Supply chain modeled as a metabolic pathway
Abstract
A new model of economic production process is proposed (in the form of a set of ODEs) based on an idea that nonconsumable factors of production facilitate the conversion of inputs to output in much the same catalytic way as do enzymes in living cells when transforming substrates into different chemical compounds. The output of a converging, multi-resource, single-product supply chain network is shown to depend on the minimum of its inputs in the form of the Leontief--Liebig production function, providing the validity of the clearing function approximation. In turn use of the clearing function is legitimate when the machine processing time is much shorter than the machine loading time.
Keyword : supply chain, production function, limiting factor, clearing function
How to Cite
Mustafin, A., & Kantarbayeva, A. (2018). Supply chain modeled as a metabolic pathway. Mathematical Modelling and Analysis, 23(3), 473-491. https://doi.org/10.3846/mma.2018.028
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
[1] J. Adams, E. Balas and D. Zawack. The shifting bottleneck procedure for job shop scheduling. Management Science, 34(3):391–401, 1988. https://doi.org/10.1287/mnsc.34.3.391.
[2] D. Armbruster. The production planning problem: clearing functions, variable lead times, delay equations and partial differential equations. In D. Armbruster and K. G. Kempf(Eds.), Decision Policies for Production Networks, pp. 289–302. Springer, London, 2012. http://dx.doi.org/10.1007/978-0-85729-644-3_12.
[3] R.U. Ayres and U.E. Simonis(Eds.). Industrial Metabolism: Restructuring for Sustainable Development. United Nations University Press, Tokyo, 1994. ISBN 9789280808414.
[4] A. Cornish-Bowden. Fundamentals of Enzyme Kinetics. Wiley-Blackwell, Weinheim, 4th edition, 2012. ISBN 9783527330744.
[5] S.A. Ebelhar, W. Chesworth and Q. Paris. Law of the Minimum. In W. Chesworth(Ed.), Encyclopedia of Soil Science, pp. 431–437. Springer Netherlands, Dordrecht, 2008. ISBN 9781402039959
[6] J.W. Fowler, G.L. Hogg and S.J. Mason. Workload control in the semiconductor industry. Production Planning & Control: The Management of Operations, 13(7):568–578, 2002. https://doi.org/10.1080/0953728021000026294.
[7] N. Georgescu-Roegen. Analytical Economics: Issues and Problems. Harvard University Press, Cambridge, MA, 1966. ISBN 9780674281639. https://doi.org/10.4159/harvard.9780674281639.
[8] R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki. The Hill equation and the origin of quantitative pharmacology. Archive for History of Exact Sciences, 66(4):427–438, 2012. https://doi.org/10.1007/s00407-012-0098- 5.
[9] E.M. Goldratt and J. Cox. The Goal: A Process of Ongoing Improvement. North River Press, Great Barrington, MA, 4th edition, 2014. ISBN 9780884271956.
[10] D. Helbing, D. Armbruster, A.S. Mikhailov and E. Lefeber. Information and material flows in complex networks. Physica A: Statistical Mechanics and its Applications, 363(1):xi–xvi, 2006. https://doi.org/10.1016/j.physa.2006.01.042.
[11] D.J. Higham. Modeling and simulating chemical reactions. SIAM Review, 50(2):347–368, 2008. https://doi.org/10.1137/060666457.
[12] P. Hochendoner, C. Ogle and W.H. Mather. A queueing approach to multi-site enzyme kinetics. Interface Focus, 4(20130077):1–11, 2014. https://doi.org/10.1098/rsfs.2013.0077.
[13] C.S. Holling. The functional response of predators to prey density and its role in mimicry and population regulation. Memoirs of the Entomological Society of Canada, 97(S45):5–60, 1965. https://doi.org/10.4039/entm9745fv.
[14] W.J. Hopp and M.L. Spearman. Factory Physics: Foundations of Manufacturing Management. McGraw-Hill, New York, NY, 3rd edition, 2008. ISBN 9780072824032.
[15] C. Kandemir-Cavas, L. Cavas, M.B. Yokes, M. Hlynka, R. Schell and K. Yurdakoc. A novel application of queueing theory on the Caulerpenyne secreted by invasive Caulerpa taxifolia (Vahl) C.Agardh (Ulvophyceae, Caulerpales): a preliminary study. Mediterranean Marine Science, 9(1):67–76, 2008. https://doi.org/10.12681/mms.144.
