In this paper we characterize optimal schedules for scheduling problems with parallel machines and unit processing times by providing necessary and sufficient conditions of optimality. We show that the optimality conditions for parallel machine scheduling are equivalent to detecting negative cycles in a specially defined graph. For a range of the objective functions, we give an insight into the underlying structure of the graph and specify the simplest types of cycles involved in the optimality conditions. Using our results we demonstrate that the optimality check can be performed by faster algorithms in comparison with existing approaches based on sufficient conditions.

Personnel rostering is a personnel scheduling problem in which shifts are assigned to employees, subject to complex organisational and contractual time-related constraints. Academic advances in this domain mainly focus on solving specific variants of this problem using intricate exact or (meta)heuristic algorithms, while little attention has been devoted to studying the underlying structure of the problems. The general assumption is that these problems, even in their most simplified form, are NP-hard. However, such claims are rarely supported with a proof for the problem under study. The present paper refutes this assumption by presenting minimum cost network flow formulations for several personnel rostering problems. Additionally, these problems are situated among the existing academic literature to obtain insights into what makes personnel rostering hard. (C) 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.

We present a constraint programming formulation for the

Personnel scheduling problems can be decomposed into two stages. In the first stage for each employee the working days have to be fixed. In the second stage for each day of the planning period an intraday scheduling problem has to be solved. It consists of the assignment of shifts to the employees who have to work on the day and for each working period of an employee a task assignment such that the demand of all tasks for personnel is covered. In Robinson et al. (Burke and Trick (Eds.), Proceedings of the 5th International Conference on the Practice and Theory of Automated Timetabling, 18th August–20th August 2004, Pittsburgh, PA, USA, pp. 561–566, 2005 ), the intraday problem has been formulated as a maximum flow problem. The assumptions are that, employees are qualified for all tasks, their shifts are given, and they are allowed to change tasks during the day. In this work, we extend the network flow model to cover the case where not all employees are qualified to perform all tasks. The model is further extended to be able to calculate shifts of employees for the given day, assuming that an earliest starting time, a latest finishing time, and a minimal working time are given. Labour cost can be also taken into account by solving a minimum cost network flow problem.

Identifying underlying structures in combinatorial optimisation problems leads to a better understanding of a problem and, consequently, to efficient solution methodologies. The present paper introduces a new network flow formulation for a large class of nurse rostering problems. By solving an integer minimum cost flow problem in a carefully constructed network, nurses’ shift schedules can be constructed in polynomial time. The performance of the new formulation is compared with a state of the art algorithm on a benchmark dataset. Computational experiments show that the new formulation performs best both in terms of solution quality and computation time. By identifying inherent combinatorial structures which can be efficiently exploited, insight is gained into the problem’s complexity, thereby laying the foundations for a theory of nurse rostering.

The cyclic job-shop problem with transportation can be used to describe optimization problems in fully automated manufacturing systems or assembly lines. We study the problem where the machines have no buffers, which rapidly decreases the number of feasible solutions and, therefore, makes it a lot harder to find those feasible solutions. After formulating the problem, we will characterize feasible solutions based on the route of the robot and their properties. With the aim of minimizing the cycle time, we have developed a tree search method to construct feasible solutions and combined it with a bounding procedure. Computational results are reported and compared to those gained by solving the problem with an LP solver.

This paper focuses on the study of cyclic job-shop problems with transportation and blocking. Within this domain, there are many real world problems like large scale productions, robotic cells, software pipelining or hoist scheduling. The aim in general is to find, for each machine, a feasible order of all the operations processed on this machine, so that an objective function is optimised. In this paper, we consider the problem of minimising the cycle time (maximising the throughput) in a job-shop environment, where the jobs are transported by a single robot between the machines. Additionally to the problem description, we will give some explanations and interpretation possibilities of the problem height, which is often omitted in the literature. As the main contribution, we will present a new integer programming formulation and show that it outperforms an existing model from the literature.

CISched aims to bring together researchers and practitioners in all aspects of Computational Intelligence applied to a wide range of scheduling problems. Due to their exponential search spaces and high complexity, scheduling problems remain highly challenging in the current research. A significant amount of research effort has been made to explore various techniques. These include evolutionary computation, neural networks, swarm intelligence, fuzzy logic, and their hybridizations, etc. This symposium aims to explore recent advances in computational intelligence techniques on solving complex scheduling problems. The programme of this symposium features 10 full papers representing the state-of-the-art research in scheduling.

