Scheduling three-operation jobs in a two-machine flow shop to minimize makespan

This paper considers a variant of the classical problem of minimizing makespan in a two-machine flow shop. In this variant, each job has three operations, where the first operation must be performed on the first machine, the second operation can be performed on either machine but cannot be preempted, and the third operation must be performed on the second machine. The NP-hard nature of the problem motivates the design and analysis of approximation algorithms. It is shown that a schedule in which the operations are sequenced arbitrarily, but without inserted machine idle time, has a worst-case performance ratio of 2. Also, an algorithm that constructs four schedules and selects the best is shown to have a worst-case performance ratio of 3/2. A polynomial time approximation scheme (PTAS) is also presented.

Two-machine flow shop scheduling problems with no-wait jobs

In this paper we provide a fairly complete complexity classification of various versions of the two-machine permutation flow shop scheduling problem to minimize the makespan in which some of the jobs have to be processed with no-wait in process. For some version, we offer a fully polynomial-time approximation scheme and a 43-approximation algorithm.

Pseudo-spectral solutions for fluid flow and heat transfer in electro-metallurgical applications

The pseudo-spectral solution method offers a flexible and fast alternative to the more usual finite element/volume/difference methods, particularly when the long-time transient behaviour of a system is of interest. Since the exact solution is obtained at the grid collocation points superior accuracy can be achieved on modest grid resolution. Furthermore, the grid can be freely adapted with time and in space, to particular flow conditions or geometric variations. This is especially advantageous where strongly coupled, time-dependent, multi-physics solutions are investigated. Examples include metallurgical applications involving the interaction of electromagnetic fields and conducting liquids with a free sutface. The electromagnetic field then determines the instantaneous liquid volume shape and the liquid shape affects in turn the electromagnetic field. In AC applications a thin "skin effect" region results on the free surface that dominates grid requirements. Infinitesimally thin boundary cells can be introduced using Chebyshev polynomial expansions without detriment to the numerical accuracy. This paper presents a general methodology of the pseudo-spectral approach and outlines the solution procedures used. Several instructive example applications are given: the aluminium electrolysis MHD problem, induction melting and stirring and the dynamics of magnetically levitated droplets in AC and DC fields. Comparisons to available analytical solutions and to experimental measurements will be discussed.

Approximation Algorithms for a Symmetric Quadratic Knapsack Problem and its Scheduling Applications

We consider a knapsack problem to minimize a symmetric quadratic function. We demonstrate that this symmetric quadratic knapsack problem is relevant to two problems of single machine scheduling: the problem of minimizing the weighted sum of the completion times with a single machine non-availability interval under the non-resumable scenario; and the problem of minimizing the total weighted earliness and tardiness with respect to a common small due date. We develop a polynomial-time approximation algorithm that delivers a constant worst-case performance ratio for a special form of the symmetric quadratic knapsack problem. We adapt that algorithm to our scheduling problems and achieve a better performance. For the problems under consideration no fixed-ratio approximation algorithms have been previously known.

An improved approximation algorithm for the two-machine flow shop scheduling problem with an interstage transporter

We study the two-machine flow shop problem with an uncapacitated interstage transporter. The jobs have to be split into batches, and upon completion on the first machine, each batch has to be shipped to the second machine by a transporter. The best known heuristic for the problem is a –approximation algorithm that outputs a two-shipment schedule. We design a –approximation algorithm that finds schedules with at most three shipments, and this ratio cannot be improved, unless schedules with more shipments are created. This improvement is achieved due to a thorough analysis of schedules with two and three shipments by means of linear programming. We formulate problems of finding an optimal schedule with two or three shipments as integer linear programs and develop strongly polynomial algorithms that find solutions to their continuous relaxations with a small number of fractional variables

An approximation algorithm for the two-machine flow shop with a single transporter

We study the two-machine flow shop problem with an uncapacitated interstage transporter. The jobs have to be split into batches, and upon completion on the first machine, each batch has to be shipped to the second machine by a transporter. The best known heuristic for the problem is a –approximation algorithm that outputs a two-shipment schedule. We design a –approximation algorithm that finds schedules with at most three shipments, and this ratio cannot be improved, unless schedules with more shipments are created. This improvement is achieved due to a thorough analysis of schedules with two and three shipments by means of linear programming. We formulate problems of finding an optimal schedule with two or three shipments as integer linear programs and develop strongly polynomial algorithms that find solutions to their continuous relaxations with a small number of fractional variables.

Schedules with due-date assignment and position-dependent processing times

Single machine scheduling problems are considered, in which the processing of jobs depend on positions of the jobs in a schedule and the due-dates are assigned either according to the CON rule (a due-date common to all jobs is chosen) or according to the SLK rule (the due-dates are computed by increasing the actual processing times of each job by a slack, common to all jobs). Polynomial-time dynamic programming algorithms are proposed for the problems with the objective functions that include the cost of assigning the due-dates, the total cost of disgarded jobs (which are not scheduled) and, possibly, the total earliness of the scheduled jobs.

