effective numerical method in solving complex multiparameter optimization problems. Several changes developed by Yi et al. made the MA excellent in terms of generating optimal solutions, as well as providing fast convergence in dealing with complicated OSP problems [8, 25, 26].

Although the aforementioned methodologies demonstrated a strong capability, to some extent, in finding the acceptable solution for the OSP problem, the complex parameters and searching processes make those methods difficult to operate and susceptible to the application environment. The complexity of the optimal sensor configuration for large-scale structures reveals the necessity for the development of efficient and robust algorithms to accurately explore the optimum solution. Recently, a new metaheuristic search algorithm, which is referred to as the firefly algorithm (FA), has been developed by Yang [27, 28]. The FA algorithm is based on the idealized behavior of the flashing characteristics of fireflies. A firefly tends to be attracted by other fireflies with high flash intensities. Previous studies indicate that the FA is particularly suited for parallel implementation and may outperform existing algorithms, such as PSO, GA, SA, and differential evolution, in terms of efficiency and success rates [28, 29]. At present, the FA has been applied to a large number of optimization problems, including continuous, combinatorial, constrained, multiobjective, and dynamic optimization [30].

However, the coding system and the movement scheme in the FA make it suitable only for global numerical optimization problems with continuous variables. In this paper, some improvements, including the coding system, the suitable distance, and the movement scheme, are introduced, and a simple discrete firefly algorithm (SDFA) is proposed based on the FA such that the outstanding optimization mechanism of the FA can be applicable in the OSP problem with discrete variables. The remaining part of this paper is organized as follows: Section 2 presents a detailed description of the SDFA after an outline of the FA. Section 3 gives a brief introduction to the GA with the aim of facilitating performance comparison between the SDFA and the GA in the next numerical simulations. Section 4 shows the comprehensive evaluation of the SDFA for OSP with different criteria employing a long-span benchmark cable-stayed bridge. Finally, conclusions are drawn in Section 5.

#### 2. Firefly Algorithm

##### 2.1. Outline of Firefly Algorithm

The FA mimics the real firefly’s swarm behaviors of communication, its search for food, and its process of finding mates. The optimization process of exploring the optimal solution is modeled in such a way that the firefly with low light intensity is attracted by the firefly with high light intensity and moves toward to it, such that the darker firefly has higher light intensity. Therefore, to establish the mathematical model of this movement, three hypotheses are adopted as follows: (1) the attractive action between two fireflies is only governed by the light intensity; (2) the light intensity of a firefly, which is deduced by the firefly’s location, is proportional to the objective function; and (3) the light intensity decreases with increasing distance, such that the brighter firefly can only attract the fireflies within its attractiveness range. Then, the movement of firefly toward firefly is formulated aswhere and represent the locations of firefly and firefly , respectively, the superscript denotes time, means the light absorption coefficient, is the distance between any two fireflies and , and is the attractiveness at . The third item in (1) is a random vector, where is a random parameter generated from the interval and denotes a vector of random numbers drawn from a Gaussian distribution. Thus, the movement of firefly defined by (1) is not always directed to firefly . More details can be found in references [25, 29, 31, 32].

The location of a firefly is simply coded using a spatial coordinate, which consists of real vectors and continuous variables. Subsequently, the distance between any two fireflies and is generally defined by the Euclidean distance or the -norm. However, it is well known that, from the view of mathematics, the OSP is a specialized knapsack problem where some specified DOFs are selected to be placed by sensors, such that the structural performance can be described effectively. Thus, the parameters that are used for optimizing are states in which those DOFs are distributed by sensors and are discrete variables. As a result, the coding system and the movement strategies in the FA are inapplicable in the OSP problem with discrete variables. It is essential to do some modifications to the original FA, such that the underlying optimization concept of the FA can be moved to the OSP problem. Here, some improvements are integrated into the FA, and the SDFA is proposed to explore the optimal sensor configuration in structural health monitoring.

##### 2.2. Simple Discrete Firefly Algorithm

Being originated from the FA, the SDFA is integrated by three parts: the coding system, the definition of distance between two fireflies, and the movement scheme. The coding system involves the code of each firefly in feasible space. The distance definition is responsible for describing the distance between two fireflies so that the movement can be realized. And the movement scheme gives the evolution rules of the SDFA. All of the three parts are introduced in next three sections.

###### 2.2.1. Coding System

In the community of applying GA in finding the optimal sensor configuration, a widely used code approach is the one-dimensional binary coding system. Each individual in the population is coded by a one-dimensional binary string. In this code system, all of the candidate DOFs are put in a line. If the th DOF is placed by a sensor, the value of the th element in the string is 1. In contrast, if the th DOF is not placed by a sensor, the value of the th element in the string is 0. The total number of ones in the string is equal to the number of sensors that needs to be placed. This coding system is intuitive and easy to be initialized and operated. Here, in the SDFA, the one-dimensional binary coding system is employed. Each firefly in the population denotes a feasible sensor configuration, and the location of each firefly is represented by a one-dimensional binary string, as shown in Table 1. In the example of Table 1, it can be found that the 2nd, 3rd, 6th, and 9th DOFs are occupied by sensors. The total numbers of candidate DOFs and sensors are 10 and 4, respectively, because the length of the string is 10 and the total number of ones is 4 in this firefly code. This coding method is very simple and intuitive, which is beneficial for the next optimization operation. When initializing the firefly population, the first th elements in a string are set to 1, and the left elements are set to 0. Then, the shuffle algorithm is applied five times, such that the fireflies can be distributed in the feasible solution space uniformly as much as possible.