THE MATHEMATICAL MODEL OF PROCESS MANAGEMENT OF MATERIALS TRANSPORTATION
Abstract
The theoretical aspects of the dynamic programming method are generalized as a mathematical method for making managerial decisions in optimization problems. A multi-stage management process based on the Bellman optimality principle in the real economic space is constructed. Functional Bellman equations that are directly adapted to a specific production problem are discussed in detail. The optimal economic and mathematical model of economic process development is obtained. This allows you to manage the economic process as a whole, to develop effective management decisions with help of which to increase the competitiveness of the enterprise. The algorithm for transportation and storage of materials, which consists of forward stroke, a process of sequentially calculating the objective function and the return, that is, restoring the optimal solution, has been developed. At the last step of the forward stroke, the optimal value of the last variable x_n^*=x_n (Y) and the optimal values of the control variables has been obtained. The economic process of managing the distribution of materials is broken into n stages, the decision is made sequentially at each stage, so a multi-step process has been obtained. The calculated performance metric for this managed system is a goal function that depends on the initial state and management X ̅(x_1,x_2,…x_n ). Functional equations have been constructed that are adapted to the problem of the distribution of materials. During the direct course at each step, all possible values of the target function have been calculated by functional equations. Each subsequent value of the objective function depends on the control at this stage and the previous value of the objective function. So, with help a computer, a table of possible conditionally optimal values of the target function and corresponding optimal controls has been constructed. At the final stage of the return stroke, the optimal value of the target function and the last optimal process control have been obtained. At the preliminary stage, depending on the optimal process control at the final stage, the conditionally optimal value of the target function and the preliminary optimal process control are found, then the following preliminary solution has been obtained in the same way. Functional equations, that are adapted to the problem of material distribution, are constructed. According to the results of the forward and backward moves of the algorithm, an optimal economic and mathematical model of material distribution was obtained.
References
1. Akylich I.L. (2011). Matematicheskoe programmirovanie v primerah i zadachah: ychebnoe posobie [Mathematical programming in examples and tasks]. Sankt-Peterburg: Izdatel’stvo Lan’, 352 p. [in Russian]
2. Bellman R. (1960). Dinamicheskoe programmirovanie [Dynamic programming]. Moskva: Inostrannaja literatyra, 400 p. [in Russian]
3. Bellman R. (1962). Nekotorie voprosi matematicheskoj teorii processov ypravlenija [Some questions of the mathematical theory of control processes]. Moskva: Inostrannaja literatyra, 336 p. [in Russian]
4. Bellman R. (1965). Ob opredelenii optimal’nih traektorij metodom dinamicheskogo programmirovanija [On determination of optimal trajectories by dynamic programming method]. Moskva: Inostrannaja literatyra, 338 p. [in Russian]
5. Bellman R. (1965). Prikladnie zadachi dinamicheskogo programmirovanija [Dynamic programming applications]. Moskva: Inostrannaja literatyra, 459 p. [in Russian]
6. Bilocerkivs’kij O.B. (2018). Matematichne modeluvannja v ekonomici ta menedgmenti: tekst lekcij [Mathematical modeling in economics and management]. Kharkiv: The text of lectures, NTU “KhPI”, 90 p. [in Ukrainian]
7. Drozdenko K.A., Kotenko A.P. (2007). Primenenie metoda dinamicheskogo programmirovanija v stohasticheskih zadachah raspredilenija resyrsov [Application of dynamic programming method in stochastic resource allocation problems]. Vestnik Samarskogo gosydarstvennogo tehnicheskogo yniversiteta. Serija “Fiziko-matematicheskie nayki”, pp. 184–185. [in Russian]
8. Efimova G.O., Rydik O.G. (2015). Metodichni vkazivki dl’a samostijnoj roboti po discipline “Doslidgennja operacij” (rozdil “Dinamichne programyvannja”) [Methodical instructions for independent work in the discipline "Operations Research" (section "Dynamic Programming")]. Odesa, 38 p. [in Ukrainian]
9. Esina V.O. (2017). Metodichni vkazivki do provedennja praktichnih zanjat’ ta samostijnoj roboti z disciplini “Optimizacijni metodi ta modeli” [Methodical instructions for conducting practical classes and independent work in the discipline "Optimization method and models"]. Kharkiv, 23 p. [in Ukrainian]
10. Kyznecov U.N., Kyzybov V.I., Voloshenko A.B. (1980). Matematicheskoe programmirovanie: ychebnoe posobie, vtoroe izdanie [Mathematical programming in examples and tasks: tutorial]. Mockva: Visshaja shkola, 300 p. [in Russian]
11. Kremer N.S., Pytko B.A., Trishin I.M., Fridman M.N. (2019). Issledovanie operacij v ekonomike: ychebnik dl’a akademicheskogo bakalavriata [Operations Research in Economics: A Textbook for Academic Baccalaureate]. Moskva: Izdatel’stvo Urajt, 438 p. [in Russian]
12. Kylish S.A., Protosenja A.G. (1985). Matematicheskie metodi i modeli v planirovanii i ypravlenii gornim proizvodstvom: ychebnoe posobie [Mathematical methods and models in the planning and management of mining: tutorial]. Moskva: Nedra, 288 p. [in Russian]
13. Norik L.A., Shevchenko A.K. (2013). Visshaja i prikladnaja matematika: ychebnoe posobie [Higher and applied mathematics: tutorial]. Kharkiv: Izdatel’stvo HNEU, 404 p. [in Ukrainian]
14. Solodovnik G.V. (2016). Determinovana model’ optimal’nogo rozpodily resyrsiv [Determined model of optimal resource allocation]. Molodij vchenij, no. 6, pp. 108–111. [in Ukrainian
