Optimal Control of the Parameters of the Production Line

Pihnastyi, Oleh (2019): Optimal Control of the Parameters of the Production Line. Published in: International Journal of Scientific and Innovative Mathematical Research , Vol. 7, No. 2 (2 February 2019): pp. 1-12.

Abstract

The problem of optimal control of the parameters of the production flow line - stocks (work in process) and the rate of processing of objects of labour for a technological operation is considered. The article presents a mathematical formulation of the problem of controlling the parameters of a production line with restrictions on work in progress and the speed of machining parts for each technological operation. The control program is determined by the specified quality criteria. An example of the calculation of the optimal control for the production line parameters is presented.

• Item Type: MPRA Paper
• Language: English
• Keywords: production line; PDE-model of production; balance equations; transition period; work in progress
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C44 - Operations Research ; Statistical Decision Theory; D - Microeconomics > D2 - Production and Organizations > D24 - Production ; Cost ; Capital ; Capital, Total Factor, and Multifactor Productivity ; Capacity; L - Industrial Organization > L2 - Firm Objectives, Organization, and Behavior > L23 - Organization of Production; Q - Agricultural and Natural Resource Economics ; Environmental and Ecological Economics > Q2 - Renewable Resources and Conservation > Q21 - Demand and Supply ; Prices

References

1. Berg R. Partial differential equations in modelling and control of manufacturing systems / R. Berg. – Netherlands, Eindhoven Univ. Technol., 2004. – 157 p.

2. Forrester J. Basics of enterprise cybernetics / J. Forrester. - M .: Progress, 1961. – 341 p.

3. Pervozvansky A. A. Mathematical methods in production control / A. A. Pervozvansky. - M .: Science,1975. – 616 p.

4. Petrov B.N. Theories of models in control processes (Information and thermodynamic aspects) / B.N. Petrov, G.M. Ulanov, I.I. Goldenblat, S.V. Ulyanov - M .: Science, 1978. – 224 p.

5. Razumov I. M. Organization and planning of machine-building production / I. M. Razumov, L. Ya. Shecharant - Moscow: Engineering, 1974. – 592 p.

6. Sokolitsyn S. А. Application of mathematical methods in economics and organization of machine-building production / S. A. Sokolitsyn. - L .: Engineering, 1970. – 345 p.

7. Control of flexible production systems. Models and algorithms / E. D. Voronin and others. / Society. ed. S.V. Emelyanov. - M .: Engineering, 1987. – 368 p.

8. V. Shkurba. Discrete production planning in the ACS / V. B. Shkurba, In .A. Boldyrev, A. A. Vyun et al. / Ed. V.M. Glushkov. - K .: Technology, 1975. – 296 p.

9. Lefeber E. Modeling, Validation and Control of Manufacturing Systems. / E.Lefeber, R.A.Berg, J.E. Rooda – Proceeding of the 2004 American Control Conference, Massachusetts, 2004. – P. 4583 – 4588.

10. Lefeber E. Controlling a re-entrant manufacturing line via the push-pull point. / E.Lefeber, D.Perdaen, D.Armbruster, K.Kempf– International Journal of Production Research 46(16), 2008. – P. 4521 – 4536.

11. Mac Gregor S.J. Handbook of Stochastic Models and Analysis of Manufacturing System Operations. / S. J. Mac Gregor, B. Tan. – New York, 2013. –Vol. 192. – Series: International Series in Operations Research & Control Science. – P. 373.

12. Pihnastyi O. M. Statistical theory of production systems / O. M Pihnastyi. - Kharkiv: KhNU, 2007. – 388 p.

13. Krasovsky A. A. Phase space and the statistical theory of dynamical systems / A. A. Krasovsky. - M .: Science,1974. – 232 p.

14. Mikhailov V.S. Theory of control / V.S. Mikhailov - K: High School, 1988. — 312 p.

15. Moiseev N. N. Elements of the theory of optimal systems / N. N. Moiseev. - M .: Science, 1974. – 526 p.

16. Pontryagin L.S. Mathematical theory of optimal processes / L.S. Pontryagin, V.G. Boltyansky, R.V. Gamkrelidze - M .: Nauka, Main publishing house of physical and mathematical literature 1983. – 392 p.

17. Pugachev V.S. Basics of the statistical theory of automatic systems / V.S. Pugachev, I.E. Kazakov, L.G. Evlanov. - M .: Mechanical Engineering, 1974. –400 p.

18. Pihnastyi O. M. Calculation of the Production Cycle Using the Statistical Theory of Production-Technical Systems / O.M. Pihnastyi, V.D. Khodusov // Supplements of the National Academy of Sciences of Ukraine. - Kiev: «Academperіodika» Publishing House. - 2009. - №12. - p. 38-44.doi.org/10.13140/RG.2.2.36267.54562

19. Bellman R. The theory of the stability of solutions of differential equations / R. Bellman – M .: Foreign Literature Publishing House, 1954. – 215 p.

20. Kazakov I.Ye. Statistical theory of control systems in the state space / I.E. Cossacks. - M .: Science, 1975. – 432 p.

