We address two offline job scheduling problems, where jobs can either be processed on a limited supply of energy-efficient machines on the edge, or offloaded to an unlimited supply of energy-inefficient machines on the cloud (called rejected in our context). The goal is to minimize the total energy consumed in processing all tasks. We consider a first scheduling problem with no due date (or deadline) constraints, and we formulate it as a scheduling problem with rejection, where the cost of rejecting a job is directly proportional to its processing time. We introduce a novel 5/4(1+ε) approximation algorithm BEKP by associating it with a Multiple Subset Sum problem for this version. Our algorithm is an improvement over the existing literature, which provides a (3/2 -1/2m) approximation for scenarios with arbitrary rejection costs. In the second scheduling problem, jobs have due date (or deadline) constraints, and the goal is to minimize the weighted number of late jobs. In our context, if a job is late, it is offloaded (rejected) to an energy-inefficient machine on the cloud, which incurs a cost directly proportional to its processing time of the job. We position this problem in the literature, and introduce a novel (1-(m-1)^m/m^m)-approximation algorithm MDP for this version, where we got our inspiration from an algorithm for the interval selection problem with a (1-m^m/(m+1)^m) approximation ratio for arbitrary rejection costs. We evaluate and discuss the effectiveness of our approaches through a series of experiments, comparing them to existing algorithms.