Solutions’ Integer Optimization for Tasks Inclusion in Packages and Package Execution Orders in the Flow Shop System Under Restrictions on the Availability of Devices with a Given Frequency

The efficiency of the processes of performing tasks of various types in Flow Shop systems is ensured by the formation of packages and optimization of their orders for performing operations with them on the devices of these systems. The process of package execution is affected by instrument failures and downtime associated with recovery. The impact of instrument failures and repairs can be reduced by pre-maintenance. During the maintenance time intervals, the devices are unavailable for the implementation of their assigned functions. By processing statistical data, the time intervals between the pre-maintenance of the devices can be determined (the periods of their availability are determined). For this reason, it is important to solve the problems of optimizing package compositions, their inclusion in the time intervals for the availability of devices, and schedules for their execution in these intervals. With small problem sizes, their solutions can be determined by using mathematical models of mixed integer linear programming.

The purpose of the work is to build a new mathematical model of mixed integer linear programming, the use of which allows us to determine the optimal solutions of the type under consideration. To achieve this goal, the methods of constructing mathematical programming models are used in the work. At the first stage, the formation of a nonlinear mathematical model of integer programming was implemented. At the second stage, in order to reduce the time required to obtain solutions, the model was linearized. To verify the model, an application was developed in the IBM ILOG CPLEX program. In the course of the research, results were obtained that showed the effectiveness of the model in solving the tasks of planning the execution of task packages in conveyor systems with limited availability of devices.

The scientific novelty of the results suggests that the model uses a method for transferring tasks between two devices directly at the end of their execution on the previous device. Their theoretical significance consists in obtaining new expressions that make it possible to determine the values of indicator variables when the function values exceed the specified boundaries. Based on these expressions, constraints are formed that require the inclusion of all tasks of different types in packages in the time intervals of instrument availability.

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