The Statement of the Task of Optimal Control of the Production Line Using the Additional Time of Equipment Operation

The production line of an enterprise with a flow method of organizing production is considered as a dynamic distributed system. The technological route for manufacturing products for many modern enterprises contains several hundreds of technological operations, in the inter-operating reserve of each of which there are thousands of products awaiting processing. Technological routes of different parts of the same type of products intersect. This leads to the fact that the distribution of objects of labour along the technological route has a significant impact on the throughput capacity of the production line. To describe such systems, a new class of production line models (PDE-model) has been introduced. Models of this class use partial differential equations to describe the behaviour of production line flow parameters. In this article, a PDE-model of the production line is built, the flow parameters of which depend on the load factor of the process equipment for each operation. For the description of a distributed dynamic system, the PDE model of the production line was used. At the same time, the single-shift mode of operation of a production enterprise is considered as a basic mode of operation.


Introduction
On the production line of the enterprise is required to process a batch of products [1]. For the process identified: a) the sequence of operations and their technological parameters; Electronic copy available at: https://ssrn.com/abstract=3492121 b) the equipment necessary to perform the operation, the parameters of its work and the layout scheme; c) the properties of the object of labor and the laws of the transfer of resources to objects of labor as a result of the impact of equipment.
It is assumed that the production technology during the production cycle does not change, that is, the parameters characterizing the operation remain unchanged. The duration of the shift is given, is eight hours. Reducing the duration of the production cycle is possible by changing the mode of loading equipment. The amount of equipment loading in the processing of a batch of products will be characterized by the shift factor of the equipment during the day Sm K . We believe that the cost of one hour of equipment operation is different for each operation and depends on the time of day. The flow parameters of the model of the controlled production process in the two-step description are interoperating reserve characterized by density     S t, 0  , and by the flow of objects of labor     S t, 1  on the technological route [2][3][4][5]. To describe the behavior of flow parameters in space and time, we use the one-dimensional coordinate space   S t, [6,7]. The coordinate S determines the place of the object of labor in the technological route. The introduced one-dimensional coordinate space   S t, allows us to construct compact models for controlling the parameters of a production line. We divide the coordinate axis ) [8]. One of the approaches to synchronize the processing performance of objects of labor in different operations of the production line is to use the main equipment in additional time between the main technological shifts (control of the shift of technological equipment for a given technological operation). If the time used by the equipment during single-shift operation is selected as the time axis of the state space, then the state of the backlog during the period between the end of the shift and the beginning of another shift in the case of using the main equipment during the second and third shift will change abruptly by the number of processed products during the second and third shift.
Let us introduce the distribution density of the cost of the above-standard costs of technological resources required to perform work on additional equipment within an hour for the technological route that characterizes the work of additional equipment (main equipment in the second or third shift, shift factor 1 
as the number of additional equipment (located in the vicinity S of the technological route coordinates) hours for a period of time . As a result of the inclusion of additional equipment that ensures the processing of objects of labor in the second and third shift at a rate equal to the rate of operation of the main equipment     , the overall rate of movement of objects of labor at the point of the technological route with the coordinate S increases by the work of the additional equipment, an additional flow of objects of labor with the total number of units and used to perform the m -th operation for the duration of the production cycle d T , we define the integral:

Relevance and practical significance
The scientific novelty consists in the development of a method for designing control systems for the parameters of the production line of enterprises with a continuous method of organizing production based on the PDE model of the control object.
In this case, the control object, the production line is represented by a dynamic system with distributed parameters along the technological route. The optimal control of the parameters of the production line is sought in the form of superpositions of delta functions.
The proposed method of designing a system for controlling the flow parameters of a production line can be used as the basis for designing highly efficient production flow control systems for enterprises manufacturing semiconductor products in the automotive industry.

Production Line Model
Line parameters for continuous production flow with a sufficiently large number of operations satisfy the balance equation system: The normative tempo     ). Control function determines the duration of the inclusion of additional equipment in the specified place of the technological route with the coordinate S at the time ).The planning interval for the line in question is equal to the interval of three shifts (daily planning interval) with one-shift operation of the main equipment. As an additional, the main equipment is used, which processes 1  Sm K the objects of labor in the second and The behavior of the parameters     of the production line is limited by the initial conditions of the distribution of objects of labor along the technological route and the purpose of management: as well as the boundary conditions determining the receipt from the warehouse of raw materials, materials for the first operation and the output of finished products from the last operation: Electronic copy available at: https://ssrn.com/abstract=3492121 In the absence of the inclusion of additional equipment parameters     S t, The notation  q t and  q t , means that the functions     in the infinitely small neighborhood to the left and to the right of q t . We believe that the regulatory parameters characterizing the operation during the production cycle d T remain unchanged in time: The conditions for the occurrence of the drive overflow process and the study of the evolution of its development for the technological route section   m , 0 S are described in detail in [4,6,[9][10][11][12]. To ensure the smooth operation of the production line, it is necessary to synchronize the rate of processing of objects of labor in individual operations within the time interval between the beginning of the q t -th and the beginning of the 1  q t -th work shift. We supplement equations (1) with the control function determining the duration of the inclusion of equipment in the position S at the time between the end of the q -th and the beginning of ). The control of the flow parameters carried Electronic copy available at: https://ssrn.com/abstract=3492121 out as a result of the use of additional equipment at the moment of time q t between the end of the q -th shift and the beginning of the ) 1 (  q -th shift is determined through the Dirac delta function   q t t   [13,14]. The equation for changing the density of inter-operating reserve (3) can be integrated over time: where the replenishment time points of the production line are

Statement of the task of optimal control of the production line
In a fairly general form, the task of building an optimal program for controlling flow parameters , when additional equipment is turned on using additional equipment, can be formulated as follows: determine the state of parameters     delivering a minimum of functionality with differential connections Electronic copy available at: https://ssrn.com/abstract=3492121 with restrictions along the trajectory on the phase variables     S t, 0  determined by the storage capacity , with constraints along the trajectory on the control [7]   initial conditions Electronic copy available at: https://ssrn.com/abstract=3492121 corresponding to filling the bunker with labor objects and emptying it, is allowed only in the time interval Integrating the balance equations in the specified time interval, we get: , what allows to write down condition of inadmissibility of overflow of the inter-operational bunker and the condition of inadmissibility so the inter-operational bunker was empty

Conclusion
Achievement of the production system with the initial distribution of objects of labor     can be implemented in a variety of ways, each of which is called a control program. In the technical problems of managing the state of the production line, the question arises of finding the most optimal program for the use of resources (optimal control). The mathematical reflection of this fact is that the control of the parameters of the production line should be chosen from the condition of minimum integral (5). This paper shows that along with traditional models for controlling the parameters of production flow lines, an important role is played by control models associated with the use of partial differential equations (PDE models). A PDE-model for controlling the parameters of a production line has been considered, taking into account restrictions on the volume of inter-operating bunkers along the technological route. The developed model allows you to determine the schedule of switching on and off the process equipment. A method for controlling the synchronization of technological operations of an industrial production line is considered. The quality criterion of the production system is written, which allows building optimal control over the parameters of a production line operating in multi-shift mode. It is shown that partial differential equations that act as differential constraints for phase variables are replaced by a system of equations for the coefficients of decomposition of production line parameters, which allowed us to obtain the control function in the form of time dependence and position (coordinates) in the technological route. ЛІТЕРАТУРА