A critical examination of node-similarity based graph matching algorithms

In this paper, we shall critically examine a special class of graph matching algorithms that follow the approach of node-similarity measurement. A high-level algorithm framework, namely node-similarity graph matching framework (NSGM framework), is proposed, from which, many existing graph matching algorithms can be subsumed, including the eigen-decomposition method of Umeyama, the polynomial-transformation method of Almohamad, the hubs and authorities method of Kleinberg, and the kronecker product successive projection methods of Wyk, etc. In addition, improved algorithms can be developed from the NSGM framework with respects to the corresponding results in graph theory. As the observation, it is pointed out that, in general, any algorithm which can be subsumed from NSGM framework fails to work well for graphs with non-trivial auto-isomorphism structure.

Approximation results for flow shop scheduling problems with machine availability constraints

This paper considers two-machine flow shop scheduling problems with machine availability constraints. When the processing of a job is interrupted by an unavailability period of a machine, we consider both the resumable scenario in which the processing can be resumed when the machine next becomes available, and the semi-resumable scenario in which some portion of the processing is repeated but the job is otherwise resumable. For the problem with several non-availability intervals on the first machine under the resumable scenario, we present a fast (3/2)-approximation algorithm. For the problem with one non-availability interval under the semi-resumable scenario, a polynomial-time approximation scheme is developed.

Single machine scheduling and due date assignment with positionally dependent processing times

We consider single machine scheduling and due date assignment problems in which the processing time of a job depends on its position in a processing sequence. The objective functions include the cost of changing the due dates, the total cost of discarded jobs that cannot be completed by their due dates and, possibly, the total earliness of the scheduled jobs. We present polynomial-time dynamic programming algorithms in the case of two popular due date assignment methods: CON and SLK. The considered problems are related to mathematical models of cooperation between the manufacturer and the customer in supply chain scheduling.

Operator non-availability periods

In the scheduling literature, the notion of machine non availability periods is well known, for instance for maintenance. In our case of planning chemical experiments, we have special periods (the week-ends, holidays, vacations) where the chemists are not available. However, human intervention by the chemists is required to handle the starting and termination of the experiments. This gives rise to a new type of scheduling problems, namely problems of finding schedules that respect the operator non availability periods. These problems are analyzed on a single machine with the makespan as criterion. Properties are described and performance ratios are given for list scheduling and other polynomial-time algorithms.

Chronotype, sleep quality and sleep duration in adult distance education: Not related to study progress

Research in traditional education shows chronotype, sleep duration and sleep quality to be related to learning performance. Research in adult students participating in distance education (DE) is scarce. This study aims to provide knowledge on these relationships in this educational setting. In an observational longitudinal study, chronotype, sleep duration (i.e., for work and free days separately) and sleep quality of 894 students were analyzed in a multiple regression analyses. Students provided information on sleep-relatedmeasures and important covariates at the start of their study and study progress was evaluated after 14 months (i.e., the number of successfully completed modules). In linewith previous research, chronotype did not predict study progress. Further, sleep duration did not predict study progress, neither as a linear nor as a polynomial term. Third, sleep quality did not predict study progress. Concluding, these results are in linewith previous research that DE provides a solution to the asynchrony problem. Findings regarding sleep duration and sleep quality are new and unexpected, asking for attention and further research. Despite the study's observational nature, findings suggest that students participating in DE may benefit from this type of education as the asynchrony problem appears not to apply here, as students can choose their own study schedule.

K1 of Chevalley groups are nilpotent

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

Computing Zeros of Analytic Functions in the Complex Plane without using Derivatives

A new approach to evaluating all multiple complex roots of analytical function f(z) confined to the specified rectangular domain of complex plane has been developed and implemented in Fortran code. Generally f (z), despite being holomorphic function, does not have a closed analytical form thereby inhibiting explicit evaluation of its derivatives. The latter constraint poses a major challenge to implementation of the robust numerical algorithm. This work is at the instrumental level and provides an enabling tool for solving a broad class of eigenvalue problems and polynomial approximations.

Learning Bayesian networks from data: An information-theory based approach

This paper provides algorithms that use an information-theoretic analysis to learn Bayesian network structures from data. Based on our three-phase learning framework, we develop efficient algorithms that can effectively learn Bayesian networks, requiring only polynomial numbers of conditional independence (CI) tests in typical cases. We provide precise conditions that specify when these algorithms are guaranteed to be correct as well as empirical evidence (from real world applications and simulation tests) that demonstrates that these systems work efficiently and reliably in practice.