Due to its complexity, its challenging features, and its practical relevance, personnel scheduling has been heavily investigated in the last few decades. However, there is a relatively low level of study on models and complexity in these important problems. In this paper, we present mathematical models which cover specific aspects in the personnel scheduling literature. Furthermore, we address complexity issues by identifying polynomial solvable and NP-hard special cases.

This book describes models and algorithms for complex scheduling problems. It presents key algorithms in detail and includes examples. This second revised edition includes new material, in particular, further generalizations of the RCPSP.

We study an inverse counterpart of the two-machine flow-shop scheduling problem that arises in the context of inverse optimization. While in the forward scheduling problem all parameters are given and the objective is to find job sequence(s) for which the value of the makespan is minimum, in the inverse scheduling the exact values of processing times are unknown and they should be selected within given boundaries so that pre-specified job sequence(s) become optimal. We derive necessary and sufficient conditions of optimality of a given solution for the general case of the flow-shop problem when the job sequences on the machines can be different. Based on these conditions we prove that the inverse flow-shop problem is NP-hard even in the case of the same job sequence on both machines and produce a linear programming formulation for a special case which can be solved efficiently.

This paper investigates an adaptive constructive method for solving nurse rostering problems. The constraints considered in the problems are categorised into three classes: those that are sequence related, those that are nurse schedule related and those that are roster related. We propose a decomposition approach (to construct solutions) that consists of two stages: (1) to construct high quality sequences for nurses by only considering the sequence constraints, and (2) to iteratively construct schedules for nurses and the overall rosters, based on the sequences built and considering the schedule and roster constraints. In the second stage of the schedule construction, nurses are ordered and selected adaptively according to the quality of the schedules they were assigned to in the last iteration. Greedy local search is carried out during and after the roster construction, in order to improve the (partial) rosters built. We show that the local search heuristic during the roster construction can further improve the constructed solutions for the benchmark problems tested.In addition, we introduce new benchmark nurse rostering datasets which are based upon real world data. The data sets represent a variety of real world constraints. The publication of this problem data to the research community is aimed at closing the gap between theory and practice in nurse scheduling research. One of the main objectives is to encourage more research on these data sets.

A flow-shop batching problem with consistent batches is considered in which the processing times of all jobs on each machine are equal to p and all batch set-up times are equal to s. In such a problem, one has to partition the set of jobs into batches and to schedule the batches on each machine. The processing time of a batch B i is the sum of processing times of operations in B i and the earliest start of B i on a machine is the finishing time of B i on the previous machine plus the set-up time s. Cheng et al. (Naval Research Logistics 47:128–144, 2000) provided an O(n) pseudopolynomial-time algorithm for solving the special case of the problem with two machines. Mosheiov and Oron (European Journal of Operational Research 161:285–291, 2005) developed an algorithm of the same time complexity for the general case with more than two machines. Ng and Kovalyov (Journal of Scheduling 10:353–364, 2007) improved the pseudopolynomial complexity to \(O(\sqrt{n})\). In this paper, we provide a polynomial-time algorithm of time complexity O(log 3 n).

This chapter presents a survey on scheduling models (project scheduling and processor scheduling) and their computational complexity.

Due to its complexity and challenging features, personnel scheduling has been heavily investigated in decades. However, there is the lack of study on models and complexity in these important problems. In this paper, we review and present recent general and specific models in the personnel scheduling literature. In particular, we address the complexity issues and present polynomial solvable and NP-complete cases for these problems.

We study a range of counterparts of the single-machine scheduling problem with the maximum lateness criterion that arise in the context of inverse optimization. While in the forward scheduling problem all parameters are given and the objective is to find the optimal job sequence for which the value of the maximum lateness is minimum, in inverse scheduling the exact values of processing times or due dates are unknown, and they should be determined so that a prespecified solution becomes optimal. We perform a fairly complete classification of the corresponding inverse models under different types of norms that measure the deviation of adjusted parameters from their given estimates.