# Algorithmic and mathematical programming approaches to scheduling problems with energy-based objectives

#### Abstract

This dissertation studies scheduling as a means to address the increasing concerns related to energy consumption and electricity cost in manufacturing enterprises. Two classes of problems are considered in this dissertation: (i) minimizing the makespan in a permutation flow shop with peak power consumption constraints (the PFSPP problem for short) and (ii) minimizing the total electricity cost on a single machine under time-of-use tariffs (the SMSEC problem for short). We incorporate the technology of dynamic speed scaling and the variable pricing of electricity into these scheduling problems to improve energy efficiency in manufacturing. The challenge in the PFSPP problem is to keep track of which jobs are running concurrently at any time so that the peak power consumption can be properly taken into account. The challenge in the SMSEC problem is to keep track of the electricity prices at which the jobs are processed so that the total electricity cost can be properly computed. ^ For the PFSPP problem, we consider both mathematical programming and combinatorial approaches. For the case of discrete speeds and unlimited intermediate storage, we propose two mixed integer programs and test their computational performance on instances arising from the manufacturing of cast iron plates. We also examine the PFSPP problem with two machines and zero intermediate storage, and investigate the structural properties of optimal schedules in this setting. ^ For the SMSEC problem, we consider both uniform-speed and speed-scalable machine environments. For the uniform-speed case, we prove that this problem is strongly NP-hard, and in fact inapproximable within a constant factor, unless P = NP. In addition, we propose an exact polynomial-time algorithm for this problem when all the jobs have the same work volume and the electricity prices follow a so-called pyramidal structure. For the speed-scalable case, in which jobs can be processed at an arbitrary speed with a trade-off between speed and energy consumption, we show that this problem is strongly NP-hard and that there is no polynomial time approximation scheme for this problem. We also present different approximation algorithms for this case and test the computational performance of these approximation algorithms on randomly generated instances.^

#### Degree

Ph.D.

#### Advisors

Nelson A. Uhan, Purdue University, Andrew L. Liu, Purdue University.

#### Subject Area

Engineering, Industrial|Operations Research

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