Clearing function in the context of the invariant manifold method
DOI:
https://doi.org/10.21638/11701/spbu10.2023.205Abstract
Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-inprogress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of CF is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio ’working machines: WIP’ guarantees the applicability of CF approximation in unsteady-state operation.
Keywords:
work in progress, production model, quasi-steady-state approximation, singular perturbation, enzyme catalysis
Downloads
Download data is not yet available.
References
Литература
Vacanti D. When will it be done?: Lean-agile forecasting to answer your customers' most important question. La Vergne, TN: Lightning Source Inc., 2020. 308 p.
Little J. A proof for the queuing formula: L = lambda W // Operations Research. 1961. Vol. 9. N 3. P. 383-387. https://doi.org/10.1287/opre.9.3.383
Little J., Graves S. Little's law // Building intuition: Insights from basic operations management models and principles / ed. by D. Chhajed, T. Lowe. London: Springer, 2008. P. 81-100. https://doi.org/10.1007/978-0-387-73699-0_5
Armbruster D., Uzsoy R. Continuous dynamic models, clearing functions, and discrete-event simulation in aggregate production planning // New directions in informatics, optimization, logistics, and production. Tutorials in operations research / ed. by P. Mirchandani. INFORMS. 2012. P. 103-126. https://doi.org/10.1287/educ.1120.0102
Graves S. A tactical planning model for a job shop // Operations Research. 1986. Vol. 34. N 4. P. 522-533. https://doi.org/10.1287/opre.34.4.522
Karmarkar U. Capacity loading and release planning with work-in-progress (WIP) and lead-times // Journal of Manufacturing and Operations Management. 1989. Vol. 2. N 1. P. 105-123. https://www.iaorifors.com/paper/3710
Karmarkar U. Manufacturing lead times, order release and capacity loading // Logistics of production and inventory / ed. by S. Graves, A. Rinnooy Kan, P. Zipkin. Amsterdam: North-Holland Publishing, 1993. P. 287-329.
Srinivasan A., Carey M., Morton T. Resource pricing and aggregate scheduling in manufacturing systems: Working Paper N 88-89-58. Pittsburgh: Graduate School of Industrial Administration, Carnegie-Mellon University, 1988. 51 p. https://econpapers.repec.org/paper/cmugsiawp/88-89-58.htm
Missbauer H., Uzsoy R. Production planning with capacitated resources and congestion. New York: Springer, 2020. 295 p. https://doi.org/10.1007/978-1-0716-0354-3_7
Missbauer H. Order release planning with clearing functions: A queueing-theoretical analysis of the clearing function concept // International Journal of Production Economics. 2011. Vol. 131. N 1. P. 399-406. https://doi.org/10.1016/j.ijpe.2009.09.003
Armbruster D. The production planning problem: Clearing functions, variable lead times, delay equations and partial differential equations // Decision policies for production networks / ed. by D. Armbruster, K. Kempf. London: Springer, 2012. P. 289-302. https://doi.org/10.1007/978-0-85729-644-3_12
Корниш-Боуден Э. Основы ферментативной кинетики / пер. с англ. и предисл. Б. И. Курганова. М.: Мир, 1979. 280 с.
Georgescu-Roegen N. Analytical economics: Issues and problems. Cambridge, MA: Harvard University Press, 1966. 434 p. https://doi.org/10.4159/harvard.9780674281639.c19
Колесова Г. И., Полетаев И. А. Некоторые вопросы исследования систем с лимитирующими факторами // Управляемые системы: сб. ст. Новосибирск: Институт математики СО АН СССР, 1969. Вып. 3. С. 71-80. http://nas1.math.nsc.ru/aim/journals/us/us3/us03_010.pdf
Романовский Ю. М., Степанова Н. М., Чернавский Д. С. Что такое математическая биофизика: Кинетические модели в биофизике. М.: Просвещение, 1971. 136 с.
Mustafin A. T. A bottleneck principle for techno-metabolic chains: Discussion Paper N 504. Kyoto: Institute of Economic Research, Kyoto University, 1999. 13 p. https://doi.org/10.48550/arXiv.1708.04177
Милованов В. П. Неравновесные социально-экономические системы: Синергетика и самоорганизация. М.: Эдиториал УРСС, 2001. 264 с.
Ponzi A., Yasutomi A., Kaneko K. A non-linear model of economic production processes // Physica A: Statistical Mechanics and its Applications. 2003. Vol. 324. N 1-2. P. 372-379. https://doi.org/10.1016/S0378-4371(02)01885-X
Niyirora J., Zhuang J. Fluid approximations and control of queues in emergency departments // European Journal of Operational Research. 2017. Vol. 261. N 3. P. 1110-1124. https://doi.org/10.1016/j.ejor.2017.03.013
Мустафин А. Т., Кантарбаева А. К. Каталитическая модель массового обслуживания на примере циклической очереди // Известия высших учебных заведений. Прикладная нелинейная динамика. 2019. Т. 27. № 5. С. 53-71. https://doi.org/10.18500/0869-6632-2019-27-5-53-71
Bradt L. The automated factory: Myth or reality? // Engineering: Cornell Quarterly. 1983. Vol. 17. N 3. P. 26-31. https://ecommons.cornell.edu/handle/1813/2422
Hopp W., Spearman M. Factory physics: Foundations of manufacturing management. 3rd ed. New York: McGraw-Hill, 2008. 720 p.
