Approximation algorithms for scheduling parallel machines with an energy constraint in green manufacturing

Motivated by current green manufacturing standards, in this paper we study a parallel-machine scheduling model in which the energy cost incurred on each machine is machine-dependent and proportional to the load of the machine. The objective is to determine a production schedule with the minimum makespan subject to the energy constraint that the total energy cost does not exceed a given bound. We provide a technical lemma that enables us to design a very efficient approximation algorithm with a worst-case bound that can arbitrarily root approach 34, improving on the existing performance ratio of 33+1 4 approximate to 1.686 in the literature. By introducing the concept of a monotonic schedule, we are able to develop the first polynomial time approximation scheme for this scheduling problem. The scheduling problem studied in this paper is an important special case of the generalized assignment problem (GAP). Our techniques and results bring new insights into research on the GAP.