Previous |  Up |  Next


stochastic programming; progressive hedging; parallel computing; steel production; heat transfer; phase change
The paper is concerned with a parallel implementation of the progressive hedging algorithm (PHA) which is applicable for the solution of stochastic optimization problems. We utilized the Message Passing Interface (MPI) and the General Algebraic Modelling System (GAMS) to concurrently solve the scenario-related subproblems in parallel manner. The standalone application combining the PHA, MPI, and GAMS was programmed in C++. The created software was successfully applied to a steel production problem which is considered by means of the two-stage stochastic PDE-constrained program with a random failure. The numerical heat transfer model for the steel production was derived with the use of the control volume method and the phase changes were taken into account with the use of the effective heat capacity. Numerical experiments demonstrate that parallel computing facility has enabled a significant reduction of computational time. The quality of the stochastic solution was evaluated and discussed. The developed system seems computationally effective and sufficiently robust which makes it applicable in other applications as well.
[1] Alquarashi, A., Etemadi, A. H., Khodaei, A.: Treatment of uncertainty for next generation power systems: State-of-the-art in stochastic optimization. Electr. Power Syst. Res. 141 (2016), 233-245. DOI 10.1016/j.epsr.2016.08.009
[2] Barttfeld, M., Alleborn, N., Durst, F.: Dynamic optimization of multiple-zone air impingement drying process. Comput. Chem. Engrg. 30 (2006), 467-489. DOI 10.1016/j.compchemeng.2005.10.016
[3] Birge, J. R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York 2011. MR 2807730
[4] Brimacombe, J. K., Sorimachi, K.: Crack formation in continuous-casting of steel. Metal. Trans. B. Proc. Metal. 8 (1977), 489-505. DOI 10.1007/bf02696937
[5] Carvalho, E. P., Martínez, J., Martínez, J. M., Pisnitchenko, F.: On optimization strategies for parameter estimation in models governed by partial differential equations. Math. Comput. Simul. 114 (2015), 14-24. DOI 10.1016/j.matcom.2010.07.020 | MR 3357814
[6] Carrasco, M., Ivorra, B., Ramos, A. M.: Stochastic topology design optimization for continuous elastic materials. Comput. Meth. Appl. Mech. Engrg. 289 (2015), 131-154. DOI 10.1016/j.cma.2015.02.003 | MR 3327148
[7] Carpentier, P. L., Gendreau, M., Bastin, F.: Long-term management of a hydroelectric multireservoir system under uncertainty using the progressive hedging algorithm. Water Resour. Res. 49 (2013), 2812-2827. DOI 10.1002/wrcr.20254
[8] Cheng, Y. M., Li, D. Z., Li, N., Lee, Y. Y., Au, S. K.: Solution of some engineering partial differential equations governed by the minimal of a functional by global optimization method. J. Mech. 29 (2013), 507-516. DOI 10.1017/jmech.2013.26
[9] Drud, A.: CONOPT - A GRG code for large sparse dynamic nonlinear optimization problems. Math. Program. 31 (1985), 153-191. DOI 10.1007/bf02591747 | MR 0777289
[10] Gade, D., Ryan, G. Hackebeil. S. M., Watson, J.-P., Wets, R. J.-B., Woodruff, D. L.: Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs. Math. Prog. 157 (2016), 47-67. DOI 10.1007/s10107-016-1000-z | MR 3492067
[11] Gonçalves, R. E. C., Finardi, E. C., Silva, E. L. da: Applying different decomposition schemes using the progressive hedging algorithm to the operation planning problem of a hydrothermal system. Electr. Power Syst. Res. 83 (2012), 19-27. DOI 10.1016/j.epsr.2011.09.006
[12] Gul, S., Denton, B. T., Fowler, J. W.: A progressive hedging approach for surgery planning under uncertainty. INFORMS J. Comput. 27 (2015), 755-772. DOI 10.1287/ijoc.2015.0658 | MR 3432659
[13] Ikeda, S., Ooka, R.: A new optimization strategy for the operating schedule of energy systems under uncertainty of renewable energy sources and demand changes. Energ. Build. 125 (2016), 75-85. DOI 10.1016/j.enbuild.2016.04.080
[14] Bergman, T. L., Lavine, A. S., Incropera, F. P., Dewitt, D. P.: Fundamentals of Heat and Mass Transfer. Seventh edition. Wiley, New York 2011.
[15] Klimeš, L.: Stochastic Programming Algorithms. Master Thesis. Brno University of Technology, 2010.
[16] Klimeš, L., Popela, P.: An implementation of progressive hedging algorithm for engineering problem. In: Proc. 16th International Conference on Soft Computing MENDEL, Brno 2010, pp. 459-464.
[17] Klimeš, L., Popela, P., Štětina, J.: Decomposition approach applied to stochastic optimization of continuous steel casting. In: Proc. 17th International Conference on Soft Computing MENDEL, Brno 2011, pp. 314-319.
[18] Klimeš, L., Mauder, T., Štětina, J.: Stochastic approach and optimal control of continuous steel casting process by using progressive hedging algorithm. In: Proc. 20th International Conference on Materials and Metallurgy METAL, Brno 2011, pp. 146-151.
