Polynomial-time approximation schemes for scheduling problems with time lags

We identify two classes of machine scheduling problems with time lags that possess Polynomial-Time Approximation Schemes (PTASs). These classes together, one for minimizing makespan and one for minimizing total completion time, include many well-studied time lag scheduling problems. The running times of these approximation schemes are polynomial in the number of jobs, but exponential in the number of machines and the ratio between the largest time lag and the smallest positive operation time. These classes constitute the first PTAS results for scheduling problems with time lags

Scheduling with Time Lags

Scheduling is essential when activities need to be allocated to scarce resources over time. Motivated by the problem of scheduling barges along container terminals in the Port of Rotterdam, this thesis designs and analyzes algorithms for various on-line and off-line scheduling problems with time lags. A time lag specifies a minimum time delay required between the execution of two consecutive operations of the same job. Time lags may be the result of transportation delays (like the time required for barges to sail from one terminal to the next), the duration of activities that do not require resources (like drying or cooling down), or intermediate processes on non-bottleneck machines between two bottleneck machines. For the on-line flow shop, job shop and open shop problems of minimizing the makespan, we analyze the competitive ratio of a class of greedy algorithms. For the off-line parallel flow shop scheduling problem with time lags of minimizing the makespan, we design algorithms with fixed worst-case performance guarantees. For two special subsets of scheduling problems with time lags, we show that Polynomial-Time Approximation Schemes (PTAS) can be constructed under certain mild conditions. For the fixed interval scheduling problem, we show that the flow shop problem is solvable in polynomial time in the case of equal time lags but that it is NP-hard in the strong sense for general time lags. The fixed interval two-machine job shop and open shop problems are shown to be solvable in polynomial time if the time lags are smaller than the processing time of any operation

Serial-batch scheduling – the special case of laser-cutting machines

The dissertation deals with a problem in the field of short-term production planning, namely the scheduling of laser-cutting machines. The object of decision is the grouping of production orders (batching) and the sequencing of these order groups on one or more machines (scheduling). This problem is also known in the literature as "batch scheduling problem" and belongs to the class of combinatorial optimization problems due to the interdependencies between the batching and the scheduling decisions. The concepts and methods used are mainly from production planning, operations research and machine learning

Algorithmic And Mathematical Programming Approaches To Scheduling Problems With Energy-Based Objectives

This dissertation studies scheduling as a means to address the increasing concerns related to energy consumption and electricity cost in manufacturing enterprises. Two classes of problems are considered in this dissertation: (i) minimizing the makespan in a permutation flow shop with peak power consumption constraints (the PFSPP problem for short) and (ii) minimizing the total electricity cost on a single machine under time-of-use tariffs (the SMSEC problem for short). We incorporate the technology of dynamic speed scaling and the variable pricing of electricity into these scheduling problems to improve energy efficiency in manufacturing.The challenge in the PFSPP problem is to keep track of which jobs are running concurrently at any time so that the peak power consumption can be properly taken into account. The challenge in the SMSEC problem is to keep track of the electricity prices at which the jobs are processed so that the total electricity cost can be properly computed. For the PFSPP problem, we consider both mathematical programming and combinatorial approaches. For the case of discrete speeds and unlimited intermediate storage, we propose two mixed integer programs and test their computational performance on instances arising from the manufacturing of cast iron plates. We also examine the PFSPP problem with two machines and zero intermediate storage, and investigate the structural properties of optimal schedules in this setting. For the SMSEC problem, we consider both uniform-speed and speed-scalable machine environments. For the uniform-speed case, we prove that this problem is strongly NP-hard, and in fact inapproximable within a constant factor, unless P = NP. In addition, we propose an exact polynomial-time algorithm for this problem when all the jobs have the same work volume and the electricity prices follow a so-called pyramidal structure. For the speed-scalable case, in which jobs can be processed at an arbitrary speed with a trade-off between speed and energy consumption, we show that this problem is strongly NP-hard and that there is no polynomial time approximation scheme for this problem. We also present different approximation algorithms for this case and test the computational performance of these approximation algorithms on randomly generated instances

Shop scheduling with availability constraints

Scheduling Theory studies planning and timetabling of various industrial and human activities and, therefore, is of constant scientific interest. Being a branch of Operational Research, Theory of Scheduling mostly deals with problems of practical interest which can be easily (from a mathematical point of view) solved by full enumeration and at the same time usually require enormous time to be solved optimally. Therefore, one attempts to develop algorithms for finding optimal or near optimal solutions of the problems under consideration in reasonable time. If the output of an algorithm is not always an optimal solution then the worst-case analysis of this algorithm is undertaken in order to estimate either a relative error or an absolute error that holds for any given instance of the problem. Scheduling problems which are usually considered in the literature assume that the processing facilities are constantly available throughout the planning period. However, in practice, the processing facility, e.g. a machine, a labour, etc. can become non-available due to various reasons, e.g. breakdowns, lunch breaks, holidays, maintenance work, etc. All these facts stimulate research in the area of scheduling with non-availability constraints. This branch of Scheduling Theory has recently received a lot of attention and a considerable number of research papers have been published. This thesis is fully dedicated to scheduling with non-availability constraints under various assumptions on the structure of the processing system and on the types of non-availability intervals