Integrated lot sizing and production scheduling formulations: an application in a refractory cement industry

Alves, Fernanda de Freitas; Nogueira, Thiago Henrique; Henriques, Rafaella de Souza; Castro, Priscila Vieira de

doi:10.1590/0104-530X2196-15

Fernanda de Freitas Alves

Departamento de Engenharia de Produção, Universidade Federal de Minas Gerais – UFMG, Avenida Antônio Carlos, 6627, Pampulha, CEP 31270-901, Belo Horizonte, MG, Brasil, e-mail: fernandafalves@ufmg.br

Thiago Henrique Nogueira

Departamento de Engenharia de Produção, Universidade Federal de Viçosa – UFV, Rodovia MG-230 – Km 7, CEP 38810-000, Rio Paranaíba, MG, Brasil, e-mail: thnogueira.ufv@gmail.com; rafaellah@ufv.br; priscila.vieira@ufv.br

Rafaella de Souza Henriques

Departamento de Engenharia de Produção, Universidade Federal de Viçosa – UFV, Rodovia MG-230 – Km 7, CEP 38810-000, Rio Paranaíba, MG, Brasil, e-mail: thnogueira.ufv@gmail.com; rafaellah@ufv.br; priscila.vieira@ufv.br

Priscila Vieira de Castro

Departamento de Engenharia de Produção, Universidade Federal de Viçosa – UFV, Rodovia MG-230 – Km 7, CEP 38810-000, Rio Paranaíba, MG, Brasil, e-mail: thnogueira.ufv@gmail.com; rafaellah@ufv.br; priscila.vieira@ufv.br

