Order constraints for single machine scheduling with non-linear cost

Abstract

International audienceTypically in a scheduling problem we are given jobs of different processing times $p_j$ and different priority weights $w_j$, and need to schedule them on a single machine in order to minimize a specific cost function. In this paper we consider the non-linear objective function $\sum w_j C_j^\beta$, where $C_j$ is the completion time of job $j$ and $\beta>0$ is some arbitrary real constant. Except for $\beta=1$ the complexity status of this problem is open. Past research mainly focused on the quadratic case ($\beta=2$) and proposed different techniques to speed up exact algorithms. This paper proposes new pruning rules and generalizations of existing rules to non-integral $\beta$. An experimental study evaluates the impact of the proposed rules on the exact algorithm A*

For the airplane refueling problem local precedence implies global precedence

“…In addition to the above rules found in the literature, Dürr and Vásquez [4] stated the conjecture local precedence implies global precedence for the scheduling problems with a penalty function f (t) := t β , β > 0. Formally, the conjecture was stated as follows.…”

“…Recently, Dürr and Vásquez [4] provided new rules and generalized existing ones for a general penalty function f (t) := t β , β > 0. An extensive experimental study analyzed the effect of these rules to the performance of the branch-and-prune search.…”

“…Sen-Dileepan-Ruparel [15] for any increasing penalty function f (t), if w i > w j and p i ≤ p j , then i ≺ g j. Dürr-Vásquez [4] for any increasing concave penalty function…”

“…Formally, the conjecture was stated as follows. [4]) Consider a penalty function f (t) := t β and fix an arbitrary constant β > 0. Then for all jobs i, j, i ≺ j implies i ≺ g j.…”

“…The conjecture was based on the statement that for any increasing penalty function f (t) and i ≺ [a,b] [4]) and the numerous instances of tests where the conjecture was verified. In addition, the same authors gave a proof for the case β = 2 and showed that the stronger implication…”