Modelos de programação estocástica no planejamento da produção de empresas moveleiras
Stochastic programming models in the production planning of furniture companies
Alem, Douglas; Morabito, Reinaldo
http://dx.doi.org/10.1590/0103-6513.075612
Production, vol.25, n3, p.657-677, 2015
Resumo
Esse trabalho aborda um problema de planejamento da produção típico de empresas moveleiras de pequeno porte, em que as demandas e os tempos de preparação dos estágios gargalos são variáveis aleatórias que podem ser aproximadas por um conjunto discreto e finito de cenários ponderados pelas correspondentes probabilidades de ocorrência. O problema com múltiplos cenários é modelado via programação estocástica de dois estágios com recurso. Para controlar a variabilidade dos custos de segundo estágio é proposto um modelo de recurso restrito que gera, progressivamente, um conjunto de soluções menos sensíveis às variações dos cenários, conforme a variabilidade é restringida a uma tolerância dada. Experiências numéricas indicam que, em muitas situações, não é muito dispendioso assegurar soluções aversas ao risco com bons níveis de serviço.
Palavras-chave
Planejamento da produção. Indústria moveleira. Programação estocástica. Aversão ao risco. Recurso restrito.
Abstract
This paper addresses a production planning problem that arises in small-scale furniture companies, where the demands and setup times of bottleneck operations are random variables that can be approximated by a discrete and finite number of scenarios that are weighted by their corresponding probabilities of occurrence. The problem is modeled under multiple scenarios via two-stage stochastic programming with recourse. To control the variability of the second-stage costs, we propose a restricted recourse model that generates a set of solutions that are less sensitive to the scenario changes because the variability is limited to a given tolerance. Numerical experiences indicate that, in some situations, risk-averse solutions with good service levels are not excessively expensive to obtain.
Keywords
Production planning. Furniture industry. Stochastic programming. Risk-aversion. Restricted recourse.
References
Aghezzaf, E. H., Sitompula, C., & Najid, N. M. (2010). Models for robust tactical planning in multi-stage production systems with uncertain demands. Computers & Operations Research, 37(5), 880-889. http://dx.doi.org/10.1016/j.cor.2009.03.012
Ahmed, S., & Sahinidis, N. V. (1998). Robust process planning under uncertainty. Industrial & Engineering Chemistry Research, 37(5), 1883-1892. http://dx.doi.org/10.1021/ie970694t
Alem, D. (2011). Programação estocástica e otimização robusta no planejamento da produção de empresas moveleiras (Tese de doutorado). Universidade de São Paulo, São Carlos.
Alem, D., & Morabito, R. (2013a). Risk-averse two- stage stochastic programs in furniture plants. OR Spectrum, 35(4), 773-806. http://dx.doi.org/10.1007/s00291-012-0312-5
Alem, D., & Morabito, R. (2013b). O problema combinado de planejamento da produção e corte de estoque sob incertezas: aplicação em fábricas de móveis de pequeno porte. Gestão & Produção, 20(1), 111-133. http://dx.doi.org/10.1590/S0104-530X2013000100009
Alonso-Ayuso, A., Carvallo, F., Escudero, L. F., Guignard, M., Pi, J., Puranmalka, R., & Weintraub, A. (2014). Medium range optimization of copper extraction planning under uncertainty in future copper prices. European Journal of Operational Research, 233(3), 711-726. http://dx.doi.org/10.1016/j.ejor.2013.08.048
Aouam, T., Rardin, R., & Abrache, J. (2010). Robust strategies for natural gas procurement. European Journal of Operational Research, 205(1), 151-158. http://dx.doi.org/10.1016/j.ejor.2009.12.015
Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88(3), 411-424. http://dx.doi.org/10.1007/PL00011380
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.