[16] U.S. Karmarkar. Manufacturing lead times, order release and capacity loading. In S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin(Eds.), Logistics of Production and Inventory, volume 4 of Handbook in Operations Research and Management Science, chapter 6, pp. 287–329. North Holland, Amsterdam, 1993. https://doi.org/10.1016/S0927-0507(05)80186-0. ISBN 0444874720
[17] G.I. Kolesova and I.A. Poletaev. Nekotorye voprosy issledovaniia sistem s limitiruiushchimi faktorami [Selected problems in research of the systems with limiting factors]. In Upravliaemye sistemy [Controllable systems], number 3, pp. 71–80. Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 1969. (In Russian)
[18] W. Leontief. The Structure of American Economy, 1919–1939; An Empirical Application of Equilibrium Analysis. Oxford University Press, New York, NY, 2nd edition, 1951.
[19] E. Levine and T. Hwa. Stochastic fluctuations in metabolic pathways. Proceedings of the National Academy of Sciences, 104(22):9224–9229, 2007. https://doi.org/10.1073/pnas.0610987104.
[20] A. Marshall. Principles of Economics. Palgrave Classics in Economics. Palgrave Macmillan UK, London, 8 edition, 2013. ISBN 9780230249295.
[21] H. Missbauer and R. Uzsoy. Optimization models of production planning problems. In K.G. Kempf, P. Keskinocak and R. Uzsoy(Eds.), Planning Production and Inventories in the Extended Enterprise: A State of the Art Handbook, Volume 1, volume 151 of International Series in Operations Research and Management Science, chapter 16, pp. 437–507. Springer, New York, NY, 2011. http://dx.doi.org/10.1007/978-1-4419-6485-4_16.
[22] L. M¨onch, J.W. Fowler and S.J. Mason. Production Planning and Control for Semiconductor Wafer Fabrication Facilities: Modeling, Analysis, and Systems, volume 52 of Operations Research/Computer Science Interfaces. Springer, New York, NY, 2013. ISBN 9781461444718.
[23] J. Monod. The growth of bacterial cultures. Annual Review of Microbiology, 3(1):371–394, 1949. https://doi.org/10.1146/annurev.mi.03.100149.002103.
[24] A. Mustafin. K printsipam promyshlennogo metabolizma [A contribution to the principles of industrial metabolism]. AlPari, (4–5):80–83, 1999. (In Russian)
[25] A. Mustafin and A. Kantarbayeva. The Leontief’s black box reverse engineered. In 22nd International Conference “Mathematical Modelling and Analysis”, May 30 – June 2, 2017, Druskininkai, Lithuania. Abstracts, p. 46, VGTU Technika, Vilnius, 2017. ISBN 9786094760228.
[26] A.T. Mustafin. A bottleneck principle for techno-metabolic chains. Discussion paper 504, Institute of Economic Research, Kyoto University, Kyoto, Japan, September 1999.
[27] G.O. Nijland, J. Schouls and J. Goudriaan. Integrating the production functions of Liebig, Michaelis–Menten, Mitscherlich and Liebscher into one system dynamics model. NJAS – Wageningen Journal of Life Sciences, 55(2):199–224, 2008. https://doi.org/10.1016/s1573-5214(08)80037-1.
[28] R.E. O’Malley. Singular Perturbation Methods for Ordinary Differential Equations, volume 89 of Applied mathematical sciences. Springer-Verlag, New York, NY, 1991. ISBN 038797556X. https://doi.org/10.1007/978-1-4612-0977-5.
[29] Yu.M. Romanovskii, N.M. Stepanova and D.S. Chernavskii. Chto takoe matematicheskaia biofizika: Kineticheskie modeli v biofizike [What is mathematical biophysics: Kinetic models in biophysics]. Prosveshchenie, Moscow, 1971. (In Russian)
[30] C. Roser, M. Nakano and M. Tanaka. Shifting bottleneck detection. In E. Y¨ucesan, C.-H. Chen, J. L. Snowdon and J. M. Charnes(Eds.), Proceedings of the 2002 Winter Simulation Conference, 8–11 Dec. 2002, San Diego, CA, pp. 1079–1086. IEEE, 2002. https://doi.org/10.1109/wsc.2002.1166360.
[31] S.S. Sana. Optimal production lot size and reorder point of a twostage supply chain while random demand is sensitive with sales teams’ initiatives. International Journal of Systems Science, 47(2):450–465, 2014. https://doi.org/10.1080/00207721.2014.886748.
[32] L.A. Segel and M. Slemrod. The quasi-steady-state assumption: a case study in perturbation. SIAM Review, 31(3):446–477, 1989. https://doi.org/10.1137/1031091.
