An efficient ILP formulation for the single machine scheduling problem

Abstract

This paper considers the problem of scheduling n jobs on a single machine. A fixed processing time and an execution interval are associated with each job. Preemption is not allowed. On the basis of analytical and numerical dominance conditions, an efficient integer linear programming formulation is proposed for this problem, aiming at minimizing the maximum lateness (L max). Experiments have been performed by means of a commercial solver that show that this formulation is effective on large problem instances. A comparison with the branch-and-bound procedure of Carlier is provided.

Minimizing the number of tardy jobs for the single machine scheduling problem: MIP-based lower and upper bounds

“…This paper describes a generic MIP for computing good-quality lower and upper bounds for the 1|r j | U j problem. The MIP formulation is inspired by the one of Briand et al [6] that solves the 1|r j |L max efficiently. Determining good-quality bounds is of interest for boosting branch-and-bound methods or for providing good initial solution to local search procedures.…”

“…In this section, the problem of finding a feasible job sequence is considered and a MIP is described, which has been originally introduced in [6]. It aims at determining the most dominant job sequence among the set S dom that is in the form…”

“…Obviously, it can be observed that the values of R k and D k variables can directly be deduced from the values of binary variables x ki . In [6], the authors state that if…”

“…Then Section 3 focuses on the problem of finding a feasible sequence to the SMSP and recalls a dominance theorem. In Section 4, a mathematical integer program (MIP) for determining a feasible job sequence, initially proposed in [6], is detailed. The following section discusses the validity of the dominance theorem stated in Section 3 for the case of the U j criterion.…”

Jackson's semi-preemptive scheduling on a single machine

“…Moreover, despite some few recent attempts to solve the 1jr j ,q j jC max [28][29][30], Carlier's algorithm is still considered as the champion 1jr j ,q j jC max solver in the scheduling community.…”