Author	Problem	Solution Method	Objective Function
Manne (1960)Manne, A. S. (1960). On the job-shop scheduling problem. Operations Research, 8(2), 219-223. http://dx.doi.org/10.1287/opre.8.2.219. http://dx.doi.org/10.1287/opre.8.2.219...	Scheduling	-	Minimize the makespan
Sousa & Wolsey (1992)Sousa, J. P., & Wolsey, L. A. (1992). A time indexed formulation of non-preemptive single machine scheduling problems. Mathematical Programming, 54(1-3), 353-367. http://dx.doi.org/10.1007/BF01586059. http://dx.doi.org/10.1007/BF01586059...	Scheduling	Cutting plane/Branch and Bound algorithm	Minimize costs/Maximize profits
Tavares (2002)Tavares, L. V. (2002). A review of the contribution of operational research of project management. European Journal of Operational Research, 136(1), 1-18. http://dx.doi.org/10.1016/S0377-2217(01)00097-2. http://dx.doi.org/10.1016/S0377-2217(01)...	Scheduling	-	-
Araujo et al. (2004)Araujo, S. A., Arenales, M. N., & Clark, A. R. (2004). Dimensionamento de lotes e programação do forno numa fundição de pequeno porte. Gestão & Produção, 11(2), 165-176. http://dx.doi.org/10.1590/S0104-530X2004000200003. http://dx.doi.org/10.1590/S0104-530X2004...	Integrated Lot Sizing and Scheduling	Local Search Algorithm	Minimize inventory costs, backorder and setup costs
Leung (2004)Leung, J. Y.-T. (2004). Handbook of scheduling: algorithms, models, and performance analysis. Boca Raton: Chapman & Hall /CRC. 1224 p.	Scheduling	-	-
Toso & Morabito (2005)Toso, E. A. V., & Morabito, R. (2005). Otimização no dimensionamento e sequenciamento de lotes de produção: estudo de caso numa fábrica de rações. Gestão & Produção, 12(2), 203-217. http://dx.doi.org/10.1590/S0104-530X2005000200006. http://dx.doi.org/10.1590/S0104-530X2005...	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize inventory costs and overtime costs
Carvalho & Santos (2006)Carvalho, C. R. V., & Santos, A. M. (2006). Dimensionamento de lote de produção em um problema de sequenciamento de uma máquina com tempo de preparação: aplicação a uma indústria química. In Anais do 38º Simpósio Brasileiro de Pesquisa Operacional (pp. 1922-1933). Goiânia: SBPO.	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize setup costs
Brahimi et al. (2006)Brahimi, N., Dauzere-Peres, S., Najid, N. M., & Nordli, A. (2006). Single item lot sizing problems. European Journal of Operational Research, 168(1), 1-16. http://dx.doi.org/10.1016/j.ejor.2004.01.054. http://dx.doi.org/10.1016/j.ejor.2004.01...	Lot Sizing	-	Minimize production costs, setup costs, inventory holding costs and backlogging costs
Santos (2006)Santos, A. M. (2006). Dimensionamento de lote de produção em um problema de seqüenciamento de uma máquina com tempo de preparação: aplicação a uma indústria química (Dissertação de mestrado). Universidade Federal de Minas Gerais, Belo Horizonte.	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize setup costs
Araujo et al. (2007)Araujo, S. A., Arenales, M. N., & Clark, A. R. (2007). Joint rolling-horizon scheduling of materials processing and lot-sizing with sequence-dependent setups. Journal of Heuristics, 13(4), 337-358. http://dx.doi.org/10.1007/s10732-007-9011-9. http://dx.doi.org/10.1007/s10732-007-901...	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic/Local Search Algorithm	Minimize a penalty-weighted sum of product backlogs, finished inventories and setup changeovers
Bustamante (2007)Bustamante, L. (2007). Minimização do custo de antecipação e atraso para o problema de sequenciamento de uma máquina com tempo de preparação dependente da sequência: aplicação em uma usina siderúrgica (Dissertação de mestrado). Universidade Federal de Minas Gerais, Belo Horizonte.	Scheduling	Branch and Bound	Minimize the sum of earliness and tardiness costs
