Approximation algorithms for two-machine flow shop scheduling with batch setup times

In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Group technology approach to the open shop scheduling problem with batch setup times

This paper studies the problem of scheduling jobs in a two-machine open shop to minimize the makespan. Jobs are grouped into batches and are processed without preemption. A batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. For this NP-hard problem, we propose a linear-time heuristic algorithm that creates a group technology schedule, in which no batch is split into sub-batches. We demonstrate that our heuristic is a -approximation algorithm. Moreover, we show that no group technology algorithm can guarantee a worst-case performance ratio less than 5/4.

Influence of a fed or fasted state on the s-IgA response to prolonged cycling in active men and women

This study investigated the effect of a fed or fasted state on the salivary immunoglobulin A (s-IgA) response to prolonged cycling. Using a randomized, crossover design, 16 active adults (8 men and 8 women) performed 2 hr of cycling on a stationary ergometer at 65% of maximal oxygen uptake on 1 occasion after an overnight fast (FAST) and on another occasion 2 hr after consuming a 2.2-MJ high-carbohydrate meal (FED). Timed, unstimulated whole saliva samples were collected immediately before ingestion of the meal, immediately preexercise, 5 min before cessation of exercise, immediately postexercise, and 1 hr postexercise. The samples were analyzed for s-IgA concentration, osmolality, and cortisol, and saliva flow rates were determined to calculate s-IgA secretion rate. Saliva flow rate decreased by 50% during exercise (p < .05), and s-IgA concentration increased by 42% (p < .05), but s-IgA secretion rate remained unchanged. There was a 37% reduction in s-IgA:osmolality postexercise (p < .05), and salivary cortisol increased by 68% (p < .05). There was no effect of FED vs. FAST on these salivary responses. The s-IgA concentration, secretion rate, and osmolality were found to be significantly lower in women than in men throughout the exercise protocol (p < .05); however, there was no difference between genders in saliva flow rate, s-IgA:osmolality ratio, or cortisol. These data demonstrate that a fed or fasted state 2 hr before exercise does not influence resting s-IgA or the response to prolonged cycling. Furthermore, these results show lower levels of s-IgA and osmolality in women than in men at rest.