[33] R.W. Shephard. Theory of Cost and Production Functions. Princeton University Press, Princeton, NJ, 2016. ISBN 9780691647524.
[34] A. Tipper. Capital-labour substitution elasticities in New Zealand: one for all industries? Working Paper 12-01, Statistics New Zealand, Wellington, New Zealand, July 2012.
[35] H.R. Varian. Microeconomic Analysis. Norton, New York, NY, 3rd edition, 1992. ISBN 9780393957358.
[2] D. Armbruster. The production planning problem: clearing functions, variable lead times, delay equations and partial differential equations. In D. Armbruster and K. G. Kempf(Eds.), Decision Policies for Production Networks, pp. 289–302. Springer, London, 2012. http://dx.doi.org/10.1007/978-0-85729-644-3_12.
[3] R.U. Ayres and U.E. Simonis(Eds.). Industrial Metabolism: Restructuring for Sustainable Development. United Nations University Press, Tokyo, 1994. ISBN 9789280808414.
[4] A. Cornish-Bowden. Fundamentals of Enzyme Kinetics. Wiley-Blackwell, Weinheim, 4th edition, 2012. ISBN 9783527330744.
[5] S.A. Ebelhar, W. Chesworth and Q. Paris. Law of the Minimum. In W. Chesworth(Ed.), Encyclopedia of Soil Science, pp. 431–437. Springer Netherlands, Dordrecht, 2008. ISBN 9781402039959
[6] J.W. Fowler, G.L. Hogg and S.J. Mason. Workload control in the semiconductor industry. Production Planning & Control: The Management of Operations, 13(7):568–578, 2002. https://doi.org/10.1080/0953728021000026294.
[7] N. Georgescu-Roegen. Analytical Economics: Issues and Problems. Harvard University Press, Cambridge, MA, 1966. ISBN 9780674281639. https://doi.org/10.4159/harvard.9780674281639.
[8] R. Gesztelyi, J. Zsuga, A. Kemeny-Beke, B. Varga, B. Juhasz and A. Tosaki. The Hill equation and the origin of quantitative pharmacology. Archive for History of Exact Sciences, 66(4):427–438, 2012. https://doi.org/10.1007/s00407-012-0098- 5.
[9] E.M. Goldratt and J. Cox. The Goal: A Process of Ongoing Improvement. North River Press, Great Barrington, MA, 4th edition, 2014. ISBN 9780884271956.
[10] D. Helbing, D. Armbruster, A.S. Mikhailov and E. Lefeber. Information and material flows in complex networks. Physica A: Statistical Mechanics and its Applications, 363(1):xi–xvi, 2006. https://doi.org/10.1016/j.physa.2006.01.042.
[11] D.J. Higham. Modeling and simulating chemical reactions. SIAM Review, 50(2):347–368, 2008. https://doi.org/10.1137/060666457.
[12] P. Hochendoner, C. Ogle and W.H. Mather. A queueing approach to multi-site enzyme kinetics. Interface Focus, 4(20130077):1–11, 2014. https://doi.org/10.1098/rsfs.2013.0077.
[13] C.S. Holling. The functional response of predators to prey density and its role in mimicry and population regulation. Memoirs of the Entomological Society of Canada, 97(S45):5–60, 1965. https://doi.org/10.4039/entm9745fv.
[14] W.J. Hopp and M.L. Spearman. Factory Physics: Foundations of Manufacturing Management. McGraw-Hill, New York, NY, 3rd edition, 2008. ISBN 9780072824032.
[15] C. Kandemir-Cavas, L. Cavas, M.B. Yokes, M. Hlynka, R. Schell and K. Yurdakoc. A novel application of queueing theory on the Caulerpenyne secreted by invasive Caulerpa taxifolia (Vahl) C.Agardh (Ulvophyceae, Caulerpales): a preliminary study. Mediterranean Marine Science, 9(1):67–76, 2008. https://doi.org/10.12681/mms.144.
[16] U.S. Karmarkar. Manufacturing lead times, order release and capacity loading. In S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin(Eds.), Logistics of Production and Inventory, volume 4 of Handbook in Operations Research and Management Science, chapter 6, pp. 287–329. North Holland, Amsterdam, 1993. https://doi.org/10.1016/S0927-0507(05)80186-0. ISBN 0444874720
[17] G.I. Kolesova and I.A. Poletaev. Nekotorye voprosy issledovaniia sistem s limitiruiushchimi faktorami [Selected problems in research of the systems with limiting factors]. In Upravliaemye sistemy [Controllable systems], number 3, pp. 71–80. Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 1969. (In Russian)
[18] W. Leontief. The Structure of American Economy, 1919–1939; An Empirical Application of Equilibrium Analysis. Oxford University Press, New York, NY, 2nd edition, 1951.
