**Alexander A. Lazarev 1, Nikolay Pravdivets 1,\* and Frank Werner <sup>2</sup>**


Received:11 June 2020; Accepted: 6 July 2020; Published: 10 July 2020

**Abstract:** 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.

**Keywords:** single-machine scheduling; minimization of maximum penalty; dual problem; inverse problem; branch and bound

**MSC:** 90B35; 90C57