21. Armbruster D. Kinetic and fluid model hierarchies for supply chains supporting policy attributes / D. Armbruster., D. Marthaler, C. Ringhofer // Bulletin of the Institute of Mathematics. – Academica Sinica, – 2006. – P. 496 – 521.

22. Ramadge P. The control of discrete event systems / P. Ramadge, W. Wonham. // Proceedings of the IEEE – 1989. – vol. 77, №1. – P. 81 – 98.

23. Harrison J. Brownian Motion and Stochastic Flow Systems. / J. Harrison. – New York, 1995. – P. 142.

24. Demutsky V.P. Theory of an Enterprise: Sustainability of Mass Production and Product Promotion / V.P. Demutsky, V.S. Pihnasta, O. M. Pihnastyi. - Kharkov .: KhNU, 2003. – 272 p.

25. Armbruster D. Continuous models for production flows. / D. Armbruster, C. Ringhofer., Jo T- J. // In Proceedings of the 2004 American Control Conference. – Boston, MA, USA. – 2004. – P. 4589 – 4594.

26. Zaruba V. Ya. The entropy of the technological process / V. Ya. Zaruba, O.M. Pihnastyi // Control of large systems: works of the international scientific-practical conference “Theory of active systems” (ТАС2011), (Moscow, November 14-16, 2011). - Moscow IMP RAS. – 2011. – vol. 2. – P. 145 – 148.

27. Serebrennikov G. G. Organization of production / G. G. Serebrennikov. - Tambov: Edition. TSTU, 2004. – 96 p.

28. Sinitsa L. M. Organization of production / L. M. Sinitsa. – М.: 2003. – 512 p.

29. Tian F. An iterative approach to item-level tactical production and inventory planning. / F.Tian, S.P.Willems, K.G.Kempf –International Journal of Production Economics, 2011. – vol. 133. – P. 439 – 450.

30. Lyapilin I. I. Introduction to the theory of kinetic equations / I. I. Lyapilin. - Yekaterinburg:2003. – 205 p.

31. Pihnasty O. M. Use of PDE - models for Building a Unified Theory of Production Lines / O.M. Pihnastyi, // The Kherson National Library Technique. Kherson: KhNTU. - 2014. - № 3 (50).– p. 405 – 412.

32. Pihnastyi O. M. On the construction of the target function of the production system / O. M. Pihnastyi // Supplements of the National Academy of Sciences of Ukraine. - Kyiv: Publishing House "Academperіodika".– 2007. – №5. – p. 50 – 55.

33. Feldbaum A. A. Methods of the theory of automatic control: a scientific publication / A. A. Feldbaum, A. G. Butkovsky. - M .: Science, 1971. ­ 744 p.

34. Lysenko Yu. G. Modeling technological flexibility of production and economic systems / Yu. G. Lysenko, N. V. Rumyantsev. - Donetsk: DonNU, 2007. – 238 p.

35. Pihnastyi O. M. Multi-stream model of the production line / O. M. Pihnastyi // Math model. - Dniprodzerzhynsk: DDTU. - 2017. - № 1. - p. 81-87. – https://doi.org/10.13140/RG.2.2.19038.13128

36. Pihnastyi O. M. The problem of optimal operative control of macroparameters of a production system with mass production. M. Pihnastyi // Dopovіdі National Academy of Ukraine. - Kiev: Publishing House "Academperіodika".– 2006. – №5 – p. 79 – 85.

37. Karcheva G. Problems and prospects of development of the banking system of Ukraine: st. sciences. Ave. / NBU; Ministry of Education and Science of Ukraine. - Sumi, 2009. – vol. 26. – p. 200–206. https://goo.gl/3HcT8P

38. Pihnastyi O. M. Review of models of controlled production processes of production lines of production systems / O.M. Pihnastyi // Scientific statements of Belgorod State University. Belgorod: BSU. - 2015. - № 34/1. p.137-152.

39. Lurie K. A. Optimal control in problems of mathematical physics / K. A. Lurie. - M .: Science. Main editors of physical and mathematical literature1975. – 480 p.

40. Armbruster D. A continuous model for supply chains with nit buffers / D. Armbruster, S. Goerlich, M. Herty // SIAM Journal on Applied Mathematics. – 2011. – P. 855 – 877.

41. Kleiner G. B. Production functions. Theory, methods, application / G. B. Kleiner. - M .: Finance and Statistics, 1986. - 239 p.

42. Valko V. I. Variation calculus and optimal control / V. I. Valko, O. V. Ermoshina, G. N. Kuvyrkin. - Moscow: Moscow State Technical University named after Bauman, 2006. - 488 p.

43. Elsgolts L.E. Differential equations and calculus of variations / L.E.Elsgolts - Moscow: Nauka, 1969. - 425 p.

44. Lavrentiev, M. Fundamentals of the calculus of variations / M. Lavrentyev, L. Lyusternik. - M .: Publishing house NKTP USSR, 1935. – vol. 2. - 399 p.

45. Armbruster D. Kinetic and fluid model hierarchies for supply chains. / D. Armbruster., D. Marthaler, C. Ringhofer // SIAM Multiscale Model Simul.– 2004. – vol. 2 №1. – P. 43 – 61.