[19] E. Levine and T. Hwa. Stochastic fluctuations in metabolic pathways. Proceedings of the National Academy of Sciences, 104(22):9224–9229, 2007. https://doi.org/10.1073/pnas.0610987104.
[20] A. Marshall. Principles of Economics. Palgrave Classics in Economics. Palgrave Macmillan UK, London, 8 edition, 2013. ISBN 9780230249295.
[21] H. Missbauer and R. Uzsoy. Optimization models of production planning problems. In K.G. Kempf, P. Keskinocak and R. Uzsoy(Eds.), Planning Production and Inventories in the Extended Enterprise: A State of the Art Handbook, Volume 1, volume 151 of International Series in Operations Research and Management Science, chapter 16, pp. 437–507. Springer, New York, NY, 2011. http://dx.doi.org/10.1007/978-1-4419-6485-4_16.
[22] L. M¨onch, J.W. Fowler and S.J. Mason. Production Planning and Control for Semiconductor Wafer Fabrication Facilities: Modeling, Analysis, and Systems, volume 52 of Operations Research/Computer Science Interfaces. Springer, New York, NY, 2013. ISBN 9781461444718.
[23] J. Monod. The growth of bacterial cultures. Annual Review of Microbiology, 3(1):371–394, 1949. https://doi.org/10.1146/annurev.mi.03.100149.002103.
[24] A. Mustafin. K printsipam promyshlennogo metabolizma [A contribution to the principles of industrial metabolism]. AlPari, (4–5):80–83, 1999. (In Russian)
[25] A. Mustafin and A. Kantarbayeva. The Leontief’s black box reverse engineered. In 22nd International Conference “Mathematical Modelling and Analysis”, May 30 – June 2, 2017, Druskininkai, Lithuania. Abstracts, p. 46, VGTU Technika, Vilnius, 2017. ISBN 9786094760228.
[26] A.T. Mustafin. A bottleneck principle for techno-metabolic chains. Discussion paper 504, Institute of Economic Research, Kyoto University, Kyoto, Japan, September 1999.
[27] G.O. Nijland, J. Schouls and J. Goudriaan. Integrating the production functions of Liebig, Michaelis–Menten, Mitscherlich and Liebscher into one system dynamics model. NJAS – Wageningen Journal of Life Sciences, 55(2):199–224, 2008. https://doi.org/10.1016/s1573-5214(08)80037-1.
[28] R.E. O’Malley. Singular Perturbation Methods for Ordinary Differential Equations, volume 89 of Applied mathematical sciences. Springer-Verlag, New York, NY, 1991. ISBN 038797556X. https://doi.org/10.1007/978-1-4612-0977-5.
[29] Yu.M. Romanovskii, N.M. Stepanova and D.S. Chernavskii. Chto takoe matematicheskaia biofizika: Kineticheskie modeli v biofizike [What is mathematical biophysics: Kinetic models in biophysics]. Prosveshchenie, Moscow, 1971. (In Russian)
[30] C. Roser, M. Nakano and M. Tanaka. Shifting bottleneck detection. In E. Y¨ucesan, C.-H. Chen, J. L. Snowdon and J. M. Charnes(Eds.), Proceedings of the 2002 Winter Simulation Conference, 8–11 Dec. 2002, San Diego, CA, pp. 1079–1086. IEEE, 2002. https://doi.org/10.1109/wsc.2002.1166360.
[31] S.S. Sana. Optimal production lot size and reorder point of a twostage supply chain while random demand is sensitive with sales teams’ initiatives. International Journal of Systems Science, 47(2):450–465, 2014. https://doi.org/10.1080/00207721.2014.886748.
[32] L.A. Segel and M. Slemrod. The quasi-steady-state assumption: a case study in perturbation. SIAM Review, 31(3):446–477, 1989. https://doi.org/10.1137/1031091.
[33] R.W. Shephard. Theory of Cost and Production Functions. Princeton University Press, Princeton, NJ, 2016. ISBN 9780691647524.
[34] A. Tipper. Capital-labour substitution elasticities in New Zealand: one for all industries? Working Paper 12-01, Statistics New Zealand, Wellington, New Zealand, July 2012.
[35] H.R. Varian. Microeconomic Analysis. Norton, New York, NY, 3rd edition, 1992. ISBN 9780393957358.