Application of Enhanced Search Technique: Chaos-Directed Genetic Algorithm in Optimal Design of Water Distribution Network

 The water distribution network (WDN), a vital component of the water supply system, is an essential urban infrastructure distributing potable water to society. Its design, a non-deterministic polynomial-hard problem, has been a widely studied complex research problem for decades, with various optimization models proposed for its optimal design. Recent advancements in enhancing the computational efficiency of stochastic optimization algorithms by introducing chaotic force have elevated the scope of formulating chaos-directed evolutionary algorithms (EAs). The present study proposes one such approach, the chaos-directed genetic algorithm (CDGA) model, to improve the search mechanism of the genetic algorithm (GA) in solving the complex optimization problem of WDN optimal design.

In one of our recent works, the influence of chaotic maps with high-dimensionality, the Henon and Lorenz maps, are explored and compared to the low-dimensional Logistic map in improving the performance of GA. With the one-dimensional Logistic map demonstrating better computational improvement of GA, the present study considers it for formulating the CDGA model. The Logistic map is a non-linear first-order difference equation. Its dynamics evolve into various possible states of system range without repetition. For the search mechanism of the optimization technique to explore different regions of search space, this particular characteristic forms the most favorable feature. Consequently, by incorporating the chaotic force of the Logistic map into GA’s evolutionary mechanism by replacing every random search phenomenon, the CDGA model is formulated. A novel method of non-sequential allocation of chaotic dynamics is employed to induce chaotic force. Notably, the method is unique, using the same initial characteristics of the Logistic equation, retaining the chaos ergodicity for the evolutionary search.

To demonstrate the computational efficiency of the CDGA model, the enormously studied benchmark problem, the Hanoi network (HN), is considered. HN is a 34-dimensional problem having a complex search space with multiple locally optimal solutions. Defining the WDN optimization problem as the single-objective design framework subjected to linear and non-linear constraints of governing laws, the principal objective is to minimize the investment cost of HN pipes. While minimizing the pipe investment cost, the constraints levied ensure that the HN is hydraulically adequate to deliver the design demands. Thus, the optimization model formulated is the integrated framework with the simulation tool to simulate the WDN's hydraulic conditions. The code for the CDGA model is written in MATLAB R2015a and combined with the simulation software, EPANET 2.0, using the EPANET-MATLAB toolkit.

The computational results demonstrate the convergence precision of the CDGA model over its traditional GA, converging to the optimal cost of 6,081,564 units, the previous best solution reported for HN in the literature. Moreover, it outperforms many stochastic optimization models reported in the literature with computational efficiency in solving HN, particularly simulated annealing, shuffled complex, shuffled frog leaping, ant colony, particle swarm, harmony search, krill herd, and cuckoo search algorithms. Hence, from the results, the study suggests formulating chaos-directed optimization algorithms to improve their traditional model's computational efficiency in solving complex optimization problems.

Keywords: Water distribution network optimal design; Evolutionary algorithms; Genetic algorithm; Chaotic maps; Logistic equation; Hanoi network