Genetic algorithms have been widely applied to solving single-objective optimization problems in water resources system analysis (Bavic et al. 1994; Wu and Simpson 1996, 1997a, 1997b and 2001; Wu et al. 2000 and 2001). In recent years, multi-objective genetic algorithms have been found to be more effective than traditional optimization techniques at solving multi-objective optimization problems. A wide range of multi-objective optimization problems have been successfully solved by using evolutionary algorithms.

There is no need to modify or simplify the system hydraulics and design criteria to fit multi-objective GA. Single-objective optimization is used to identify the optimal or near-optimal solutions according to the sole objective function. As soon as a solution is found better than the current-best solution, it is accepted. Multi-objective optimization is to locate the non-inferior (or non-dominated) solutions in solution space. Solution A is called non-inferior to solution B if and only if solution A is no worse than solution B in all the objectives. It is also said that solution A dominates solution B or that solution A is a non-dominated solution. A global non-dominated solution is defined as the solution that is no worse than any other feasible solutions in all the objectives. There exist multiple global non-dominated solutions. The task of a multi-objective optimization is to search for all the global non-dominated or non-inferior solutions also known as the Pareto-optimal set or Pareto-optimal front.

Conventionally, a multi-objective optimization problem was transformed into a single-objective optimization problem by using two approaches including weighted sum of objectives and e-constraint method (Cohon, 1978). Weighted sum approach applies a set of weighting factors to all the objectives and sums up the weighted objectives to construct a composite single objective. It is expected that the optimization of a composite objective is equivalent to the optimization of the original multiple objectives, but the optimal solution depends on the chosen weights and it can only search for a single optimal solution rather than Pareto-optimal solutions in one run. The constraint method chooses one of the objective functions and treats the other objective functions as constraints. Each of the constraints is limited to a prescribed value. It transforms a multi-objective optimization problem into a single-objective optimization. The optimal solution resulted by the constraint method, however, depends on the pre-defined constraint limits. Pareto-optimal solutions can be obtained by performing multiple runs of the single-objective optimization problem using different weighting factors or constraint limits. The more combinations of weighting factors or constraint limits, the more optimization runs are required, the greater the computational cost. In contrast, multi-objective genetic algorithm concurrently optimizes all the objective functions in one run without any fix-up on objective functions. It provides an effective method for handling multi-objective optimization.

The goal of single-objective optimization is to search for an optimal solution. Multi-objective optimization has two goals during the search process. One goal is to find a set of Pareto-optimal solutions as close as possible to Pareto-optimal front. The second goal is to maintain a set of Pareto-optimal solutions as diverse as possible. Searching for Pareto-optimal solutions is certainly the primary task for multi-objective optimization. A solution of single-objective optimization problem is evaluated by the objective value, which directly contributes to the fitness of the corresponding genotype solution. However, the fitness of a solution for multi-objective optimization problem is determined by the solution dominance that can be defined as the number of solutions dominated among the current population of solutions. The stronger the dominance, the greater the fitness is assigned to a solution. While identifying Pareto-optimal solutions is important, maintaining the diversity of Pareto-optimal solutions is also essential. Dealing with multi-objective optimization, such as minimizing cost and maximizing benefit for a water distribution system, it is anticipated that optimal trade-off solutions are found and uniformly distributed for the entire range of cost budget. This is normally achieved by using a method of fitness sharing or solution clustering.

To effectively solve the problem of cost-benefit trade-off optimal design, as formulated in the early section, fast messy genetic algorithm (Goldberg et al. 1993) has been extended to handle the multi-objective functions. The multi-objective fast messy GA has been integrated with the WaterGEMS hydraulic network solver. The integrated approach (Wu et al. 2002) provides a powerful design optimization tool to assist hydraulic engineers to practically and efficiently design a water distribution system. It offers capability of three levels of optimization design analysis, including minimum cost design, maximum benefit design and cost-benefit trade-off design optimization.