Scheduling with resource allocations originates from various real-life systems [1–۴], where the processing or setup times are controllable by resource allocations such as money, energy, fuel or additional manpower. Since more and more people from industrial or academic areas are concerned about improving the resource efficiency, deep investigation into this subfield has high practical significance.

Since its appearance in [5,6], lots of researches have been devoted to this subfield of scheduling. The following lists two criteria (other criteria can be found in [7]) that can be used to classify researches in the subfield. On the one hand, according to the machine settings, the relevant researches can be categorized as follows: single machine scheduling [8–۱۳], parallel machine scheduling [14], flow-shop scheduling [15,16], and batch-scheduling [17–۲۲]. On the other hand, researches in this subfield can also be classified into the following categories: scheduling without or with resource constraints. The former problem setting receives more attention in relevant researches [17,18]. The latter setting assumes that the allocations for each operation should not exceed the maximal amount. Such constraints are called technological constraints hereafter.