Beyer, H. G., & Sendhoff, B. (2007). Robust optimization: a comprehensive survey. Computers Methods in Applied Mechanics Engineering, 196(33-34), 3190-3218. http://dx.doi.org/10.1016/j.cma.2007.03.003
Carnieri, C., Guillermo, A., & Gavinho, L. (1994). Solution procedures for cutting lumber into furniture parts. European Journal of Operational Research, 73(3), 495-501. http://dx.doi.org/10.1016/0377-2217(94)90244-5
Dupacová, J., Consigli, G. & Wallace, S. W. (2000). Scenarios for multistage stochastic programs. Annals of Operations Research, 100(1-4), 25-53. http://dx.doi.org/10.1023/A:1019206915174
Foronda, S., & Carino, H. (1991). A heuristic approach to the lumber allocation and manufacturing in hardwood dimension and furniture manufacturing. European Journal of Operational Research, 54(2), 151-162. http://dx.doi.org/10.1016/0377-2217(91)90294-6
Geng, N., Jiang, Z., & Chen, F. (2009). Stochastic programming based capacity planning for semiconductor wafer fab with uncertain demand and capacity. European Journal of Operational Research, 198(3), 899-908. http://dx.doi.org/10.1016/j.ejor.2008.09.029
Gollmer, R., Neise, F., & Schultz, R. (2008). Stochastic programs with frst-order dominance constraints induced by mixed-integer linear recourse. SIAM Journal on Optimization, 19(2), 552-571. http://dx.doi.org/10.1137/060678051
Gramani, M., & França, P. (2006). The combined cutting stock and lot-sizing problem in industrial processes. European Journal of Operational Research, 174(1), 509-521. http://dx.doi.org/10.1016/j.ejor.2004.12.019
Guigues, V., & Sagastizábal, C. (2012). The value of rolling-horizon policies for risk-averse hydro-thermal planning. European Journal of Operational Research, 217(1), 129-140. http://dx.doi.org/10.1016/j.ejor.2011.08.017
ILOG. (2008). Solver CPLEX. New York: IBM. Recuperado em 20 de junho de 2012, de http://www.ilog.com/products/cplex/
Jia, Z., & Ierapetritou, M. G. (2004). Short-term scheduling under uncertainty using MILP sensitivity analysis. Industrial & Engineering Chemistry Research, 43(14), 3782-3791. http://dx.doi.org/10.1021/ie0306731
Kall, P., & Wallace, S. (1994). Stochastic programming. New York: Wiley.
Khor, C. S., Elkamel, A., Ponnambalamb, K., & Douglas, P. L. (2008). Two-stage stochastic programming with fixed recourse via scenario planning with economic and operational risk management for petroleum refinery planning under uncertainty. Chemical Engineering & Processing, 47(9-10), 1744-1764. http://dx.doi.org/10.1016/j.cep.2007.09.016
Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science, 37(5), 519-531. http://dx.doi.org/10.1287/mnsc.37.5.519
Kuhn, S., & Schultz, R. (2009). Risk neutral and risk averse power optimization in electricity networks with dispersed generation. Mathematical Methods in Operations Research, 69(2), 353-367. http://dx.doi.org/10.1007/s00186-008-0264-3
Laguna, M. (1998). Applying robust optimisation to capacity expansion of one location in telecommunications with demand uncertainty. Management Science, 44(11), 101-110. http://dx.doi.org/10.1287/mnsc.44.11.S101
Lai, K. K., & Ng, W. L. (2005). A stochastic approach to hotel revenue optimization. Computers & Operations Research, 32(5), 1059-1072. http://dx.doi.org/10.1016/j.cor.2003.09.012
Lai, K., Wang, M., & Liang, L. (2007). A stochastic approach to professional services firms’ revenue optimization. European Journal of Operational Research, 182(3), 971-982. http://dx.doi.org/10.1016/j.ejor.2006.09.038
Leung, S., & Wu, Y. (2004). A robust optimization model for stochastic aggregate production planning. Production Planning & Control, 15(5), 502-514. http://dx.doi.org/10.1080/09537280410001724287
Leung, S., Tsang, S., Ng, W., & Wu, Y. (2007). A robust optimization model for multi-site production planning problem in an uncertain environment. European Journal of Operational Research, 181(1), 224-238. http://dx.doi.org/10.1016/j.ejor.2006.06.011
Li, Z., & Ierapetritou, M. G. (2008). Robust optimization for process scheduling under uncertainty. Industrial & Engineering Chemistry Research, 47(12), 4148-4157. http://dx.doi.org/10.1021/ie071431u
Ma, Z., Kwon, R. H., & Lee, C. G. (2010). A stochastic programming winner determination model for truckload procurement under shipment uncertainty. Transportation Research Part E, 46(1), 49-60. http://dx.doi.org/10.1016/j.tre.2009.02.002
Markowitz, H. (1959). Portfolio selection: efficient diversification of investments. New York: John Wiley & Sons.
