Single machine group scheduling with resources dependent setup time

Aiming at the problem of the single machine group scheduling with continuous resources, the model of single machine group scheduling with resources dependent setup time is set up. In this model, the jobs in the same group can not be separated, the setup time of a group is a linear non-increasing function of the amount of resources consumed, and the jobs in the same group have a time dependent learning effect. Three versions of the problem are discussed. In the first one, the objective is to minimize the total resources consumption under the makespan constrains, whereas in the second one, the objective is to minimize the makespan. In the third one, the objective is to minimize the sum of weighted makespan and total weighted resources consumption under the total resources consumption constrains. The optimal allocation methods are presented respectively. It also illustrates the validity of the algorithms by examples.