A Strong Preemptive Relaxation for Weighted Tardiness and Earliness/Tardiness Problems on Unrelated Parallel Machines

Abstract

Research on due date oriented objectives in the parallel machine environment is at best scarce compared to objectives such as minimizing the makespan or the completion time related performance measures. Moreover, almost all existing work in this area is focused on the identical parallel machine environment. In this study, we leverage on our previous work on the single machine total weighted tardiness (TWT) and total weighted earliness/tardiness (TWET) problems and develop a new preemptive relaxation for the TWT and TWET problems on a bank of unrelated parallel machines. The key contribution of this paper is devising a computationally effective Benders decomposition algorithm for solving the preemptive relaxation formulated as a mixed integer linear program. The optimal solution of the preemptive relaxation provides a tight lower bound. Moreover, it offers a near-optimal partition of the jobs to the machines, and then we exploit recent advances in solving the non-preemptive single machine TWT and TWET problems for constructing non-preemptive solutions of high quality to the original problem. We demonstrate the effectiveness of our approach with instances up to 5 machines and 200 jobs.

An exact extended formulation for the unrelated parallel machine total weighted completion time problem

“…More recent surveys on parallel machine scheduling include (Mokotoff, 2001) and (Blazewicz et al, 2007, Chapter 5). A detailed discussion of the parallel machine scheduling literature on additive due date related performance measures is presented by Şen and Bülbül (2015).…”

“…It turns out that compared to custom B&B procedures, exact solution methods that rely on mathematical programming based decomposition techniques are far more promising for parallel machine scheduling problems with additive objective functions. This is true for both the total (weighted) completion time and the due date related performance measures (Şen and Bülbül, 2015). The underlying reason for this phenomenon is rooted in the tight lower bounds attained by good mathematical programming formulations of parallel machine scheduling problems.…”

“…This observation and the success of the mathematical programming based solution methods for parallel machine scheduling problems detailed above motivates the work in this paper. In this sense we follow suit with Chen and Powell (1999); van den Akker et al (1999) and apply a decomposition approach to a mathematical programming formulation that is demonstrated to provide tight lower bounds for the TWT and TWET objectives in the unrelated parallel machine environment (Şen and Bülbül, 2015). In the next section, we present our own formulation for Rm-TWCT and prove its correctness before introducing the solution algorithm based on Benders decomposition in Section 3.…”

“…The problem of determining the best schedule with this preemptive scheme is then formulated as an assignment or a transportation problem, in which a job j with an integer processing time p j is allocated a total of p j unit-length intervals in the planning horizon and no machine executes more than a single unit-length job at a time. Building upon and extending this body of work, Şen and Bülbül (2015) propose the preemptive formulation (TR − A) below for the unrelated parallel machine environment:…”

“…As discussed in-depth by Şen and Bülbül (2015), these constraints remove the drawback of the preemptive formulation in (Kedad-Sidhoum et al, 2008) stemming from having the unit-length jobs of a given job distributed over multiple machines; however, they also destroy the desirable polyhedral structure, and we no longer have a transportation problem on our hands. It turns out that (TR − A) is an MILP formulation of pseudo-polynomial size, which grows very quickly with increasing m, n, and long processing times.…”