Morabito, R., & Arenales, M. (2000). Optimizing the cutting of stock plates in a furniture company. International Journal of Production Research, 38(12), 2725-2742. http://dx.doi.org/10.1080/002075400411457
Mulvey, J., Vanderbei, R., & Zenios, S. (1995). Robust optimization of large scale systems. Operations Research, 43(2), 264-281. http://dx.doi.org/10.1287/opre.43.2.264
Pan, F., & Nagi, R. (2010). Robust supply chain design under uncertain demand in agile manufacturing. Computers & Operations Research, 37(4), 668-683. http://dx.doi.org/10.1016/j.cor.2009.06.017
Pousinho, H., Mendes, V., & Catalão, J. (2011). A risk-averse optimization model for trading wind energy in a market environment under uncertainty. Energy, 36(8), 4935-4942. http://dx.doi.org/10.1016/j.energy.2011.05.037
Pousinho, H., Mendes, V., & Catalão, J. (2012). Scheduling of a hydro producer considering head-dependency, price scenarios and risk-aversion. Energy Conversion and Management, 56, 96-103. http://dx.doi.org/10.1016/j. enconman.2011.11.020
Rangel, S., & Figueiredo, A. G. (2008). O problema de corte de estoque em indústrias de móveis e pequeno e médio portes. Pesquisa Operacional, 28(3), 451-472. http://dx.doi.org/10.1590/S0101-74382008000300004
Rosenthal, R. (2008). Gams: a user’s guide. Washington: GAMS. Recuperado em 20 de junho de 2012, de http://www.gams.com/docs/ document.htm
Santos, S. G., Araujo, S. A., & Rangel, M. S. (2011). Integrated cutting machine programming and lot sizing in furniture industry. Pesquisa Operacional para o Desenvolvimento, 3(1), 249-266.
Schultz, R., & Tiedemann, S. (2006). Conditional value-at-risk in stochastic programs with mixed-integer recourse. Mathematical Programming, 105(2-3), 365-386. http://dx.doi.org/10.1007/s10107-005-0658-4
Shapiro, A., Dentcheva, D., & Ruszczynski, A. (2009). Lectures on stochastic programming: modeling and theory. Philadelphia: SIAM. http://dx.doi.org/10.1137/1.9780898718751
Suh, M. H., & Lee, T. Y. (2001). Robust optimization method for the economic term in chemical process design and planning. Industrial & Engineering Chemistry Research, 40(25), 5950-5959. http://dx.doi.org/10.1021/ie0005147
Ukkusuri, S. V., Ramadurai, G., & Patil, G. (2010). A robust transportation signal control problem accounting for traffic dynamics. Computers & Operations Research, 37(5), 869-879. http://dx.doi.org/10.1016/j.cor.2009.03.017
Vladimirou, H., & Zenios, S. (1997). Stochastic linear programs with restricted recourse. European Journal of Operational Research, 101(1), 177-192. http://dx.doi.org/10.1016/0377-2217(95)00370-3
Yan, S., & Tang, C. H. (2009). Inter-city bus scheduling under variable market share and uncertain market demands. Omega, 37(1), 178-192. http://dx.doi.org/10.1016/j.omega.2006.11.008
Yu, C., & Li, H. (2000). A robust optimization model for stochastic logistic problems. International Journal of Production Economics, 64(1-3), 385-397. http://dx.doi.org/10.1016/S0925-5273(99)00074-2
Zanjani, M. K., Ait-Kadi, D., & Nourelfath, M. (2009). Robust production planning in a manufacturing environment with random yield: a case in sawmill production planning. European Journal of Operational Research, 201(3), 882-891. http://dx.doi.org/10.1016/j.ejor.2009.03.041
Ahmed, S., & Sahinidis, N. V. (1998). Robust process planning under uncertainty. Industrial & Engineering Chemistry Research, 37(5), 1883-1892. http://dx.doi.org/10.1021/ie970694t
Alem, D. (2011). Programação estocástica e otimização robusta no planejamento da produção de empresas moveleiras (Tese de doutorado). Universidade de São Paulo, São Carlos.
Alem, D., & Morabito, R. (2013a). Risk-averse two- stage stochastic programs in furniture plants. OR Spectrum, 35(4), 773-806. http://dx.doi.org/10.1007/s00291-012-0312-5
Alem, D., & Morabito, R. (2013b). O problema combinado de planejamento da produção e corte de estoque sob incertezas: aplicação em fábricas de móveis de pequeno porte. Gestão & Produção, 20(1), 111-133. http://dx.doi.org/10.1590/S0104-530X2013000100009
Alonso-Ayuso, A., Carvallo, F., Escudero, L. F., Guignard, M., Pi, J., Puranmalka, R., & Weintraub, A. (2014). Medium range optimization of copper extraction planning under uncertainty in future copper prices. European Journal of Operational Research, 233(3), 711-726. http://dx.doi.org/10.1016/j.ejor.2013.08.048
Aouam, T., Rardin, R., & Abrache, J. (2010). Robust strategies for natural gas procurement. European Journal of Operational Research, 205(1), 151-158. http://dx.doi.org/10.1016/j.ejor.2009.12.015
Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88(3), 411-424. http://dx.doi.org/10.1007/PL00011380
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.
