3.3. Optimization Techniques:
Four techniques viz. MoGA, MOPSO, PS, and MOEA/D were compared on three groundwater model domain sizes. MATLAB provides inbuilt functions for the implementation of MoGA and PS. The following subsections discuss the details of the optimization algorithms.
3.3.1 Multi-objective Genetic Algorithm: In MATLAB, ‘gamultiobj’ is the function used for the Multi-objective GA. This function uses a controlled and elitist genetic algorithm (The MathWork Inc.), which is a variant of NSGA-II (D. Kalyanmoy, 2001). This algorithm increases the diversity by favoring a variety of individuals even if they have a lower fitness value. After the initialization, generic operations (crossover, mutation, and selection) are iteratively performed. Consequently, crossover fraction, mutation rate, number of generations, and population size are major parameters affecting the performance of the algorithm. For the ANN model, these parameters were set to 0.85, 0.015, 2000, and 200 respectively, and for the simulation model, these were set as 0.85, 0.015, 150, and 25 respectively.
3.3.2 Multi-objective Particle Swarm Optimization: This heuristic algorithm is inspired by the movement of a bird flock. Each bird or particle has a position and local velocity in the feasible solution domain. The exploration of the domain by these particles is governed by both local and global velocity, which are computed based on the personal best and the global best solution obtained in the domain. In Coello et. al. (2004), the most widely used implementation of MOPSO introduced a mutation operator that enriches the exploration capability of the algorithm. Víctor Martínez-Cagigal (2020) provides a MATLAB implementation of Coello et. al. (2004), which is used in this work. The c1 and c2 were set to 2 for both ANN and simulation models. Maximum iterations and number of particles, for the ANN model, were set to 1000 and 100 respectively. For the simulation model, these were 150 and 25.
3.3.3 Pareto Search: The Pareto search algorithm by MATLAB, finds the non-dominated solution by the use of Pareto search in a set of points (archives and iterates). The algorithm uses the poll to find better solutions and if no better solution is available, then in the next iteration half multiply the mesh size. Theoretically, the algorithm converges to points near the true Pareto front (The MathWork Inc.). In MATLAB, paretosearch function is used for its implementation. Also, it takes fewer function evaluations than gamultiobj and it performs better or equal, in comparison to NSGA-II if there are no non-linear constraints (The MathWork Inc.). No parameter tuning is required for the Pareto search algorithm, though stopping criteria based upon tolerance and time can be adjusted.
3.3.4 Multi-objective Evolutionary Algorithm Based on Decomposition: MOEA/D solves a multi-objective by decomposing it into multiple scalar sub-problems and solving them simultaneously. The solution of these sub-problems is evaluated based on its neighboring sub-problems, which makes its computational cost lower in each generation in comparison to NSGA-II (Zhang et.al. 2007). Polynomial mutation (order n) is used as the mutation function. Several sub-problems, maximum iterations, percentage of the neighborhood, and mutation rate are the major parameters for MOEA/D. For the final ANN model, these parameters were set to 1500, 100, 40%, and 1/n respectively. For the simulation model, these were set to 100, 20, 40%, and 1/n respectively.
The use of these algorithms is computationally expensive with the groundwater simulation models, which require few seconds for a single run and require days to complete the single solution. Therefore, an ANN model is also developed to speed up the evaluation of cost function. The efficiency ANN-Optimization model is majored with the performance of the actual simulation-optimization algorithm by performing a feasible number of simulation runs. The ANN model was trained with a dataset generated by simulation models itself. The parameters of optimization algorithms are set to improve the solution set at the cost of increasing the number of simulations which is accommodated by the ANN model.