Abstract—Truss optimization problem has been vastly studied during the past 30 years and many different methods have been proposed for this problem. Even though most of these methods assume that the design variables are continuously valued, in reality, the design variables of optimization problems such as cross-sectional areas are discretely valued. In this paper, an improved hill climbing and an improved simulated annealing algorithm have been proposed to solve the truss optimization problem with discrete values for cross- sectional areas. Obtained results have been compared to other methods in the literature and the comparison represents that the proposed methods can be used more efficiently than other proposed methods. Keywords—Size Optimization of Trusses, Hill Climbing, Simulated Annealing. I. INTRODUCTION N recent years, many algorithms have been developed to SSsolve the structural engineering optimization problem. These optimization techniques can be categorized into classical and heuristic search methods. Linear programming, non-linear programming and optimality criteria are examples of classical optimization methods [1]. Most of the proposed methods for structural optimization problem are based on the assumption that the design variables are continuously valued. In reality, however, the design variables such as cross- sectional areas are discretely valued and they are chosen from a list of discrete variables [2]. In both continuous and discrete cases, different methods have been proposed to solve the problem. Evolutionary algorithms such as genetic algorithms have been mostly used in both cases. Rajeev and Krishnamoorthy in 1992 [3] and then Wu and Chow [4] in 1995 proposed a genetic algorithm approach for discrete optimization of trusses. Cheng [5] in 2010 integrated the concepts of genetic algorithms and the finite element method to propose an efficient algorithm for optimal design of steel truss arch bridges. Other methods have been also used to solve the problem. Li et al. [6] in 2006 used Morteza Kazemi Torbaghan is with the Department of civil engineering, kashmar branch, Islamic Azad University, Kashmar, Iran (corresponding author Morteza Kazemi Torbaghan, phone: +989153054418; e-mail: Kazemi@iaukashmar.ac.ir). Seyed Mehran Kazemi is with the Computer Science Department, University of British Columbia (e-mail: smkazemi@cs.ubc.ca). Rahele Zhiani is with the Department of Chemistry, Neyshabur branch, Islamic Azad University, Neyshabur, Iran (e-mail: R_Zhiani2006@yahoo.com). Fakhriye Hamed is with the Department of civil engineering, kashmar branch, Islamic Azad University, kashmar, Iran. (e-mail: h.faxriye@gmail.com). a particle swarm optimization algorithm to solve the problem. Other stochastic search techniques based on natural phenomena were suggested by Saka [7] in 2007. Lamberty [8] in 2008 proposed an efficient simulated annealing method for design optimization of trusses. Assari et al. [9] in 2012 presented an improved big bang – big crunch algorithm for size optimization of trusses. There are also lots of other methods applied to this optimization problem. In this paper, an improved hill climbing and also an improved simulated annealing algorithm have been proposed to solve the size optimization of trusses assuming discrete values for cross-sectional areas. The proposed algorithms use hill climbing and simulated annealing iteratively to find the optimum design for a given truss. The rest of this paper is organized as follows: in section II, the problem of size optimization for trusses has been formulated. Section III explains hill climbing algorithm. Section IV uses the explanation in section III to explain simulated annealing algorithm. In section V, the proposed algorithms to improve hill climbing and simulated annealing has been described. Section VI is related to experimental results. Finally, section VII shows the conclusion and summarizes the paper. II. PROBLEM FORMULATION The optimization problem is the minimization of the weight of the structure subject to stress, displacement and minimum member size constrains. The objective function is: where ɣ i is the material density of the member, L i is the length and A i is the cross-sectional area of the i-th bar. The problem is subject to tensile and compressive stress constraints, bounds on displacements, and side constraints on the areas, as follows: σ min ≤ σ i ≤ σ max i = 1, …, n δ min ≤ δ i ≤ δ max i = 1, …, m A i {Available areas} i = 1, …, n g where n is the number of members making up the structure, m is the number of nodes, ng is the number of groups (number of design variables), σ i and δ i are the stress and nodal deflection, respectively, and A i is the cross-sectional area [9]. Improved Hill Climbing and Simulated Annealing Algorithms for Size Optimization of Trusses Morteza Kazemi Torbaghan, Seyed Mehran Kazemi, Rahele Zhiani, and Fakhriye Hamed I World Academy of Science, Engineering and Technology International Journal of Civil and Environmental Engineering Vol:7, No:2, 2013 135 International Scholarly and Scientific Research & Innovation 7(2) 2013 scholar.waset.org/1307-6892/7358 International Science Index, Civil and Environmental Engineering Vol:7, No:2, 2013 waset.org/Publication/7358