Abstract
The study was carried out with the aim of solving the problem of replacing equipment and creating a mathematical model of the problem of controlled optimization, as the most complete and realistic description of the production process. The Bellman principle of optimality was applied in the study, since when solving dynamic programming problems, this principle is general. For the solution, the Bellman functional equation was chosen as the initial mathemat cal model: 
 
 
 
 
 
 The advantage of such a mathematical model is the possibility of modifying the problem. At the first step, it is proposed to sell the replaced equipment at the price (R(tk)-Z(tk))p, 0<p<1, p is the markdown coefficient for the sale. Further, the installed equipment may not be new and purchased at a price ((R(tk)-Z(tk))q, 0<q<1. Here q is the discount factor for purchase and is set at the input. Not new equipment is cheaper than new and sale of replaced equipment also contributes to profits New components must be added to the control vector - the age of the equipment being installed (not new) The control structure becomes more complex and realistic The effect of the "curse of dimensionality" does not appear during numerous tests of the program Three modifications of the problem have been developed on equipment replacement. Replaced equipment is thrown away. You can not throw it away, but sell it at a price (Rk(tk)-Zk(tk))p, where p is a markdown coefficient 0<p<1. You can increase profits if you buy not new equipment The paper considers the option of adding another component to the control vector - "rejuvenating" repair of equipment. After such repair, the age of the equipment becomes, for example, one year less. Repairs are taken from the graph of profit earned versus repair cost, which is obtained from the results of the program runs. Combining the above modifications into a single whole is a mathematical model of the problem of replacing equipment, which is the basis of the program.
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