The present study focuses on exploring optimal transfer solutions for the Demonstration and Experiment of Space Technology for INterplanetary voYage (DESTINY) mission, which was proposed as an Institute of Space and Astronautical Science (ISAS) Epsilon-class small program in 2013 based on the "Space Science & Exploration Roadmap" proposed by the ISAS and later approved by the government committee on space policy. During the DESTINY mission, the spacecraft will first be injected into a low elliptical orbit by an Epsilon rocket, and its altitude will then be increased to approach the Moon using an ion engine system. Next, the spacecraft will be injected via a transfer orbit to the L2 Halo orbit of the Sun-Earth system by way of gravity assist from the Moon. As the spacecraft revolves around the Earth several hundred times, it gradually increases its altitude. Thus, the launch time and thrust profile must be chosen properly. Note that there are several conflicting requirements, including the reduction of fuel consumption, total flight time, and maximum eclipse time, that must be taken into consideration. To satisfy these requirements, many-objective evolutionary computation is applied to find better orbital designs.