Online scheduling to minimize modified total tardiness with an availability constraint

We consider online scheduling problems to minimize modified total tardiness. The problems are online in the sense that jobs arrive over time. For each job J(j), its processing time p(j), due date d(j) and weight w(j) become known at its arrival time (or release time) r(j). Preemption is not allowed. We first show that there is no finite competitive ratio for problem 1 vertical bar online, r(j), d(j)vertical bar Sigma w(j)T(j). So we focus on problem 1 vertical bar online, r(j), d(j)vertical bar Sigma w(j)(T-j + d(j) j and show that D-SWFT (Delayed Shortest Weighted Processing Time) algorithm is 3-competitive. We further study two problems 1 vertical bar online, r(j), d(j), h(1), N - res vertical bar Sigma w(j)(T-j + d(j)) and 1 vertical bar online, r(j), d(j), h(1), N - res vertical bar Sigma w(j)(T-j + d(j)), where res and N - res denote resumable and non-resumable models respectively, and h(1) denotes a non-available time interval vertical bar s, alpha s vertical bar with s > 0 and alpha >= 1. We give a lower bound of I + a for both problems and prove that M - D - SWPT (Modified D-SWPT) is 3 alpha and 6 alpha-competitive in the resumable and nonresumable models, respectively. Moreover, we extend the upper bounds to the scenario of parallel machine scheduling with uniform job weight and an assumption that all machines have the same non-available time interval vertical bar s, alpha s vertical bar. A lower bound of min{alpha, 1 + alpha/m} is given as well for the scenario. (C) 2009 Elsevier B.V. All rights reserved.