Scheduling on a single machine under time-of-use electricity tariffs

We consider the problem of scheduling jobs on a single machine to minimize the total electricity cost of processing these jobs under time-of-use electricity tariffs. For the uniform-speed case, in which all jobs have arbitrary power demands and must be processed at a single uniform speed, we prove that the non-preemptive version of this problem is inapproximable within a constant factor unless P=NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {P} = \text {NP}$$\end{document}. On the other hand, when all the jobs have the same workload and the electricity prices follow a so-called pyramidal structure, we show that this problem can be solved in polynomial time. For the speed-scalable case, in which jobs can be processed at an arbitrary speed with a trade-off between speed and power demand, we show that the non-preemptive version of the problem is strongly NP-hard. We also present different approximation algorithms for this case, and test the computational performance of these approximation algorithms on randomly generated instances. In addition, for both the uniform-speed and speed-scaling cases, we show how to compute optimal schedules for the preemptive version of the problem in polynomial time.