Powerset residuated algebras

We present an algebraic approach to canonical embeddings of arbitrary residuated algebras into powerset residuated algebras. We propose some construction of powerset residuated algebras and prove a representation theorem for symmetric residuated algebras

Scheduling Algorithms for Procrastinators

This paper presents scheduling algorithms for procrastinators, where the speed that a procrastinator executes a job increases as the due date approaches. We give optimal off-line scheduling policies for linearly increasing speed functions. We then explain the computational/numerical issues involved in implementing this policy. We next explore the online setting, showing that there exist adversaries that force any online scheduling policy to miss due dates. This impossibility result motivates the problem of minimizing the maximum interval stretch of any job; the interval stretch of a job is the job's flow time divided by the job's due date minus release time. We show that several common scheduling strategies, including the "hit-the-highest-nail" strategy beloved by procrastinators, have arbitrarily large maximum interval stretch. Then we give the "thrashing" scheduling policy and show that it is a \Theta(1) approximation algorithm for the maximum interval stretch.Comment: 12 pages, 3 figure

Single machine scheduling with time-dependent linear deterioration and rate-modifying maintenance

We study single machine scheduling problems with linear time-dependent deterioration effects and maintenance activities. Maintenance periods (MPs) are included into the schedule, so that the machine, that gets worse during the processing, can be restored to a better state. We deal with a job-independent version of the deterioration effects, that is, all jobs share a common deterioration rate. However, we introduce a novel extension to such models and allow the deterioration rates to change after every MP. We study several versions of this generalized problem and design a range of polynomial-time solution algorithms that enable the decision-maker to determine possible sequences of jobs and MPs in the schedule, so that the makespan objective can be minimized. We show that all problems reduce to a linear assignment problem with a product matrix and can be solved by methods very similar to those used for solving problems with positional effects

Combining time and position dependent effects on a single machine subject to rate-modifying activities

We introduce a general model for single machine scheduling problems, in which the actual processing times of jobs are subject to a combination of positional and time-dependent effects, that are job-independent but additionally depend on certain activities that modify the processing rate of the machine, such as, maintenance. We focus on minimizing two classical objectives: the makespan and the sum of the completion times. The traditional classification accepted in this area of scheduling is based on the distinction between the learning and deterioration effects on one hand, and between the positional effects and the start-time dependent effects on the other hand. Our results show that in the framework of the introduced model such a classification is not necessary, as long as the effects are job-independent. The model introduced in this paper covers most of the previously known models. The solution algorithms are developed within the same general framework and their running times are no worse than those available earlier for problems with less general effects

Refined conditions for V-shaped optimal sequencing on a single machine to minimize total completion time under combined effects

We address single machine scheduling problems for which the actual processing times of jobs are subject to various effects, including a positional effect, a cumulative effect and their combination. We review the known results on the problems to minimize the makespan, the sum of the completion times and their combinations and identify the problems for which an optimal sequence cannot be found by simple priority rules such as SPT (Shortest Processing Time) and/or LPT (Longest Processing Time). Typically, these are problems to minimize the sum of the completion times under a deterioration effect, and we verify under which conditions for these problems an optimal permutation is V-shaped (an LPT subsequence followed by an SPT subsequence). We demonstrate that previously used techniques for proving that an optimal sequence is V- shaped are not properly justified. We use the corrected method to describe a wide range of problems with a pure positional effect and a combination of a cumulative effect with a positional effect for which an optimal sequence is V-shaped. On other hand, we show that even the refined approach has its limitations

Examples of changes in organ building in the 1st half of the 20th century as shown by the selected instruments preserved in the present Lubuskie Voivodship

Lubuskie Voivodship is a varied and valuable area in terms of sacred architecture and church organs. The development of sacred building industry, industrialization of organ-building, and the use of the steam engine in the 19th century enabled organ manufacturers to expand their production in that area. That process reached its peak at the turn of the 19th and 20th centuries and caused major changes in organ instruments as such, that have been preserved in differing degrees to this day. The first half of the 19th century was the time of significant development in organ-building in terms of technology and sound aesthetics. It affected such aspects as specification, pipe scaling, voicing, system of action, and windchest construction. In the first two decades, apart from taking advantage of the previously worked out solutions, organ companies focused on the “moderniz-ation” of their instruments. It was, however, still carried out basing on tubular-pneumatic action and cone valve chests. The following decade brought some new ideas in the field of organ spe-cifications. Attempts to make electrical action more common were followed by turning back to the tracker action and slider chests. The Organ Revival Movement (Orgelbewegung) helped to break the monopoly of large organ companies focused on series production. However, these new trends in organ-building could not spread immediately due to the upcoming economic crises and the two world wars. The instruments preserved in the Lubuskie region document the most important transition in German organ-building in the first half of the 20th century and provide interesting research mate-rial. The instruments discussed in this article, selected from different locations (Drezdenko, Strzelce Krajeńskie, Gorzów Wlkp., Międzyrzecz, Iłowa, Słonów), present a clear development line over the years. Their choice was dictated by the accessibility of source materials, complemented by the author’s own research

Selected problems and methods for the probabilistic analysis of sequencing tasks

W artykule przedstawiono problem szeregowania zadań na równoległych procesorach oraz wybrane metody jego probabilistycznej analizy. Umieszczone w tekście przykłady dotyczą algorytmów LS i LPT badanych w oparciu o kryterium Cmax.The article contains no abstrac