We study two-machine shop-scheduling problems provided that lower and upper bounds on durations of n jobs are given before scheduling. An exact value of the job duration remains unknown until completing the job. The objective is to minimize the makespan (schedule length). We address the issue of how to best execute a schedule if the job duration may take any real value from the given segment. Scheduling decisions may consist of two phases: an off-line phase and an on-line phase. Using information on the lower and upper bounds for each job duration available at the off-line phase, a scheduler can determine a minimal dominant set of schedules (DS) based on sufficient conditions for schedule domination. The DS optimally covers all possible realizations (scenarios) of the uncertain job durations in the sense that, for each possible scenario, there exists at least one schedule in the DS which is optimal. The DS enables a scheduler to quickly make an on-line scheduling decision whenever additional information on completing jobs is available. A scheduler can choose a schedule which is optimal for the most possible scenarios. We developed algorithms for testing a set of conditions for a schedule dominance. These algorithms are polynomial in the number of jobs. Their time complexity does not exceed . Computational experiments have shown the effectiveness of the developed algorithms. If there were no more than 600 jobs, then all 1000 instances in each tested series were solved in one second at most. An instance with 10,000 jobs was solved in 0.4 s on average. The most instances from nine tested classes were optimally solved. If the maximum relative error of the job duration was not greater than , then more than of the tested instances were optimally solved. If the maximum relative error was equal to , then of the tested instances from the nine classes were optimally solved.
This study addresses a two-machine job-shop scheduling problem with fixed lower and upper bounds on the job processing times. An exact value of the job duration remains unknown until completing the job. The objective is to minimize a schedule length (makespan). It is investigated how to best execute a schedule, if the job processing time may be equal to any real number from the given (closed) interval. Scheduling decisions consist of the off-line phase and the on-line phase of scheduling. Using the fixed lower and upper bounds on the job processing times available at the off-line phase, a scheduler may determine a minimal dominant set of schedules (minimal DS), which is based on the proven sufficient conditions for a schedule dominance. The DS optimally covers all possible realizations of the uncertain (interval) processing times, i.e., for each feasible scenario, there exists at least one optimal schedule in the minimal DS. The DS enables a scheduler to make the on-line scheduling decision, if a local information on completing some jobs becomes known. The stability approach enables a scheduler to choose optimal schedules for most feasible scenarios. The on-line scheduling algorithms have been developed with the asymptotic complexity for n given jobs. The computational experiment shows the effectiveness of these algorithms.
1 September 2012 in Cybernetics and Systems Analysis
Development of agriculture in Russia and Belarus is based on the practical implementation of "smart" systems in agriculture based on the use of modern wireless, intelligent technologies and Internet of Things. This review presents research articles (mainly, in Russian) published in the period of 2013 – 2022 on the use of cloud technologies and Internet of Things for the development of agriculture in Russia and Belarus. An analysis of the use of cloud technologies and Internet of Things in the modern world is given on the basis of research articles and reviews published in English in the period of 2017 – 2022. The main directions of digitalization of modern agriculture are listed. The uses of cloud technologies and Internet of Things in agriculture are described along with promising directions for further research and applications.