Yamashita & Morabito (2007)Yamashita, D. S., & Morabito, R. (2007). Um algoritmo exato para o problema de programação de projetos com custo de disponibilidade de recursos e múltiplos modos. Pesquisa Operacional, 27(1), 27-49.	Scheduling	Branch and Bound	Minimize total cost allocation of project resources
Toledo et al. (2007)Toledo, C. F., França, P. M., Morabito, R., & Kimms, A. (2007). Um modelo de otimização para o problema integrado de dimensionamento de lotes e programação da produção em fábricas de refrigerantes. Pesquisa Operacional, 27(1), 155-186. http://dx.doi.org/10.1590/S0101-74382007000100009. http://dx.doi.org/10.1590/S0101-74382007...	Integrated Lot Sizing and Scheduling	Branch and Cut/Linear Relaxation	Minimize production costs, setup costs and inventory holding costs
Allahverdi et al. (2008)Allahverdi, A., Ng, C. T., Cheng, T. C. E., & Kovalyov, M. Y. (2008). A survey of scheduling problems with setup times or costs. European Journal of Operational Research, 187(3), 985-1032. http://dx.doi.org/10.1016/j.ejor.2006.06.060. http://dx.doi.org/10.1016/j.ejor.2006.06...	Scheduling	-	Minimize setup costs
Chen & Askin (2009)Chen, J., & Askin, R. G. (2009). Project selection, scheduling and resource allocation with time dependent returns. European Journal of Operational Research, 193(1), 23-34. http://dx.doi.org/10.1016/j.ejor.2007.10.040. http://dx.doi.org/10.1016/j.ejor.2007.10...	Scheduling	Implicit enumeration algorithm	Maximize profits
Ferreira et al. (2009)Ferreira, D., Morabito, R., & Rangel, S. (2009). Solution approaches for the soft drink integrated production lot sizing and scheduling problem. European Journal of Operational Research, 196(2), 697-706. http://dx.doi.org/10.1016/j.ejor.2008.03.035. http://dx.doi.org/10.1016/j.ejor.2008.03...	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize the total sum of product inventory, demand backorder, machine changeover and tank changeover costs
Toso et al. (2009)Toso, E. A., Morabito, R., & Clark, A. R. (2009). Lot sizing and sequencing optimization at an animal-feed plant. Computers & Industrial Engineering, 57(3), 813-821. http://dx.doi.org/10.1016/j.cie.2009.02.011. http://dx.doi.org/10.1016/j.cie.2009.02....	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize the costs of inventory, overtime and setup costs
Bernardes et al. (2010)Bernardes, E. D., Araujo, S. A., & Rangel, S. (2010). Reformulação para um problema integrado de dimensionamento e sequenciamento de lotes. Pesquisa Operacional, 30(3), 637-655.	Integrated Lot Sizing and Scheduling	Branch and Cut	Minimize inventory costs, backorder and setup costs
Stadtler (2010)Stadtler, H. (2010). Multi-level single machine lot-sizing and scheduling with zero lead times. European Journal of Operational Research, 209(3), 241-252. http://dx.doi.org/10.1016/j.ejor.2010.09.022. http://dx.doi.org/10.1016/j.ejor.2010.09...	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize inventory costs and setup costs
Henriques et al. (2010)Henriques, R. S., Carvalho, C. R. V., & Cunha, A. S. (2010). Um modelo com variáveis indexadas no tempo para a integração do Dimensionamento de Lotes e Sequenciamento em uma máquina com tempos de preparação. In Anais do XLII Simpósio Brasileiro de Pesquisa Operacional. Bento Gonçalves: SBPO.	Integrated Lot Sizing and Scheduling	Branch and Bound	Minimize inventory costs
