Polynomial Time Approximation Scheme for Two Parallel Machines Scheduling with a Common Due Date to Maximize Early Work

Malgorzata Sterna,Kateryna Czerniachowska

doi:10.1007/s10957-017-1147-7

Malgorzata Sterna, Kateryna Czerniachowska

Open Access

https://doi.org/10.1007/s10957-017-1147-7

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Abstract

We study the scheduling problem with a common due date on two parallel identical machines and the total early work criterion. The problem is known to be NP-hard. We prove a few dominance properties of optimal solutions of this problem. Their proposal was inspired by the results of some auxiliary computational experiments. Test were performed with the dynamic programming algorithm and list algorithms. Then, we propose the polynomial time approximation scheme, based on structuring problem input. Moreover, we discuss the relationship between the early work criterion and the related late work criterion. We compare the computational complexity and approximability of scheduling problems with both mentioned objective functions.

Highlights

The scheduling theory provides models and methods helpful in solving practical problems
We returned to the studies on the late work minimization/early work maximization on parallel identical machines, which is a classical scheduling model, finding many practical applications
We collected results on early/late work scheduling problems, which have been published in the literature since the last survey paper on this subject appeared

Summary

Introduction

The scheduling theory provides models and methods helpful in solving practical problems (cf., e.g., [1,2]). We analyze the early work criterion, which maximizes the amount of work executed before the due date (i.e. the total size of early parts of jobs) and the late work criterion, strictly related to this objective function, minimizing late parts of jobs [3] These performance measures find many practical applications. We proved a few dominance properties of optimal solutions of this problem, which allowed us to propose the polynomial time approximation scheme These properties were disclosed based on the results of some auxiliary computational experiments performed for dynamic programming [9] and list algorithms. We studied the relationship between the late and early work, which has not been addressed formally before We showed that these two criteria are equivalent when the optimal solutions are considered, but they have different nature, when the existence of approximation algorithms with a bounded approximation ratio is taken into account.

Related Work

Problem Formulation

Dominance Properties

Polynomial Time Approximation Scheme

Conclusions