This paper explores the application of generative AI for systematic and inventive problem-solving in engineering design. Utilizing a multi-directional prompting approach, the study investigates the ability of AI chatbots to generate and evaluate innovative concepts based on numerous elementary solution principles. The research involved two sets of experiments with graduate and undergraduate students solving seven engineering design problems. The findings indicate that while generative AI can quickly generate a large number of ideas, it often overestimates the feasibility and usefulness of its solutions and tends to create overly complex designs. Comparisons of AI evaluations with those conducted by human participants revealed significant differences, highlighting the need for human oversight to ensure practical and contextually relevant outcomes. The experiments also revealed performance differences among various AI models, confirming a bias in self-assessment. Despite these challenges, integrating generative AI with multidirectional prompting using elementary inventive principles, primarily based on TRIZ methodology, proved effective in fostering innovative solutions. The study highlights the potential of simultaneously applying different AI models alongside human expertise to leverage the strengths of both large language models and human technical creativity.