On the Dual and Inverse Problems of Scheduling Jobs to Minimize the Maximum Penalty

Abstract

<jats:p>In this paper, we consider the single-machine scheduling problem with given release dates and the objective to minimize the maximum penalty which is NP-hard in the strong sense. For this problem, we introduce a dual and an inverse problem and show that both these problems can be solved in polynomial time. Since the dual problem gives a lower bound on the optimal objective function value of the original problem, we use the optimal function value of a sub-problem of the dual problem in a branch and bound algorithm for the original single-machine scheduling problem. We present some initial computational results for instances with up to 20 jobs.</jats:p>

Approximation of the Objective Function of Single-Machine Scheduling Problem

“…In particular, for w 1 = 1 there exists a solution such that w j = X(1, j) and such that w j = Ỹ(1, j). In other words, the section of the set of solutions by the plane w 1 = 1 lies inside the parallelepiped described by the inequalities (13) and has at least one common point with each of its faces. Therefore, by Lemma 3 the center of this parallelepiped is an interior point of the set of solutions to the initial system of inequalities and is a solution to the initial system.…”

“…The goal is to schedule the jobs to proceed on the machine, minimizing some objective functions. A wide variety of studies of this problem can be found for such objective functions as the total or maximum lateness [8,9], the weighted number of tardy jobs [10], the total (weighted) completion time [11,12] or any arbitrary non-decreasing function of the completion time [13]. The idea is that the objective function is known and should be maximized or minimized.…”

Advances and Novel Approaches in Discrete Optimization

“…Lazarev et al [15] consider the single-machine problem with given release dates and the objective to minimize the maximum job penalty. While this problem is NP-hard in the strong sense, they introduce a dual and an inverse problem, which can be both polynomially solved.…”