On rescheduling due to machine disruption while to minimize the total weighted completion time

TL;DR

This paper addresses a single machine rescheduling problem caused by unexpected machine unavailability, aiming to minimize total weighted completion time and schedule deviation, with solutions including an exact algorithm and an approximation scheme.

Contribution

It introduces a novel rescheduling model that balances minimizing total weighted completion time with limiting schedule deviations, providing the first exact and approximation algorithms for this NP-hard problem.

Findings

A pseudo-polynomial time exact algorithm is developed.

A fully polynomial time approximation scheme is proposed.

The approximation scheme is proven to be the best possible for this problem.

Abstract

We investigate a single machine rescheduling problem that arises from an unexpected machine unavailability, after the given set of jobs has already been scheduled to minimize the total weighted completion time. Such a disruption is represented as an unavailable time interval and is revealed to the production planner before any job is processed; the production planner wishes to reschedule the jobs to minimize the alteration to the originally planned schedule, which is measured as the maximum time deviation between the original and the new schedules for all the jobs. The objective function in this rescheduling problem is to minimize the sum of the total weighted completion time and the weighted maximum time deviation, under the constraint that the maximum time deviation is bounded above by a given value. That is, the maximum time deviation is taken both as a constraint and as part of the objective function. We present a pseudo-polynomial time exact algorithm and a fully polynomial time approximation scheme, the latter of which is the best possible given that the general problem is NP-hard.