Approximation algorithms for bi-objective parallel-machine scheduling in green manufacturing

We consider bi-objective parallel-machine scheduling in green manufacturing to minimize the makespan and total processing cost. Each machine has a different constant processing cost per unit time. For the objective of minimizing the makespan, given a total cost budget, we provide an approximation algorithm with a worst-case ratio of 33+14≈1.686, which improves the previous bound of 2. For the objective of minimizing the total processing cost, subject to all the jobs must be completed before a given common deadline, we provide an approximation algorithm with a worst-case ratio of 2+r3, where r is the ratio of the maximum to the minimum processing cost per unit time on a machine.