An Approximate Pareto Set for Minimizing the Maximum Lateness and Makespan on Parallel Machines

We consider the two-parallel machines scheduling problem, with the aim of minimizing the maximum lateness and the makespan. Formally, the problem is defined as follows. We have to schedule a set J of n jobs on two identical machines. Each job i in J has a processing time p_i and a delivery time q_i. Each machine can only perform one job at a given time. The machines are available at time t=0 and each of them can process at most one job at a given time. The problem is to find a sequence of jobs, with the objective of minimizing the maximum lateness L_max and the makespan C_max. With no loss of generality, we consider that all data are integers and that jobs are indexed in non-increasing order of their delivery times: q_1 >= q_2 >= ... >= q_n. This paper proposes an exact algorithm (based on a dynamic programming) to generate the complete Pareto Frontier in a pseudo-polynomial time. Then, we present an FPTAS (Fully Polynomial Time Approximation Scheme) to generate an approximate Pareto Frontier, based on the conversion of the dynamic programming. The proposed FPTAS is strongly polynomial. Some numerical experiments are provided in order to compare the two proposed approaches.