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We are given a set of parallel jobs that have to be executed on a set of speed-scalable processors varying their speeds dynamically. Running a job at a slower speed is more energy-efficient, however, it takes a longer time and affects the performance. Every job is characterized by the processing volume and the number or the set of the required processors. Our objective is to minimize the maximum completion time so that the energy consumption is not greater than a given energy budget. For various particular cases, we propose polynomial-time approximation algorithms, consisting of two stages. At the first stage, we give an auxiliary convex program. By solving this problem, we get processing times of jobs and a lower bound on the makespan. Then, at the second stage, we transform our problem into the corresponding scheduling problem with the constant speed of processors and construct a feasible schedule. We also obtain an “almost exact” solution for the preemptive settings based on a configuration linear program.

The paper is concerned with the two-machine flow shop, where each job requires an additional resource (referred to as storage space) from the start of its first operation till the end of its second operation. The storage requirement of a job is determined by the processing time of its first operation. At any point in time, the total consumption of this additional resource cannot exceed a given limit (referred to as the storage capacity). The goal is to minimise the makespan, i.e. to minimise the time needed for the completion of all jobs. This problem is NP-hard in the strong sense. The paper analyses how the parameter - a lower bound on the storage capacity specified in terms of the processing times, affects the computational complexity.