Васильева А. Б., Бутузов В. Ф. Асимптотические методы в теории сингулярных возмущений. М.: Высшая школа, 1990. 207 с.
Verhulst F. Methods and applications of singular perturbations: Boundary layers and multiple timescale dynamics. New York: Springer, 2005. 328 p.
Shchepakina E., Sobolev V., Mortel M. Singular perturbations. Introduction to system order reduction methods with applications. New York: Springer, 2014. 212 p. https://doi.org/10.1007/978-3-319-09570-7
Тихонов А. Н. Системы дифференциальных уравнений, содержащие малые параметры при производных // Математический сборник. 1952. Т. 31(73). № 3. С. 575-586. https://www.mathnet.ru/links/448f715c3bfde9444fe634ffabd73075/sm5548.pdf
Johnson K., Goody R. The original Michaelis constant: Translation of the 1913 Michaelis-Menten paper // Biochemistry. 2011. Vol. 50. N 39. P. 8264-8269. https://doi.org/10.1021/bi201284u
Holling C. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly // The Canadian Entomologist. 1959. Vol. 91. N 5. P. 293-320. https://doi.org/10.4039/ent91293-5
Monod J. The growth of bacterial cultures // Annual Review of Microbiology. 1949. Vol. 3. N 1. P. 371-394. https://doi.org/10.1146/annurev.mi.03.100149.002103
Whitt W. Time-varying queues // Queueing Models and Service Management. 2018. Vol. 1. N 2. P. 79-164. http://140.120.49.88/index.php/qmsm/article/view/33
References
Vacanti D. When will it be done?: Lean-agile forecasting to answer your customers' most important question. La Vergne, TN, Lightning Source Inc. Publ., 2020, 308 p.
Little J. A proof for the queuing formula: L = lambda W. Operations Research, 1961, vol. 9, no. 3, pp. 383-387. https://doi.org/10.1287/opre.9.3.383
Little J., Graves S. Little's law. Building intuition: Insights from basic operations management models and principles. Ed. by D. Chhajed, T. Lowe. London, Springer Publ., 2008, pp. 81-100. https://doi.org/10.1007/978-0-387-73699-0_5
Armbruster D., Uzsoy R. Continuous dynamic models, clearing functions, and discrete-event simulation in aggregate production planning. New directions in informatics, optimization, logistics, and production. Tutorials in operations research. Ed. by P. Mirchandan. INFORMS Publ., 2012, pp. 103-126. https://doi.org/10.1287/educ.1120.0102
Graves S. A tactical planning model for a job shop. Operations Research, 1986, vol. 34, no. 4, pp. 522-533. https://doi.org/10.1287/opre.34.4.522
Karmarkar U. Capacity loading and release planning with work-in-progress (WIP) and lead-times. Journal of Manufacturing and Operations Management, 1989, vol. 2, no. 1, pp. 105-123. https://www.iaorifors.com/paper/3710
Karmarkar U. Manufacturing lead times, order release and capacity loading. Logistics of production and inventory. Ed. by S. Graves, A. Rinnooy Kan, P. Zipkin. Amsterdam, North-Holland Publ., 1993, pp. 287-329.
Srinivasan A., Carey M., Morton T. Resource pricing and aggregate scheduling in manufacturing systems. Working Paper no. 88-89-58. Pittsburgh, Graduate School of Industrial Administration, Carnegie-Mellon University Publ., 1988, 51 p. https://econpapers.repec.org/paper/cmugsiawp/88-89-58.htm
Missbauer H., Uzsoy R. Production planning with capacitated resources and congestion. New York, Springer Publ., 2020, 295 p. https://doi.org/10.1007/978-1-0716-0354-3_7
Missbauer H. Order release planning with clearing functions: A queueing-theoretical analysis of the clearing function concept. International Journal of Production Economics, 2011, vol. 131, no. 1, pp. 399-406. https://doi.org/10.1016/j.ijpe.2009.09.003
Armbruster D. The production planning problem: Clearing functions, variable lead times, delay equations and partial differential equations. Decision policies for production networks. Ed. by D. Armbruster, K. Kempf. London, Springer Publ., 2012, pp. 289-302. https://doi.org/10.1007/978-0-85729-644-3_12
Cornish-Bowden A. Fundamentals of enzyme kinetics. 4th ed. Weinheim, Wiley-Blackwell Publ., 2012, 498 p. (Rus. ed.: Cornish-Bowden A. Osnovy fermentativnoi kinetiki. Moscow, Mir Publ., 1979, 280 p.)
