Fast divide-and-conquer algorithms for preemptive scheduling problems with controllable processing times - a polymatroid optimization approach

We consider a variety of preemptive scheduling problems with controllable processing times on a single machine and on identical/uniform parallel machines, where the objective is to minimize the total compression cost. In this paper, we propose fast divide-and-conquer algorithms for these scheduling problems. Our approach is based on the observation that each scheduling problem we discuss can be formulated as a polymatroid optimization problem. We develop a novel divide-and-conquer technique for the polymatroid optimization problem and then apply it to each scheduling problem. We show that each scheduling problem can be solved in $ \O({\rm T}_{\rm feas}(n) \times\log n)$ time by using our divide-and-conquer technique, where n is the number of jobs and Tfeas(n) denotes the time complexity of the corresponding feasible scheduling problem with n jobs. This approach yields faster algorithms for most of the scheduling problems discussed in this paper.

Fifty years of scheduling: a survey of milestones

Scheduling has become a major field within operational research with several hundred publications appearing each year. This paper explores the historical development of the subject since the mid-1950s when the landmark publications started to appear. A discussion of the main topics of scheduling research for the past five decades is provided, highlighting the key contributions that helped shape the subject. The main topics covered in the respective decades are combinatorial analysis, branch and bound, computational complexity and classification, approximate solution algorithms and enhanced scheduling models.

Shop-scheduling problems with fixed and non-fixed machine orders of the jobs

The paper deals with the determination of an optimal schedule for the so-called mixed shop problem when the makespan has to be minimized. In such a problem, some jobs have fixed machine orders (as in the job-shop), while the operations of the other jobs may be processed in arbitrary order (as in the open-shop). We prove binary NP-hardness of the preemptive problem with three machines and three jobs (two jobs have fixed machine orders and one may have an arbitrary machine order). We answer all other remaining open questions on the complexity status of mixed-shop problems with the makespan criterion by presenting different polynomial and pseudopolynomial algorithms.

Heuristics for short route job shop scheduling problems

We consider two “minimum”NP-hard job shop scheduling problems to minimize the makespan. In one of the problems every job has to be processed on at most two out of three available machines. In the other problem there are two machines, and a job may visit one of the machines twice. For each problem, we define a class of heuristic schedules in which certain subsets of operations are kept as blocks on the corresponding machines. We show that for each problem the value of the makespan of the best schedule in that class cannot be less than 3/2 times the optimal value, and present algorithms that guarantee a worst-case ratio of 3/2.

Models of core/periphery structures

A common but informal notion in social network analysis and other fields is the concept of a core/periphery structure. The intuitive conception entails a dense, cohesive core and a sparse, unconnected periphery. This paper seeks to formalize the intuitive notion of a core/periphery structure and suggests algorithms for detecting this structure, along with statistical tests for testing a priori hypotheses. Different models are presented for different kinds of graphs (directed and undirected, valued and nonvalued). In addition, the close relation of the continuous models developed to certain centrality measures is discussed.

A multilevel approach to the travelling salesman problem

We motivate, derive, and implement a multilevel approach to the travelling salesman problem.The resulting algorithm progressively coarsens the problem, initialises a tour, and then employs either the Lin-Kernighan (LK) or the Chained Lin-Kernighan (CLK) algorithm to refine the solution on each of the coarsened problems in reverse order.In experiments on a well-established test suite of 80 problem instances we found multilevel configurations that either improved the tour quality by over 25% as compared to the standard CLK algorithm using the same amount of execution time, or that achieved approximately the same tour quality over seven times more rapidly. Moreover, the multilevel variants seem to optimise far better the more clustered instances with which the LK and CLK algorithms have the most difficulties.

Mathematical modelling of the behaviour of granular material in a computational fluid dynamics framework using micro-mechanical models

In this paper, a Computational Fluid Dynamics framework is presented for the modelling of key processes which involve granular material (i.e. segregation, degradation, caking). Appropriate physical models and sophisticated algorithms have been developed for the correct representation of the different material components in a granular mixture. The various processes, which arise from the micromechanical properties of the different mixture species can be obtained and parametrised in a DEM / experimental framework, thus enabling the continuum theory to correctly account for the micromechanical properties of a granular system. The present study establishes the link between the micromechanics and continuum theory and demonstrates the model capabilities in simulations of processes which are of great importance to the process engineering industry and involve granular materials in complex geometries.

Multilevel refinement strategies for the capacity vehicle routing problem

We discuss the application of the multilevel (ML) refinement technique to the Vehicle Routing Problem (VRP), and compare it to its single-level (SL) counterpart. Multilevel refinement recursively coarsens to create a hierarchy of approximations to the problem and refines at each level. A SL algorithm, which uses a combination of standard VRP heuristics, is developed first to solve instances of the VRP. A ML version, which extends the global view of these heuristics, is then created, using variants of the construction and improvement heuristics at each level. Finally some multilevel enhancements are developed. Experimentation is used to find suitable parameter settings and the final version is tested on two well-known VRP benchmark suites. Results comparing both SL and ML algorithms are presented.