### Book chapter

## Lower and Upper Bounds for the Preemptive Single Machine Scheduling Problem with Equal Processing Times

The preemptive single machine scheduling problem of minimizing the total weighted completion time with equal processing times and arbitrary release dates is one of the four single machine scheduling problems with an open computational complexity status. In this paper we present lower and upper bounds for the exact solution of this problem based on the assignment problem. We also investigate properties of these bounds and worst-case behavior.

The assembly process is extremely complex for aircraft and its management requires to address numerous optimization problems related to the assignment of tasks to workstations, staffing problem for each workstation and finally the assignment of tasks to operators at each workstation. This paper treats the latter problem dealing with the assignment of tasks to operators under ergonomic constraints. The problem of optimal tasks scheduling in aircraft assembly line is modelled as Resource-Constrained Project Scheduling Problem (RCPSP). The objective of this research is to assign tasks to operators and to find an optimal schedule of task processing under economic and ergonomic constraints. Two different models to solve this problem are presented and evaluated on an industrial case study.

The preemptive single machine scheduling problem of minimizing the total weighted completion time with arbitrary processing times and release dates is an important NP-hard problem in scheduling theory. In this paper we present an efficient high-quality heuristic for this problem based on the WSRPT (Weighted Shortest Remaining Processing Time) rule. The running time of the suggested algorithm increases only as a square of the number of jobs. Our computational study shows that very large size instances might be treated within extremely small CPU times and the average error is always less than 0.1%.

In this paper, the single-track railway scheduling problem with two stations and several segments of the track is considered. Two subsets of trains are given, where trains from the first subset go from the first station to the second station, and trains from the second subset go in the opposite direction. The speed of trains over each segment is the same. A polynomial time reduction from the problem under consideration to a special case of the single-machine equal-processing-time scheduling problem with setup times is presented. Different polynomial time algorithms are developed for special cases with divers objective functions under various constraints. Moreover, several theoretical results which can be ranked in a series of similar investigations of NP-hardness of equal-processing-time single-machine scheduling problems without precedence relations are obtained.

In this paper, the single-track railway scheduling problem with two stations and several segments of the track is considered. Two subsets of trains are given, where trains from the first subset go from the first station to the second station, and trains from the second subset go in the opposite direction. The speed of trains over each segment is the same. A polynomial time reduction from the problem under consideration to a special case of the single-machine equal-processing-time scheduling problem with setup times is presented. Different polynomial time algorithms are developed for special cases with divers objective functions under various constraints. Moreover, several theoretical results which can be ranked in a series of similar investigations of NP-hardness of equal-processing-time single-machine scheduling problems without precedence relations are obtained.

The cell formation problem (CFP) is an NP-hard optimization problem considered for cell manufacturing systems. Because of its high computational complexity several heuristics have been developed for solving this problem. In this paper we present a branch and bound algorithm which provides exact solutions of the CFP. This algorithm finds optimal solutions for 13 problems of the 35 popular benchmark instances from the literature.

In order to implement the human-centric manufacturing and sustainability concepts in industry, an important effort should be done in order to model working conditions for human operators and improve them. Several studies have been conducted for mass production assembly lines where short cycle times make the work content highly repetitive. However, the case of low-volume production with long cycle times and different impacts on human operators has been rarely considered in the literature. In this paper, we develop a model to take into account the associated ergonomic risks in assembly lines with long cycle times. An optimization method is also developed in order to schedule tasks and assign the required tasks to a set of human operators taking into account the existing ergonomic risks.