Georgescu-Roegen N. Analytical economics: Issues and problems. Cambridge, MA, Harvard University Press, 1966, 434 p. https://doi.org/10.4159/harvard.9780674281639.c19
Kolesova G. I., Poletaev I. A. Nekotorye voprosy issledovaniia sistem s limitiruiushchimi faktorami [Selected problems in research of the systems with limiting factors]. Upravliaemye sistemy. Sb. st., vyp. 3 [ Controllable systems. Collected papers, iss. 3]. Novosibirsk, Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences Publ., 1969, pp. 71-80. http://nas1.math.nsc.ru/aim/journals/us/us3/us03_010.pdf (In Russian)
Romanovskii Yu. M., Stepanova N. M., Chernavskii D. S. Chto takoe matematicheskaia biofizika: Kineticheskie modeli v biofizike [ What is mathematical biophysics: The kinetic models in biophysics ]. Moscow, Prosveshchenie Publ., 1971, 136 p. (In Russian)
Mustafin A. T. A bottleneck principle for techno-metabolic chains. Discussion Paper no. 504. Kyoto, Institute of Economic Research, Kyoto University Press, 1999, 13 p. https://doi.org/10.48550/arXiv.1708.04177
Milovanov V. P. Neravnovesnye sotsial'no-ekonomicheskie sistemy: Sinergetika i samoorganizatsiia [ Nonequilibrium social-economic systems: Synergetics and self-organization ]. Moscow, Editorial URSS Publ., 2001, 264 p. (In Russian)
Ponzi A., Yasutomi A., Kaneko K. A non-linear model of economic production processes. Physica A: Statistical Mechanics and its Applications, 2003, vol. 324, no. 1-2, pp. 372-379. https://doi.org/10.1016/S0378-4371(02)01885-X
Niyirora J., Zhuang J. Fluid approximations and control of queues in emergency departments. European Journal of Operational Research, 2017, vol. 261, no. 3, pp. 1110-1124. https://doi.org/10.1016/j.ejor.2017.03.013
Mustafin A. T., Kantarbayeva A. K. Kataliticheskaiia model' massovogo obsluzhivaniia na primere tsiklicheskoi ocheredi [A catalytic model of service as applied to the case of a cyclic queue]. Izvestiia vysshikh uchebnykh zavedenii. Prikladnaia nelineinaia dinamika [ Proceedings of Higher Educational Institutes. Applied Nonlinear Dynamics ], 2019, vol. 27, no. 5, pp. 53-71. https://doi.org/10.18500/0869-6632-2019-27-5-53-71. (In Russian)
Bradt L. The automated factory: Myth or reality? Engineering: Cornell Quarterly, 1983, vol. 17, no. 3, pp. 26-31. https://ecommons.cornell.edu/handle/1813/2422
Hopp W., Spearman M. Factory physics: Foundations of manufacturing management. 3rd ed. New York, McGraw-Hill Publ., 2008, 720 p.
Vasil'eva A. B., Butuzov V. F. Asimptoticheskie metody v teorii singuliarnykh vozmushchenii [ Asymptotic methods in singular perturbations theory ]. Moscow, Vysshaia shkola Publ., 1990, 207 p. (In Russian)
Verhulst F. Methods and applications of singular perturbations: Boundary layers and multiple timescale dynamics. New York, Springer Publ., 2005, 328 p.
Shchepakina E., Sobolev V., Mortel M. Singular perturbations. Introduction to system order reduction methods with applications. New York, Springer Publ., 2014, 212 p. https://doi.org/10.1007/978-3-319-09570-7
Tikhonov A. N. Sistemy differentsial'nykh uravnenii, soderzhashchie malye parametry pri proizvodnykh [Systems of differential equations containing small parameters at the derivatives]. Sbornik Mathematics, 1952, vol. 31(73), no. 3, pp. 575-586. https://www.mathnet.ru/links/448f715c3bfde9444fe634ffabd73075/sm5548.pdf (In Russian)
Johnson K., Goody R. The original Michaelis constant: Translation of the 1913 Michaelis - Menten paper. Biochemistry, 2011, vol. 50, no. 39, pp. 8264-8269. https://doi.org/10.1021/bi201284u
Holling C. The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. The Canadian Entomologist, 1959, vol. 91, no. 5, pp. 293-320. https://doi.org/10.4039/ent91293-5
Monod J. The growth of bacterial cultures. Annual Review of Microbiology, 1949, vol. 3, no. 1, pp. 371-394. https://doi.org/10.1146/annurev.mi.03.100149.002103
Whitt W. Time-varying queues. Queueing Models and Service Management, 2018, vol. 1, no. 2, pp. 79-164. http://140.120.49.88/index.php/qmsm/article/view/33
Downloads
Published
2023-07-27
How to Cite
Mustafin, A. T., & Kantarbayeva, A. K. (2023). Clearing function in the context of the invariant manifold method. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 19(2), 185—198. https://doi.org/10.21638/11701/spbu10.2023.205
Issue
Section
Applied Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
