Self-Adaptive Penalty Approach Compared with Other Constraint-Handling Techniques for Pipeline Optimization

Optimal design and rehabilitation of a water distribution system is a constrained nonlinear optimization problem. A penalty function is often employed to transform a constrained into a nonconstrained optimization problem within the framework of a genetic algorithm search. A penalty factor is used for defining the penalty function and calculating the penalty cost for the solutions with constraint violation. Effective penalty factors vary from one optimization model to another. This paper introduces a self-adaptive penalty approach to artificially evolve both the penalty factor and the design solutions. Solution robustness is proposed and quantified, along with solution fitness, for representing the boundary between the infeasible and feasible solutions of a high-dimension optimization problem. The solution space of multiple dimensions is mapped onto a 2D space, providing significant insight into the self-adaptive penalty approach, which is compared with other constraint-handling techniques tested on a benchmark example. The results show that this approach is more effective than other penalty methods used for searching for optimal and near-optimal solutions. The self-adaptive penalty approach relieves modelers from tuning the penalty function and facilitates a practical optimization modeling for water distribution design.