Ferreira et al. (2010)Ferreira, D., Morabito, R., & Rangel, S. (2010). Relax and fix heuristics to solve one-stage one-machine lot-scheduling models for small-scale soft drink plants. Computers & Operations Research, 37(4), 684-691. http://dx.doi.org/10.1016/j.cor.2009.06.007. http://dx.doi.org/10.1016/j.cor.2009.06....	Integrated Lot Sizing and Scheduling	Relax-and-fix heuristic	Minimize the total sum of the inventory, backorder and machine changeover costs
Defalque et al. (2011)Defalque, C. M., Rangel, S., & Ferreira, D. (2011). Usando o ATSP na modelagem do problema integrado de produção de bebidas. TEMA. Tendências em Matemática Aplicada e Computacional, 12(3), 195-209. http://dx.doi.org/10.5540/tema.2011.012.03.0195. http://dx.doi.org/10.5540/tema.2011.012....	Integrated Lot Sizing and Scheduling	Branch and Cut	Minimize inventory costs, backorder and setup costs
Clark et al. (2011)Clark, G. E. A., Almada-Lobo, B., & Almeder, C. (2011). Lot sizing and scheduling: industrial extensions and research opportunities. International Journal of Production Research, 49(9), 2457-2461. http://dx.doi.org/10.1080/00207543.2010.532908. http://dx.doi.org/10.1080/00207543.2010....	Integrated Lot Sizing and Scheduling	-	-
Ramos & Oliveira (2011)Ramos, R. S., & Oliveira, F. B. (2011). Uma abordagem ao problema de sequenciamento em uma máquina com penalidades por antecipação e atraso da produção por meio de algoritmos evolutivos. Conexão Ciência, 6(2), 21-37.	Scheduling	Hybrid evolutionary algorithm	Minimize the sum of earliness and tardiness costs
Shim et al. (2011)Shim, I.-S., Kim, H.-C., Doh, H.-H., & Lee, D.-H. (2011). A two-stage heuristic for single machine capacitated lot-sizing and scheduling with sequence-dependent setup costs. Computers & Industrial Engineering, 61(4), 920-929. http://dx.doi.org/10.1016/j.cie.2011.06.002. http://dx.doi.org/10.1016/j.cie.2011.06....	Integrated Lot Sizing and Scheduling	Adapted heuristic	Minimize the sum of setup and inventory holding costs
Stadtler & Sahling (2012)Stadtler, H., & Sahling, F. (2012). A lot-sizing and scheduling model for multi-stage flow lines with zero lead times. European Journal of Operational Research, 225(3), 404-419. http://dx.doi.org/10.1016/j.ejor.2012.10.011. http://dx.doi.org/10.1016/j.ejor.2012.10...	Integrated Lot Sizing and Scheduling	Relax-and-fix/Relax-and-optimize heuristics	Minimize inventory holding costs, setup costs and penalty costs for backorders
Pinedo (2012)Pinedo, M. L. (2012). Scheduling: theory, algorithms, and systems. 4 ed. Nova York: Springer.	Scheduling	-	-
Rego (2013)Rego, M. F. (2013). Algoritmos multiobjetivos para o problema de sequenciamento de tarefas em uma máquina com tempo de preparação dependente da sequência e da família (Dissertação de mestrado). Universidade Federal de Ouro Preto, Ouro Preto.	Scheduling	Multi-objective algorithms	Minimize makespan and total weighted lateness
Molina et al. (2013)Molina, F., Morabito, R., & Araujo, S. A. (2013). Modelos matemáticos para problemas de dimensionamento de lotes com restrições de capacidade e custos de transporte. Gestão & Produção, 20(3), 573-586. http://dx.doi.org/10.1590/S0104-530X2013000300006. http://dx.doi.org/10.1590/S0104-530X2013...	Lot Sizing	Modified Lagrangian Relaxation	Minimize inventory holding costs, backorder, setup costs and transportation costs
Seeanner & Meyr (2013)Seeanner, F., & Meyr, H. (2013). Multi-stage simultaneous lot-sizing and scheduling for flow line production. OR-Spektrum, 35(1), 33-73. http://dx.doi.org/10.1007/s00291-012-0296-1. http://dx.doi.org/10.1007/s00291-012-029...	Integrated Lot Sizing and Scheduling	Relax-and-fix/LP-and-fix heuristics	Minimize the sum of inventory holding costs, setup costs, production costs, costs for standby, external purchase, overtime and holding of Work-In-Progress stock

