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.