  To fully leverage the multi-task optimization paradigm for accelerating the
solution of expensive scheduling problems, this study has effectively tackled
three vital concerns. The primary issue is identifying auxiliary tasks that
closely resemble the original expensive task. We suggested a sampling strategy
based on job importance, creating a compact matrix by extracting crucial rows
from the entire problem specification matrix of the expensive task. This matrix
serves as an economical auxiliary task. Mathematically, we proved that this
economical auxiliary task bears similarity to its corresponding expensive task.
The subsequent concern revolves around making auxiliary tasks more
cost-effective. We determined the sampling proportions for the entire problem
specification matrix through factorial design experiments, resulting in a more
compact auxiliary task. With a reduced search space and shorter function
evaluation time, it can rapidly furnish high-quality transferable information
for the primary task. The last aspect involves designing transferable deep
information from auxiliary tasks. We regarded the job priorities in the (sub-)
optimal solutions to the economical auxiliary task as transferable invariants.
By adopting a partial solution patching strategy, we augmented specificity
knowledge onto the common knowledge to adapt to the target expensive task. The
strategies devised for constructing task pairs and facilitating knowledge
transfer, when incorporated into various evolutionary multitasking algorithms,
were utilized to address expensive instances of permutation flow shop
scheduling. Extensive experiments and statistical comparisons have validated
that, with the collaborative synergy of these strategies, the performance of
evolutionary multitasking algorithms is significantly enhanced in handling
expensive scheduling tasks.