Input Data	Distribution Value
Processing Time ( $p_{i}$ ) – Hours	U(1,4)
Setup Time ( $S t_{i j}$ ) – Hours	U(0,2)
Demand ( $d_{i t}$ ) – Units	U(0,5 $α_{1}$ )
Cost ( $H_{i} a n d B_{i}$ ) – R$	U(0, $\bar{j}$ )

		MBILP-CH						MBILP-DH						MBILP-DHP
Class	Product Family	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$
1	2	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.8	100.0%	0.0%	7.5	100.0%
1	3	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	2.4	100.0%	0.0%	51.6	100.0%
1	4	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	5.2	100.0%	0.0%	109.2	100.0%
1	5	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	9.0	100.0%	0.0%	315.4	100.0%
1	6	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	15.2	100.0%	-	-	0.0%
1	7	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	20.9	100.0%	-	-	0.0%
1	10	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	-	-	0.0%	-	-	0.0%
1	15	99.9%	0.1	100.0%	100.0%	3600.0	0.0%	84.8%	0.1	100.0%	40.0%	1212.9	67.0%	-	-	0.0%	-	-	0.0%
1	20	99.9%	0.1	100.0%	100.0%	3600.0	0.0%	57.6%	0.2	100.0%	26.0%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
Average		99.9%	0.0	100.0%	22.2%	800.1	78.0%	15.8%	0.1	100.0%	7.3%	534.8	85.0%	0.0%	8.9	67.0%	0.0%	120.9	44.0%
Standard Deviation		0.0%	0.0	0.0	44.1%	1587.4	0.4	32.1%	0.0	0.0	15.0	1217.4	0.3	0.0%	7.8	0.5	0.0%	136.2	0.5
Class	Product Family	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$
2	2	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.0	100.0%	0.0%	0.8	100.0%	0.0%	16.7	100.0%
2	3	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	2.4	100.0%	5.0%	41.1	100.0%
2	4	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	5.1	100.0%	0.0%	297.0	100.0%
2	5	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	9.0	100.0%	0.0%	552.7	100.0%
2	6	33.3%	0.1	100.0%	0.0%	0.1	100.0%	33.3%	0.1	100.0%	0.0%	0.1	100.0%	33.3%	14.2	100.0%	-	-	0.0%
2	7	33.3%	0.1	100.0%	0.0%	0.3	100.0%	33.3%	0.1	100.0%	0.0%	0.7	100.0%	33.3%	20.3	100.0%	-	-	0.0%
2	10	99.9%	0.1	100.0%	78.7%	3600.0	0.0%	41.5%	0.1	100.0%	7.7%	2026.0	67.0%	-	-	0.0%	-	-	0.0%
2	15	99.9%	0.1	100.0%	99.3%	3600.0	0.0%	12.6%	0.1	100.0%	4.3%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
2	20	99.9%	0.1	100.0%	100.0%	3600.0	0.0%	7.9%	0.1	100.0%	3.0%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
Average		40.7%	0.0	100.0%	30.9%	1200.1	67.0%	14.3%	0.1	100.0%	1.7%	1025.2	74.0%	11.1%	8.6	67.0%	1.3%	226.9	44.0%
Standard Deviation		46.4%	0.0	0.0	46.7%	1799.9	0.5	17.1%	0.0	0.0	2.8%	1603.3	0.4	17.2%	7.5	0.5	2.5%	251.5	0.5
Class	Product Family	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$	${\bar{G A P}}_{Re l a x}$	${\bar{T}}_{Re l a x}$	$% I n s . Re s o l .$	${\bar{G A P}}_{I n t e g e r}$	${\bar{T}}_{I n t e g e r}$	$% I n s . Re s o l .$
3	2	0.0%	0.0	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.8	100.0%	0.0%	8.0	100.0%
3	3	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	0.1	100.0%	0.0%	2.4	100.0%	0.0%	75.8	100.0%
3	4	33.2%	0.1	100.0%	0.0%	0.5	100.0%	2.3%	0.1	100.0%	0.0%	3.4	100.0%	2.3%	5.0	100.0%	33.3%	353.6	67.0%
3	5	75.6%	0.0	100.0%	0.0%	3.3	100.0%	39.4%	0.1	100.0%	0.0%	131.0	100.0%	39.4%	9.1	100.0%	32.0%	3547.5	33.0%
3	6	99.9%	0.0	100.0%	0.0%	6.2	100.0%	18.0%	0.1	100.0%	0.0%	2.5	100.0%	18.0%	14.7	100.0%	-	-	0.0%
3	7	99.8%	0.1	100.0%	0.0%	51.7	100.0%	6.6%	0.1	100.0%	0.0%	82.9	100.0%	6.6%	21.1	100.0%	-	-	0.0%
3	10	99.9%	0.1	100.0%	9.7%	3261.8	33.0%	4.2%	0.2	100.0%	0.7%	2876.3	33.0%	-	-	0.0%	-	-	0.0%
3	15	99.8%	0.1	100.0%	39.3%	3600.0	0.0%	3.0%	0.1	100.0%	0.7%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
3	20	99.8%	0.1	100.0%	53.3%	3600.0	0.0%	1.6%	0.1	100.0%	0.7%	3600.0	0.0%	-	-	0.0%	-	-	0.0%
Average		67.6%	0.0	100.0%	11.4%	1169.3	70.0%	8.3%	0.1	100.0%	0.2%	1144.0	70.0%	11.0%	8.8	67.0%	16.3%	996.2	33.0%
Standard Deviation		44.1%	0.0	0.0	20.4%	1741.3	0.5	12.9%	0.0	0.0	0.3%	1674.7	0.5	15.4%	7.8	0.5	18.9%	1707.4	0.4
Model Average		69.4%	0.0	100.0%	21.5%	1056.5	72.0%	12.8%	0.1	100.0%	3.1%	901.4	77.0%	7.4%	8.8	67.0%	5.9%	448.0	41.0%
Model Standard Deviation		26.2%	0.0	0.0	14.5%	109.8	0.0	10.1%	0.0	0.0	7.8%	246.0	0.1	9.5%	0.2	0.0	10.2%	875.8	0.0

Formulations	Binary Variables	Constraints
MBILP-CH	O( $\bar{j}$ ²)	O( $\bar{j}$ ²)
MBILP-DH	O( $\bar{j}$ ³)	O( $\bar{j}$ ³)
MBILP-DHP	O( $\bar{j} \bar{s}$ )	O( $\bar{j}$ ^{2 $\bar{s}$ 2})

Brasil

Brasil

Integrated lot sizing and production scheduling formulations: an application in a refractory cement industry

Abstract