Shift-Based Penalty for Evolutionary Constrained Multiobjective Optimization and its Application

This article presents a new constraint-handling technique (CHT), called shift-based penalty (ShiP), for solving constrained multiobjective optimization problems. In ShiP, infeasible solutions are first shifted according to the distributions of their neighboring feasible solutions. The degree of shift is adaptively controlled by the proportion of feasible solutions in the current parent and offspring populations. Then, the shifted infeasible solutions are penalized based on their constraint violations. This two-step process can encourage infeasible solutions to approach/enter the feasible region from diverse directions in the early stage of evolution, and guide diverse feasible solutions toward the Pareto optimal solutions in the later stage of evolution. Moreover, ShiP can achieve an adaptive transition from both diversity and feasibility in the early stage of evolution to both diversity and convergence in the later stage of evolution. ShiP is flexible and can be embedded into three well-known multiobjective optimization frameworks. Experiments on benchmark test problems demonstrate that ShiP is highly competitive with other representative CHTs. Further, based on ShiP, we propose an archive-assisted constrained multiobjective evolutionary algorithm (CMOEA), called ShiP+, which outperforms two other state-of-the-art CMOEAs. Finally, ShiP is applied to the vehicle scheduling of the urban bus line successfully.