### Efficiency Criteria as a Solution to the Uncertainty in the Choice of Population Size in Population-Based Algorithms Applied to Water Network Optimization

Daniel Mora-Melià, Jimmy Gutiérrez-Bahamondes, Pedro Iglesias-Rey, F. Martínez-Solano
2016 Water
Different Population-based Algorithms (PbAs) have been used in recent years to solve all types of optimization problems related to water resource issues. However, the performances of these techniques depend heavily on correctly setting some specific parameters that guide the search for solutions. The initial random population size P is the only parameter common to all PbAs, but this parameter has received little attention from researchers. This paper explores P behaviour in a pipe-sizing
more » ... pipe-sizing problem considering both quality and speed criteria. To relate both concepts, this study applies a method based on an efficiency ratio E. First, specific parameters in each algorithm are calibrated with a fixed P. Second, specific parameters remain fixed, and the initial population size P is modified. After more than 600,000 simulations, the influence of P on obtaining successful solutions is statistically analysed. The proposed methodology is applied to four well-known benchmark networks and four different algorithms. The main conclusion of this study is that using a small population size is more efficient above a certain minimum size. Moreover, the results ensure optimal parameter calibration in each algorithm, and they can be used to select the most appropriate algorithm depending on the complexity of the problem and the goal of optimization. 2 of 17 focuses on WDN design, specifically the optimal sizing of pipes, ensuring that requirements regarding demands, pressures at nodes and velocities in lines are met. Optimal sizing of pipes includes nonlinear equations between flow losses and head, as well as discrete variables such as pipe diameter. Thus, the selected problem is a non-linear integer problem, where a set of solutions is selected among a discrete set of feasible diameters, i.e., a combinatorial optimization problem. Therefore, this problem can be interpreted as an NP-hard type problem , i.e., it cannot be solved using known methods in a deterministic polynomial manner. Previously, researchers developed optimization methods to solve this problem, including approaches that reduce the complexity of the original non-linear problem. In this sense, deterministic optimization techniques such as linear programming  and non-linear programming  were used to optimally design water distribution networks. Unfortunately, these methods have limitations. On one hand, linear programming simplifications assumed that all functions are linear. However, these simplifications reduce the accuracy of the final solution. On the other hand, nonlinear programming typically falls into local minima of the objective function, which depends on the starting point of convergence. To overcome these disadvantages new techniques were needed. More recently, stochastic optimization techniques, such as meta-heuristic algorithms, have been applied in the field of water resources. In this way, they have been used for a variety of purposes, as model calibration , optimal planning, design and operation of water systems [5,6], best management practice (BMP) models , etc. These algorithms allow full consideration of system's nonlinearity and widely explore the search space to obtain a good solution in a reasonable time, minimizing or maximizing an objective function and trying to avoid being trapped at local minima or maxima. Consequently, the application of these techniques extends the field search and the capacity to obtain better solutions. In addition, meta-heuristic algorithms require relatively little knowledge about the problem to solve, that is to say, they can be considered as problem-independent techniques    . For these reasons, meta-heuristic techniques are often used as alternatives to traditional optimization methods. Meta-heuristics can be divided into two groups: single point methods and population based-algorithms. Single point methods try to improve upon a specific solution by exploring its neighbourhood. Some examples of this class are Simulated Annealing  or Tabu Search  . Population-based methods combine a number of solutions to generate new solutions better than the previous ones. These algorithms have shown satisfactory capabilities to solve NP-hard optimization problems and they are the most well-established class of meta-heuristics for solving hydraulic engineering problems. Specifically, Population-based algorithms such as Genetic Algorithms (GA) [13, 14] , Memetic Algorithms , Harmony Search [16,17], Shuffled Frog Leaping Algorithms (SFLA)  and even hybrid techniques , have been successfully applied to the optimal sizing of pipes problem. Generally, all these algorithms share some basic principles. They use operators to calculate neighbour solutions that improve the quality of the initial solution, which is usually generated randomly. These selection mechanisms are guided by several specific parameters in each technique, and proper calibration of these parameters is essential for good algorithm performance, in terms of finding the best solution [20, 21] . However, researchers have concluded that there are no universally accepted values for the "best" algorithm parameter setting, since the behaviour of each algorithm also depends on the optimization problem and the computational effort invested in solving the problem. In this sense, Maier et al.  made a thorough study about what is the current situation and future challenges in the implementation of Population-based algorithms to solve problems related to water resources. Among the research challenges, the development of knowledge about what is the impact of algorithm operators on searching behaviour is highlighted. Among all the parameters specified before starting a PbA, population size is likely the calibration parameter that has received the least attention from researchers. However, this choice is one of the most