Beyer, H. G., & Sendhoff, B. (2007). Robust optimization: a comprehensive survey. Computers Methods in Applied Mechanics Engineering, 196(33-34), 3190-3218. http://dx.doi.org/10.1016/j.cma.2007.03.003
Carnieri, C., Guillermo, A., & Gavinho, L. (1994). Solution procedures for cutting lumber into furniture parts. European Journal of Operational Research, 73(3), 495-501. http://dx.doi.org/10.1016/0377-2217(94)90244-5
Dupacová, J., Consigli, G. & Wallace, S. W. (2000). Scenarios for multistage stochastic programs. Annals of Operations Research, 100(1-4), 25-53. http://dx.doi.org/10.1023/A:1019206915174
Foronda, S., & Carino, H. (1991). A heuristic approach to the lumber allocation and manufacturing in hardwood dimension and furniture manufacturing. European Journal of Operational Research, 54(2), 151-162. http://dx.doi.org/10.1016/0377-2217(91)90294-6
Geng, N., Jiang, Z., & Chen, F. (2009). Stochastic programming based capacity planning for semiconductor wafer fab with uncertain demand and capacity. European Journal of Operational Research, 198(3), 899-908. http://dx.doi.org/10.1016/j.ejor.2008.09.029
Gollmer, R., Neise, F., & Schultz, R. (2008). Stochastic programs with frst-order dominance constraints induced by mixed-integer linear recourse. SIAM Journal on Optimization, 19(2), 552-571. http://dx.doi.org/10.1137/060678051
Gramani, M., & França, P. (2006). The combined cutting stock and lot-sizing problem in industrial processes. European Journal of Operational Research, 174(1), 509-521. http://dx.doi.org/10.1016/j.ejor.2004.12.019
Guigues, V., & Sagastizábal, C. (2012). The value of rolling-horizon policies for risk-averse hydro-thermal planning. European Journal of Operational Research, 217(1), 129-140. http://dx.doi.org/10.1016/j.ejor.2011.08.017
ILOG. (2008). Solver CPLEX. New York: IBM. Recuperado em 20 de junho de 2012, de http://www.ilog.com/products/cplex/
Jia, Z., & Ierapetritou, M. G. (2004). Short-term scheduling under uncertainty using MILP sensitivity analysis. Industrial & Engineering Chemistry Research, 43(14), 3782-3791. http://dx.doi.org/10.1021/ie0306731
Kall, P., & Wallace, S. (1994). Stochastic programming. New York: Wiley.
Khor, C. S., Elkamel, A., Ponnambalamb, K., & Douglas, P. L. (2008). Two-stage stochastic programming with fixed recourse via scenario planning with economic and operational risk management for petroleum refinery planning under uncertainty. Chemical Engineering & Processing, 47(9-10), 1744-1764. http://dx.doi.org/10.1016/j.cep.2007.09.016
Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science, 37(5), 519-531. http://dx.doi.org/10.1287/mnsc.37.5.519
Kuhn, S., & Schultz, R. (2009). Risk neutral and risk averse power optimization in electricity networks with dispersed generation. Mathematical Methods in Operations Research, 69(2), 353-367. http://dx.doi.org/10.1007/s00186-008-0264-3
Laguna, M. (1998). Applying robust optimisation to capacity expansion of one location in telecommunications with demand uncertainty. Management Science, 44(11), 101-110. http://dx.doi.org/10.1287/mnsc.44.11.S101
Lai, K. K., & Ng, W. L. (2005). A stochastic approach to hotel revenue optimization. Computers & Operations Research, 32(5), 1059-1072. http://dx.doi.org/10.1016/j.cor.2003.09.012
Lai, K., Wang, M., & Liang, L. (2007). A stochastic approach to professional services firms’ revenue optimization. European Journal of Operational Research, 182(3), 971-982. http://dx.doi.org/10.1016/j.ejor.2006.09.038
Leung, S., & Wu, Y. (2004). A robust optimization model for stochastic aggregate production planning. Production Planning & Control, 15(5), 502-514. http://dx.doi.org/10.1080/09537280410001724287
Leung, S., Tsang, S., Ng, W., & Wu, Y. (2007). A robust optimization model for multi-site production planning problem in an uncertain environment. European Journal of Operational Research, 181(1), 224-238. http://dx.doi.org/10.1016/j.ejor.2006.06.011
Li, Z., & Ierapetritou, M. G. (2008). Robust optimization for process scheduling under uncertainty. Industrial & Engineering Chemistry Research, 47(12), 4148-4157. http://dx.doi.org/10.1021/ie071431u
Ma, Z., Kwon, R. H., & Lee, C. G. (2010). A stochastic programming winner determination model for truckload procurement under shipment uncertainty. Transportation Research Part E, 46(1), 49-60. http://dx.doi.org/10.1016/j.tre.2009.02.002
Markowitz, H. (1959). Portfolio selection: efficient diversification of investments. New York: John Wiley & Sons.