[19] Marca, M. La, Armbruster, D., Herty, M., Ringhofer, C.: Control of continuum models of production systems. IEEE Trans. Automat. Control 55 (2010), 2511-2526. DOI 10.1109/tac.2010.2046925 | MR 2721893
[20] Lamghari, A., Dimitrakopoulos, R.: Progressive hedging applied as a metaheuristic to schedule production in open-pit mines accounting for reserve uncertainty. Eur. J. Oper. Res. 253 (2016), 843-855. DOI 10.1016/j.ejor.2016.03.007 | MR 3490823
[21] Liu, J., Liu, C.: Optimization of mold inverse oscillation control parameters in continuous casting process. Mater. Manuf. Process. 30 (2015), 563-568. DOI 10.1080/10426914.2015.1004696
[22] Mills, K. C., Ramirez-Lopez, P., Lee, P. D., Santillana, B., Thomas, B. G., Morales, R.: Looking into continuous casting mould. Ironmak. Steelmak. 41 (2014), 242-249. DOI 10.1179/0301923313z.000000000255
[23] Mauder, T., Kavička, F., Štětina, J., Franěk, Z., Masarik, M.: A mathematical & stochatic modelling of the concasting of steel slabs. In: Proc. International Conference on Materials and Metallurgy, Hradec nad Moravicí 2009, pp. 41-48.
[24] Mauder, T., Novotný, J.: Two mathematical approaches for optimal control of the continuous slab casting process. In: Proc. 16th International Conference on Soft Computing MENDEL, Brno 2010, pp. 41-48.
[25] Rockafellar, R. T., Wets, R. J.-B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16 (1991), 119-147. DOI 10.1287/moor.16.1.119 | MR 1106793
[26] Ruszczynski, A., Shapiro, A.: Stochastic Programming Models. Handbooks in Operations Research and Management Science, Volume 10: Stochastic Programming, Elsevier, Amsterdam 2003. DOI 10.1016/s0927-0507(03)10001-1 | MR 2051791
[27] Shioura, A., Shakhlevich, N. V., Strusevich, V. A.: Application of submodular optimization to single machine scheduling with controllable processing times subject to release dates and deadlines. INFORMS J. Comput. 28 (2016), 148-161. DOI 10.1287/ijoc.2015.0660 | MR 3461551
[28] Stefanescu, D. M.: Science and Engineering of Casting Solidification. Second edition. Springer, New York 2009.
[29] Štětina, J., Klimeš, L., Mauder, T.: Minimization of surface defects by increasing the surface temperature during the straightening of a continuously cast slab. Mater. Tehnol. 47 (2013), 311-316.
[30] Ugail, H., Wilson, M. J.: Efficient shape parametrisation for automatic design optimisation using a partial differential equation formulation. Comput. Struct. 81 (2003), 2601-2609. DOI 10.1016/s0045-7949(03)00321-3
[31] Varaiya, P., Wets, R. J.-B.: Stochastic dynamic optimization approaches and computation. In: Proc. 13th International Symposium on Mathematical Programming, Tokio 1989, pp. 309-331. DOI 10.1007/978-3-642-82450-0_11 | MR 1114320
[32] Veliz, F. B., Watson, J. P., Weintraub, A., Wets, R. J.-B., Woodruff, D. L.: Stochastic optimization models in forest planning: a progressive hedging solution approach. Ann. Oper. Res. 232 (2015), 259-274. DOI 10.1007/s10479-014-1608-4 | MR 3383965
[33] Waanders, B. G. V., Carnes, B. R.: Optimization under adaptive error control for finite element based simulations. Comput. Mech. 47 (2011), 49-63. DOI 10.1007/s00466-010-0530-0 | MR 2756370
[34] Wets, R. J.-B.: The aggretation principle in scenario analysis and stochastic optimization. In: Algorithms and Model Formulations in Mathematical Programming (S. W. Wallace, ed.), Springer, Berlin 1989. DOI 10.1007/978-3-642-83724-1_4 | MR 0996646
[35] Yang, Z., Qui, H. L., Luo, X. W., Shen, D.: Simulating schedule optimization problem in steelmaking continuous casting process. Int. J. Simul. Model. 14 (2015), 710-718. DOI 10.2507/ijsimm14(4)co17
[36] Yang, J., Ji, Z. P., Liu, S., Jia, Q.: Multi-objective optimization based on pareto optimum in secondary cooling and EMS of continuous casting. In: Proc. International Conference on Advanced Robotics and Mechatronics (ICARM), Macau 2016, pp. 283-287. DOI 10.1109/icarm.2016.7606933
[37] Žampachová, E., Popela, P., Mrázek, M.: Optimum beam design via stochastic programming. Kybernetika 46 (2010), 571-582. MR 2676092
[38] Zarandi, M. H. F., Dorry, F., Moghadam, F. S.: Steelmaking-continuous casting scheduling problem with interval type 2 fuzzy random due dates. In: Proc. IEEE Conference on Norbert Wiener in the 21st Century (21CW), Boston 2014. DOI 10.1109/norbert.2014.6893896
Partner of
EuDML logo