Scheduling a Single Machine with Primary and Secondary Objectives

Abstract

We study a scheduling problem in which jobs with release times and due dates are to be processed on a single machine. With the primary objective to minimize the maximum job lateness, the problem is strongly NP-hard. We describe a general algorithmic scheme to minimize the maximum job lateness, with the secondary objective to minimize the maximum job completion time. The problem of finding the Pareto-optimal set of feasible solutions with these two objective criteria is strongly NP-hard. We give the dominance properties and conditions when the Pareto-optimal set can be formed in polynomial time. These properties, together with our general framework, provide the theoretical background, so that the basic framework can be expanded to (exponential-time) implicit enumeration algorithms and polynomial-time approximation algorithms (generating the Pareto sub-optimal frontier with a fair balance between the two objectives). Some available in the literature experimental results confirm the practical efficiency of the proposed framework.

Polynomial Algorithm for Constructing Pareto-Optimal Schedules for Problem 1∣rj∣Lmax,Cmax

“…This paper deals with a problem with two objective functions L max and C max , which in general case can be referred as 1|r j |L max , C max . This problem was considered in [15], where authors consider some dominance properties and conditions when the Pareto-optimal set can be formed in polynomial time. Definition 1.1 For any instance A of the problem, each permutation τ of the jobs of the set N is uniquely defines early schedule π A τ .…”

A Brief Look at Multi-Criteria Problems: Multi-Threshold Optimization versus Pareto-Optimization

“…The objective is to find a feasible schedule σ in which the maximum job lateness L max is the minimum possible one. This problem 1|r j |L max is, in fact, equivalent to the abovementioned one 1|r j , q j |C max with job delivery times, and hence, it is also strongly NP-hard [6].…”

“…With the Pareto-optimization approach, we need to solve two relevant problems: (1) among all feasible schedules with a given maximum job lateness, find one with the minimum makespan, and (2) vice versa, among all feasible schedules with a given makespan, find one with the minimum maximum job lateness. Both of these problems are strongly NP-hard [6].…”

“…Now it remains to verify condition (6), i.e., we wish to know if, among all schedules from the set S A 1 , there is one satisfying condition (6). In general, it may take an exponential time to answer this question for an arbitrary value A 2 since the corresponding decision problem is NP-complete.…”

Special Issue on Algorithms for Scheduling Problems

“…The last paper accepted for this special issue is [11], which addresses a scheduling problem, where jobs with given release times and due dates have to be processed on a single machine. The primary criterion of minimizing the maximum lateness of the given jobs makes this problem strongly NP-hard.…”