Вісник № 01. Системний аналіз, управління та інформаційні технології
Постійне посилання колекціїhttps://repository.kpi.kharkov.ua/handle/KhPI-Press/42344
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Документ Total weighted tardiness minimization for tasks with a common due date on parallel machines in case of agreeable weights and processing times(НТУ "ХПІ", 2019) Pavlov, Alexander Anatolievich; Misura, Elena Borisovna; Melnikov, Oleg ValentinovichWe consider n tasks scheduling problem on m identical parallel machines by the criterion of minimizing the total weighted tardiness of tasks. All tasks arrive for processing at the same time. Weights and processing times are agreeable, that is, a greater weight of a task corre sponds to a shorter processing time. In addition, we have arbitrary start times of machines for tasks processing. The times may be less or greater than the due date or to coincide with it. The problem in this formulation is addressed for the first time. It can be used to provide planning and decision making in systems with a network representation of technological processes and limited resources. We give efficient PSC-algorithm with 𝑂(𝑚𝑛 log 𝑛) complexity that includes the polynomial component and the approximation algorithm based on permutations of tasks. The polynomial component contains sufficient signs of optimality of the obtained solutions and allows to obtain an exact solution by polynomial subalgorithm. In the case when the sufficient signs of optimality do not fulfill, we obtain approximate solution with an estimate of deviation from the optimum for each individual problem instance of any practical dimension. We show that a schedule obtained as a result of the problem solving can be split into two schedules: the schedule on machines which start time is less than or equal to the due date, and the schedule on machines which start after the due date. Optimization is only done in the first schedule. The second schedule is optimal by construction. Statistical studies of the PSC-algorithm showed its high efficiency. We solved problems with dimensions up to 40,000 tasks and up to 30 machines. The average time to solve the problem by the algorithm using the most efficient types of permutations was 27.3 ms for this dimension. The average frequency of an optimal solution obtaining amounted to 90.3 %. The average deviation from an optimum was no more than 0.000251.Документ Combinatorial optimization under uncertainty and formal models of expert estimation(НТУ "ХПІ", 2019) Pavlov, Alexander AnatolievichPreviously, the author formalized the concepts of uncertainty, compromise solution, compromise criteria and conditions for a quite general class of combinatorial optimization problems. The functional of the class’ problems contains linear convolution of weights and arbitrary numerical characteristics of a feasible solution. It was shown that the efficiency of the presented algorithms for the uncertainty resolution is largely determined by the efficiency of solving the combinatorial optimization problem in a deterministic formulation. A part of the formulated compromise criteri a and conditions uses expert weights. Previously, the author and his disciples also formulated combinatorial optimization models, optimality criteria, criteria for decisions’ consistency. The models allow to evaluate and justify the degree of stability and reliability of the estimated values of empirical coefficients using a formally ill-conditioned empirical pairwise comparison matrix of arbitrary dimension. The matrix may contain zero elements. The theoretical research and statistical experiments allowed to choose the most efficient of these optimization models. In this article, on the base of earlier results by the author and his disciples, we formalize and substantiate the efficiency of the proposed sequential procedure for expert estimation of weights that determine compromise criteria and conditions. The procedure is an integral part of the algorithm introduced by the author to solve combinatorial optimization problems under uncertainty of the mentioned class. We give unified algorithm for efficient uncertainty resolution that includes original and efficient formal procedure for expert coefficients’ estimation using empirical matrices of pairwise comparisons.