Morabito, R., & Arenales, M. (2000). Optimizing the cutting of stock plates in a furniture company. International Journal of Production Research, 38(12), 2725-2742. http://dx.doi.org/10.1080/002075400411457
Mulvey, J., Vanderbei, R., & Zenios, S. (1995). Robust optimization of large scale systems. Operations Research, 43(2), 264-281. http://dx.doi.org/10.1287/opre.43.2.264
Pan, F., & Nagi, R. (2010). Robust supply chain design under uncertain demand in agile manufacturing. Computers & Operations Research, 37(4), 668-683. http://dx.doi.org/10.1016/j.cor.2009.06.017
Pousinho, H., Mendes, V., & Catalão, J. (2011). A risk-averse optimization model for trading wind energy in a market environment under uncertainty. Energy, 36(8), 4935-4942. http://dx.doi.org/10.1016/j.energy.2011.05.037
Pousinho, H., Mendes, V., & Catalão, J. (2012). Scheduling of a hydro producer considering head-dependency, price scenarios and risk-aversion. Energy Conversion and Management, 56, 96-103. http://dx.doi.org/10.1016/j. enconman.2011.11.020
Rangel, S., & Figueiredo, A. G. (2008). O problema de corte de estoque em indústrias de móveis e pequeno e médio portes. Pesquisa Operacional, 28(3), 451-472. http://dx.doi.org/10.1590/S0101-74382008000300004
Rosenthal, R. (2008). Gams: a user’s guide. Washington: GAMS. Recuperado em 20 de junho de 2012, de http://www.gams.com/docs/ document.htm
Santos, S. G., Araujo, S. A., & Rangel, M. S. (2011). Integrated cutting machine programming and lot sizing in furniture industry. Pesquisa Operacional para o Desenvolvimento, 3(1), 249-266.
Schultz, R., & Tiedemann, S. (2006). Conditional value-at-risk in stochastic programs with mixed-integer recourse. Mathematical Programming, 105(2-3), 365-386. http://dx.doi.org/10.1007/s10107-005-0658-4
Shapiro, A., Dentcheva, D., & Ruszczynski, A. (2009). Lectures on stochastic programming: modeling and theory. Philadelphia: SIAM. http://dx.doi.org/10.1137/1.9780898718751
Suh, M. H., & Lee, T. Y. (2001). Robust optimization method for the economic term in chemical process design and planning. Industrial & Engineering Chemistry Research, 40(25), 5950-5959. http://dx.doi.org/10.1021/ie0005147
Ukkusuri, S. V., Ramadurai, G., & Patil, G. (2010). A robust transportation signal control problem accounting for traffic dynamics. Computers & Operations Research, 37(5), 869-879. http://dx.doi.org/10.1016/j.cor.2009.03.017
Vladimirou, H., & Zenios, S. (1997). Stochastic linear programs with restricted recourse. European Journal of Operational Research, 101(1), 177-192. http://dx.doi.org/10.1016/0377-2217(95)00370-3
Yan, S., & Tang, C. H. (2009). Inter-city bus scheduling under variable market share and uncertain market demands. Omega, 37(1), 178-192. http://dx.doi.org/10.1016/j.omega.2006.11.008
Yu, C., & Li, H. (2000). A robust optimization model for stochastic logistic problems. International Journal of Production Economics, 64(1-3), 385-397. http://dx.doi.org/10.1016/S0925-5273(99)00074-2
Zanjani, M. K., Ait-Kadi, D., & Nourelfath, M. (2009). Robust production planning in a manufacturing environment with random yield: a case in sawmill production planning. European Journal of Operational Research, 201(3), 882-891. http://dx.doi.org/10.1016/j.ejor.2009.03.041
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