## 2 Method
### Harmony Search Method  
The use of meta-heuristic techniques for optimisation problems in various fields of science and engineering has increased in recent years, especially in structural engineering. For example, the harmony search and flower pollination algorithms have been employed for the optimum design of truss systems [27, 28], steel frames [29], cylindrical reinforced concrete walls [30], plane stress systems [31], PID controlled active tuned mass damper [32], retaining walls [33], and for the stacking sequence optimisation of laminated composite plates [34]. In addition to harmony search and flower pollination, techniques such as random forest [35, 36], gene expression programming [37], supervised machine learning [38], neuro-swarm optimisation [39] and the imperialist competitive algorithm [40] have also been applied to various engineering problems.  
The harmony search algorithm was developed by [PERSON] et al. [41] and has been widely adopted for the optimisation of a water network design [42], a slope stability analysis [43], heat and power systems [44], job shop scheduling [45], team orienteering [46], and vehicle routing [47]. The method was initially designed with discrete valued data for musical composition and was then further developed for application to the optimisation of continuous valued solution vectors, e.g., those encountered in the dimensioning of structural components. A parameter-setting-free version of the harmony search algorithm was developed by [PERSON] and [PERSON] [48]; this algorithm is more accessible and efficient because the proper selection of algorithm-specific parameters is a difficult task.  
The harmony search optimisation algorithm requires a predetermined number of design variables and an objective quantity to be maximised or minimised. The design variables of a doubly symmetric plate girder are the width of the flange (\(b_{f}\)), thickness of the flange (\(t_{f}\)), depth of the web plate (\(D\)), thickness of the web plate (\(t_{w}\)), and stiffener spacing (\(a\)), and the objective quantity to be minimised is the cross-sectional area of the girder beam. The harmony search optimisation process starts with the generation of a certain number of design variable combinations, each of which is called a candidate solution vector. This initial population of candidate solutions is randomly generated within predefined design constraints. From any given population of solutions, the solution vectors that deliver the best and worst results are identified. In the next step, based on certain rules, a new candidate solution is generated and compared to the members of the previously generated population. If the new candidate solution performs better than the worst performing solution vector in the population, the newly generated solution vector is incorporated into the population and the previous worst-performing solution vector is removed from the population. This procedure is repeated for a predetermined number of iterations, and the convergence of the result is observed.  
The process of generating the new candidate solution vector relies on the harmony memory consideration rate (\(H\)\(M\)\(R\)) and the pitch adjustment rate (\(PAR\)). In this analysis, these parameters were treated as variable values and calculated using \(H\)\(M\)\(R\)  
constraints given in Equations (7) and (8), the minimum sizes are defined for the flange and the web. These geometric constraints are in the form of lower bounds for the flange, and web dimensions and are chosen to be of the same order of magnitude as the specimen dimensions in the experiment conducted by [PERSON] et al. (2014). These lower bounds are listed in Table 2.