An original statement and solution algorithms are presented for one of the key problems in the scheduling theory. The problem of optimal scheduling for a parallel system consists in the generation or control of schedules to minimize the schedule length or the losses from schedule disruptions in the completion of jobs on machines. This problem is NP-hard and cannot be solved exactly for any real-life number of dimensions. A series of modifications of the efficient parametric algorithm is proposed to find an approximate solution, which are an extension of a similar algorithm for optimal scheduling on unrelated parallel machines with release dates using the performance criterion (Stakh). Software implementations of the algorithm modifications have been tested on the data of a generating problem by the Stakh criterion; the corresponding statistics is provided.

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The paper presents the problem of controlling the input and output material flows of an industrial enterprise, supplemented by the condition for the choice of sales prices and adjusted in terms of the number of products sold. This economic and mathematical model finds optimal solutions for the management problem at enterprises, namely, the problem of forming a production program according to the criterion of the maximum net profit at the end of the planning period. In the constraints of the model, both production components and constraints on resources and the logic of input and output material flows are systematically taken into account. The considered model and the given control problems are investigated using a unified approach that allows working with logical conditions of any complexity and setting the corresponding formal optimization problems. The results of testing the algorithm on test data close to industrial (real) dimensions are also given.

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